total_json.jsonl

{"table/equation": "\\begin{table}[t] \\centering\n\n    \\begin{tabular}{|c|c|c|c|}\n\n\t\\hline\n\n\t\\hline\n\n\t$\\nu_{\\mu}$ &  $\\bar \\nu_{\\mu}$ & $\\nu_e$ &  $\\bar \\nu_e$\\\\\n\n\t\\hline\n\n\t669.0 &13.7 & 5.9& 0.3\\\\\n\n\t\\hline\\hline\n\n    \\end{tabular}\n\n    \\caption{\\label{tab:rates}\\sl%\n\n      Nominal performance of the CNGS reference beam~\\cite{fluxes}.\n\n      The total non-oscillated CC event rates are calculated assuming\n\n      $10^{19}$ pot and 1 Kton lead target mass.}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[t] \\centering\n\n    \\begin{tabular}{|c|c|c|c|c|c|}\n\n\t\\hline\\hline\n\n\t$(\\theta_{13}; \\theta_{14}; \\theta_{24}; \\theta_{34})$ & $N_\\tau$ & background & $(\\theta_{13}; \\theta_{14}; \\theta_{24}; \\theta_{34})$ & $N_\\tau$ & background \\\\\n\n\t\\hline\n\n\t$(5^\\circ;  5^\\circ;  5^\\circ; 20^\\circ)$    &     8.9    & 1.0 & $(10^\\circ; 5^\\circ;  5^\\circ; 20^\\circ)$    &     8.5    & 1.0 \\\\  \n\n\t$(5^\\circ;  5^\\circ;  5^\\circ; 30^\\circ)$    &     6.9    & 1.0 & $(10^\\circ; 5^\\circ;  5^\\circ; 30^\\circ)$    &     6.5    & 1.0 \\\\\n\n\t$(5^\\circ;  5^\\circ; 10^\\circ; 20^\\circ)$    &     8.3    & 1.0 & $(10^\\circ; 5^\\circ; 10^\\circ; 20^\\circ)$    &     7.9    & 1.0 \\\\\n\n\t$(5^\\circ;  5^\\circ; 10^\\circ; 30^\\circ)$    &    10.5    & 1.0 & $(10^\\circ; 5^\\circ; 10^\\circ; 30^\\circ)$    &    10.3    & 1.0 \\\\\n\n\t3 families                          &    15.1    & 1.0 &          3 families                          &    14.4    & 1.0 \\\\\n\n\t\\hline \\hline\n\n    \\end{tabular}\n\n    \\caption{\\label{tab:rates3}\\sl%\n\n      Event rates and expected background for the $\\nu_\\mu \\to\n\n      \\nu_\\tau$ channel in the OPERA detector, for different values of\n\n      $\\theta_{14}, \\theta_{24}$ and $\\theta_{34}$ in the (3+1)\n\n      scheme. The other unknown angle, $\\theta_{13}$ has been fixed\n\n      to: $\\theta_{13} = 5^\\circ, 10^\\circ$. The CP-violating phases\n\n      are: $\\delta_1 = \\delta_2 = 0$; $\\delta_3 = 90^\\circ$.  As a\n\n      reference, the expected value in the case of standard\n\n      three-family oscillation (\\textit{i.e.}, for $\\theta_{i4} = 0$) is shown\n\n      for maximal CP-violating phase $\\delta$. The rates are computed\n\n      according to Eq.~\\eqref{eq:signaltau}.}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[t] \\centering\n\n    \\begin{tabular}{|c|c|c|c|c|c|}\n\n\t\\hline\\hline\n\n\t$(\\theta_{13}; \\theta_{14}; \\theta_{24}; \\theta_{34})$ & $N_e$ & $\\nu_e^{CC}$ & $\\nu_\\mu^{NC}$ & $\\tau \\to e$ & $\\nu_\\mu^{CC}$  \\\\ \\hline\n\n\t$(5^\\circ; 5^\\circ; 5^\\circ; 20^\\circ)$   &  3.5 & 19.4 & 5.3 & 2.8 & 0.9 \\\\\n\n\t$(5^\\circ; 5^\\circ; 5^\\circ; 30^\\circ)$   &  3.5 & 19.4 & 5.3 & 2.1 & 0.9 \\\\\n\n\t$(5^\\circ; 5^\\circ; 10^\\circ; 20^\\circ)$  &  2.4 & 19.4 & 5.3 & 2.3 & 0.9 \\\\\n\n\t$(5^\\circ; 5^\\circ; 10^\\circ; 30^\\circ)$  &  2.4 & 19.4 & 5.3 & 2.4 & 0.9 \\\\\n\n\t3 families                        &  3.7 & 19.7 & 5.3 & 4.6 & 0.9 \\\\\n\n\t\\hline\n\n\t$(10^\\circ; 5^\\circ; 5^\\circ; 20^\\circ)$  & 10.6 & 19.4 & 5.3 & 2.7 & 0.9 \\\\\n\n\t$(10^\\circ; 5^\\circ; 5^\\circ; 30^\\circ)$  & 10.4 & 19.4 & 5.3 & 2.0 & 0.9 \\\\\n\n\t$(10^\\circ; 5^\\circ; 10^\\circ; 20^\\circ)$ &  8.8 & 19.4 & 5.3 & 2.2 & 0.9 \\\\\n\n\t$(10^\\circ; 5^\\circ; 10^\\circ; 30^\\circ)$ &  8.6 & 19.4 & 5.3 & 2.4 & 0.9 \\\\\n\n\t3 families                        & 15.1 & 19.7 & 5.3 & 4.8 & 0.9 \\\\\n\n\t\\hline \\hline\n\n    \\end{tabular}\n\n    \\caption{\\label{tab:rates2}\\sl%\n\n      Event rates and expected background for the $\\nu_\\mu \\to \\nu_e$\n\n      channel in the OPERA detector, for different values of \n\n      $\\theta_{14}, \\theta_{24}$ and $\\theta_{34}$ in the (3+1)\n\n      scheme. The other unknown angle, $\\theta_{13}$, has been fixed\n\n      to: $\\theta_{13} = 5^\\circ, 10^\\circ$. The CP-violating phases\n\n      are: $\\delta_1 = \\delta_2 = 0$; $\\delta_3 = 90^\\circ$.  As a\n\n      reference, the expected value in the case of standard\n\n      three-family oscillation(\\textit{i.e.}, for $\\theta_{i4} = 0$) is shown\n\n      for maximal CP-violating phase $\\delta$.  The rates are computed\n\n      according to Eq.~\\eqref{eq:signale}. Backgrounds have been\n\n      computed following Ref.~\\cite{Komatsu:2002sz}.}\n\n\\end{table}"}
{"table/equation": "\\begin{equation}\n\n    \\eta_s = \\sum_{i \\,\\in\\, \\text{sol}} |U_{s i}|^2\n\n    \\qquad\\text{and}\\qquad\n\n    c_s = \\sum_{j \\,\\in\\, \\text{atm}} |U_{s j}|^2\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:3+1param}\n\n    U_\\text{SBL} =\n\n    R_{14}(\\theta_{14}) \\; R_{24}(\\theta_{24}) \\; R_{34}(\\theta_{34}) \\; \n\n    R_{23}(\\theta_{23} ,\\, \\delta_3) \\; R_{13}(\\theta_{13} ,\\, \\delta_2) \\;\n\n    R_{12}(\\theta_{12} ,\\, \\delta_1) \\,.\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:3+1param2}\n\n    U_\\text{atm} =\n\n    R_{34}(\\theta_{34}) \\; R_{24}(\\theta_{24}) \\;\n\n    R_{23}(\\theta_{23} ,\\, \\delta_3) \\;\n\n    R_{14}(\\theta_{14}) \\; R_{13}(\\theta_{13} ,\\, \\delta_2) \\; \n\n    R_{12}(\\theta_{12} ,\\, \\delta_1) \\,.\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:pee}\n\n    P_{ee} = 1 - \\sin^2 2 \\theta_{14} \\sin^2 \\frac{\\Delta_\\Sbl L }{2} \n\n    - c^4_{14} \\sin^2 2 \\theta_{13} \\sin^2 \\frac{\\Delta_\\Atm L}{2} \\, ,\n\n\\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    P(\\nu_\\alpha \\to \\nu_\\beta) &= \\delta_{\\alpha \\beta}\n\n    - 4 \\Re \\left[ U_{\\alpha 3} U^\\star_{\\beta 3} \\left(\n\n    \\delta_{\\alpha\\beta} - U^\\star_{\\alpha 3} U_{\\beta 3}\n\n    - U^\\star_{\\alpha 4} U_{\\beta 4} \\right) \\right]\n\n    \\sin^2 \\frac{\\Delta_{23} L }{2}\n\n    \\\\\n\n    &- 2 \\Re \\left[ \\delta_{\\alpha\\beta} U_{\\alpha 4} U^\\star_{\\beta 4}\n\n    - |U_{\\alpha 4}|^2 |U_{\\beta 4}|^2 \\right]\n\n    \\pm 2 \\Im \\left[ U_{\\alpha 3} U^\\star_{\\beta 3}\n\n    U^\\star_{\\alpha 4} U_{\\beta 4} \\right] \\sin \\Delta_{23} L \\,,\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    P_{\\mu\\mu} &= 1 - 2 c^2_{14} s^2_{24} (1 - c^2_{14} s^2_{24})\n\n    - 4 \\big\\lbrace s^2_{23} c^2_{24} \\left[ c^2_{24} (c^2_{23}\n\n    - s^2_{13}) - s^2_{14} s^2_{24} \\right]\n\n    \\\\    \n\n    &- 2 c^3_{24} s_{23} (1 - 2 s^2_{23}) s_{13} s_{14} s_{24}\n\n    \\cos(\\delta_2 - \\delta_3) \\big\\rbrace\n\n    \\sin^2 \\frac{\\Delta_\\Atm L }{2} \\,.\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:pmumu}\n\n    P_{\\mu\\mu} = 1 - 2 s^2_{24}\n\n    - 4 s^2_{23} \\left[ c^2_{23} (1 - 2 s^2_{24}) - s^2_{13} \\right]\n\n    \\sin^2 \\frac{\\Delta_\\Atm L }{2} \\,.\n\n\\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    \\label{eq:pmus}\n\n    P_{\\mu s} &= 2 c^2_{34} s^2_{24} + \\big\\lbrace\n\n    \\sin^2 2 \\theta_{23} (c^4_{13} c^2_{24} s^2_{34}\n\n    - c^2_{34} s^2_{24})\n\n    \\\\\n\n    &+ 2  c_{34} \\sin 2 \\theta_{23} s_{34} \\left[\n\n    s_{24} (1 -2 s^2_{23}) \\cos \\delta_3\n\n    + 2 s_{23} s_{13} s_{14} \\cos \\delta_2 \\right] \\big\\rbrace\n\n    \\sin^2 \\frac{\\Delta_\\Atm L }{2}\n\n    \\\\\n\n    &\\pm c_{34} \\sin 2 \\theta_{23} s_{24} s_{34}\n\n    \\sin\\delta_3 \\sin\\Delta_\\Atm L \\,.\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    \\label{eq:pmue}\n\n    P_{\\mu e} &= 4 \\left\\lbrace s^2_{23} s^2_{13}\n\n    [1 - s^2_{13} - s^2_{14} - s^2_{24}] \n\n    + s_{23} s_{13} s_{14} s_{24} \\cos(\\delta_2 - \\delta_3) \\right\\rbrace\n\n    \\sin^2 \\frac{\\Delta_\\Atm L}{2}\n\n    \\\\\n\n    & \\pm 2 s_{23} s_{13} s_{14} s_{24} \\sin(\\delta_2 - \\delta_3)\n\n    \\sin\\Delta_\\Atm L + 2 s^2_{14} s^2_{24} \\,.\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    \\label{eq:pmutau}\n\n    P_{\\mu \\tau} &= 2 s^2_{24} s^2_{34} + \\big\\lbrace \\sin^2 2 \\theta_{23}\n\n    [c^4_{13} c^2_{24} c^2_{34} - s^2_{24} s^2_{34}]\n\n    \\\\\n\n    &- 4 \\sin 2 \\theta_{23} s_{13} s_{14} [s_{23} s_{34} \\cos\\delta_2\n\n    + c_{23} s_{24} \\cos(\\delta_2 - \\delta_3)]\n\n    \\\\\n\n    &+ 2 \\sin 2 \\theta_{23} s_{24} s_{34} c^2_{13} c^2_{24} c_{34}\n\n    [c^2_{14} - 2 c^2_{13} s^2_{23}] \\cos\\delta_3 \\big\\rbrace\n\n    \\sin^2 \\frac{\\Delta_\\Atm L}{2}\n\n    \\\\\n\n    &\\mp \\sin 2 \\theta_{23} s_{24} s_{34} c^2_{13} c^2_{14} c^2_{24} c_{34}\n\n    \\sin\\delta_3 \\sin\\Delta_\\Atm L \\,.\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    P^{3\\nu}_{\\mu\\tau} = P_{\\mu\\tau} (\\theta_{i4} = 0)\n\n    \\simeq c^4_{13} \\sin^2 2 \\theta_{23}\n\n    \\sin^2 \\frac{\\Delta_\\Atm L}{2} \\,.\n\n\\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    \\Delta P_{\\mu \\tau} &\\equiv P_{\\mu \\tau} - P^{3\\nu}_{\\mu\\tau}\n\n    \\\\\n\n    &= \\left\\lbrace - (s^2_{24} + s^2_{34}) \\sin^2 2 \\theta_{23} \n\n    + 2 \\sin 2 \\theta_{23} (1 - 2 s^2_{13}) s_{24} s_{34}\n\n    \\cos\\delta_3 \\right\\rbrace \\sin^2 \\frac{\\Delta_\\Atm L}{2}\n\n    \\\\\n\n    &\\mp \\sin 2 \\theta_{23} s_{24} s_{34} \\sin\\delta_3\n\n    \\sin \\Delta_\\Atm L + \\dots\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    \\label{eq:dpmue}\n\n    \\Delta P_{\\mu e} &\\equiv P_{\\mu e} - P_{\\mu e}(\\theta_{i4} = 0)\n\n    \\\\\n\n    &= s_{23} s_{13} s_{14} s_{24} \\cos(\\delta_2 - \\delta_3)\n\n    \\sin^2 \\frac{\\Delta_\\Atm L}{2}\n\n    \\\\\n\n    &\\pm 2 s_{23} s_{13} s_{14} s_{24}\n\n    \\sin(\\delta_2 - \\delta_3) \\sin \\Delta_\\Atm L + \\dots\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    N = \\int \\frac{d \\phi_{\\nu_\\alpha}(E)}{dE} \\,\n\n    \\sigma_{\\nu_\\alpha}(E) \\, dE \\,,\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:signaltau}\n\n    N_{\\mu \\tau} = A \\int \\frac{ d\\phi_{\\nu_{\\mu}}(E)}{dE} \\,\n\n    P_{\\mu \\tau}(E) \\sigma_{\\nu_\\tau}^{CC}(E) \\,\n\n    \\varepsilon_{\\mu \\tau} \\, dE \\,.\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:signale}\n\n    N_{\\mu e} = A \\int \\frac{ d\\phi_{\\nu_{\\mu}}(E)}{dE} \\,\n\n    P_{\\mu e}(E) \\sigma_{\\nu_{e}}^{CC}(E) \\,\n\n    \\varepsilon_{\\mu e}(E) \\, dE \\,,\n\n\\end{equation}"}
{"table/equation": "\\begin{table}[t] \\centering\n\n    \\begin{tabular}{|c|c|c|c|}\n\n\t\\hline\n\n\t\\hline\n\n\t$\\nu_{\\mu}$ &  $\\bar \\nu_{\\mu}$ & $\\nu_e$ &  $\\bar \\nu_e$\\\\\n\n\t\\hline\n\n\t669.0 &13.7 & 5.9& 0.3\\\\\n\n\t\\hline\\hline\n\n    \\end{tabular}\n\n    \\caption{\\label{tab:rates}\\sl%\n\n      Nominal performance of the CNGS reference beam~\\cite{fluxes}.\n\n      The total non-oscillated CC event rates are calculated assuming\n\n      $10^{19}$ pot and 1 Kton lead target mass.}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[t] \\centering\n\n    \\begin{tabular}{|c|c|c|c|c|c|}\n\n\t\\hline\\hline\n\n\t$(\\theta_{13}; \\theta_{14}; \\theta_{24}; \\theta_{34})$ & $N_\\tau$ & background & $(\\theta_{13}; \\theta_{14}; \\theta_{24}; \\theta_{34})$ & $N_\\tau$ & background \\\\\n\n\t\\hline\n\n\t$(5^\\circ;  5^\\circ;  5^\\circ; 20^\\circ)$    &     8.9    & 1.0 & $(10^\\circ; 5^\\circ;  5^\\circ; 20^\\circ)$    &     8.5    & 1.0 \\\\  \n\n\t$(5^\\circ;  5^\\circ;  5^\\circ; 30^\\circ)$    &     6.9    & 1.0 & $(10^\\circ; 5^\\circ;  5^\\circ; 30^\\circ)$    &     6.5    & 1.0 \\\\\n\n\t$(5^\\circ;  5^\\circ; 10^\\circ; 20^\\circ)$    &     8.3    & 1.0 & $(10^\\circ; 5^\\circ; 10^\\circ; 20^\\circ)$    &     7.9    & 1.0 \\\\\n\n\t$(5^\\circ;  5^\\circ; 10^\\circ; 30^\\circ)$    &    10.5    & 1.0 & $(10^\\circ; 5^\\circ; 10^\\circ; 30^\\circ)$    &    10.3    & 1.0 \\\\\n\n\t3 families                          &    15.1    & 1.0 &          3 families                          &    14.4    & 1.0 \\\\\n\n\t\\hline \\hline\n\n    \\end{tabular}\n\n    \\caption{\\label{tab:rates3}\\sl%\n\n      Event rates and expected background for the $\\nu_\\mu \\to\n\n      \\nu_\\tau$ channel in the OPERA detector, for different values of\n\n      $\\theta_{14}, \\theta_{24}$ and $\\theta_{34}$ in the (3+1)\n\n      scheme. The other unknown angle, $\\theta_{13}$ has been fixed\n\n      to: $\\theta_{13} = 5^\\circ, 10^\\circ$. The CP-violating phases\n\n      are: $\\delta_1 = \\delta_2 = 0$; $\\delta_3 = 90^\\circ$.  As a\n\n      reference, the expected value in the case of standard\n\n      three-family oscillation (\\textit{i.e.}, for $\\theta_{i4} = 0$) is shown\n\n      for maximal CP-violating phase $\\delta$. The rates are computed\n\n      according to Eq.~\\eqref{eq:signaltau}.}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[t] \\centering\n\n    \\begin{tabular}{|c|c|c|c|c|c|}\n\n\t\\hline\\hline\n\n\t$(\\theta_{13}; \\theta_{14}; \\theta_{24}; \\theta_{34})$ & $N_e$ & $\\nu_e^{CC}$ & $\\nu_\\mu^{NC}$ & $\\tau \\to e$ & $\\nu_\\mu^{CC}$  \\\\ \\hline\n\n\t$(5^\\circ; 5^\\circ; 5^\\circ; 20^\\circ)$   &  3.5 & 19.4 & 5.3 & 2.8 & 0.9 \\\\\n\n\t$(5^\\circ; 5^\\circ; 5^\\circ; 30^\\circ)$   &  3.5 & 19.4 & 5.3 & 2.1 & 0.9 \\\\\n\n\t$(5^\\circ; 5^\\circ; 10^\\circ; 20^\\circ)$  &  2.4 & 19.4 & 5.3 & 2.3 & 0.9 \\\\\n\n\t$(5^\\circ; 5^\\circ; 10^\\circ; 30^\\circ)$  &  2.4 & 19.4 & 5.3 & 2.4 & 0.9 \\\\\n\n\t3 families                        &  3.7 & 19.7 & 5.3 & 4.6 & 0.9 \\\\\n\n\t\\hline\n\n\t$(10^\\circ; 5^\\circ; 5^\\circ; 20^\\circ)$  & 10.6 & 19.4 & 5.3 & 2.7 & 0.9 \\\\\n\n\t$(10^\\circ; 5^\\circ; 5^\\circ; 30^\\circ)$  & 10.4 & 19.4 & 5.3 & 2.0 & 0.9 \\\\\n\n\t$(10^\\circ; 5^\\circ; 10^\\circ; 20^\\circ)$ &  8.8 & 19.4 & 5.3 & 2.2 & 0.9 \\\\\n\n\t$(10^\\circ; 5^\\circ; 10^\\circ; 30^\\circ)$ &  8.6 & 19.4 & 5.3 & 2.4 & 0.9 \\\\\n\n\t3 families                        & 15.1 & 19.7 & 5.3 & 4.8 & 0.9 \\\\\n\n\t\\hline \\hline\n\n    \\end{tabular}\n\n    \\caption{\\label{tab:rates2}\\sl%\n\n      Event rates and expected background for the $\\nu_\\mu \\to \\nu_e$\n\n      channel in the OPERA detector, for different values of \n\n      $\\theta_{14}, \\theta_{24}$ and $\\theta_{34}$ in the (3+1)\n\n      scheme. The other unknown angle, $\\theta_{13}$, has been fixed\n\n      to: $\\theta_{13} = 5^\\circ, 10^\\circ$. The CP-violating phases\n\n      are: $\\delta_1 = \\delta_2 = 0$; $\\delta_3 = 90^\\circ$.  As a\n\n      reference, the expected value in the case of standard\n\n      three-family oscillation(\\textit{i.e.}, for $\\theta_{i4} = 0$) is shown\n\n      for maximal CP-violating phase $\\delta$.  The rates are computed\n\n      according to Eq.~\\eqref{eq:signale}. Backgrounds have been\n\n      computed following Ref.~\\cite{Komatsu:2002sz}.}\n\n\\end{table}"}
{"table/equation": "\\begin{equation}\n\n    \\eta_s = \\sum_{i \\,\\in\\, \\text{sol}} |U_{s i}|^2\n\n    \\qquad\\text{and}\\qquad\n\n    c_s = \\sum_{j \\,\\in\\, \\text{atm}} |U_{s j}|^2\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:3+1param}\n\n    U_\\text{SBL} =\n\n    R_{14}(\\theta_{14}) \\; R_{24}(\\theta_{24}) \\; R_{34}(\\theta_{34}) \\; \n\n    R_{23}(\\theta_{23} ,\\, \\delta_3) \\; R_{13}(\\theta_{13} ,\\, \\delta_2) \\;\n\n    R_{12}(\\theta_{12} ,\\, \\delta_1) \\,.\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:3+1param2}\n\n    U_\\text{atm} =\n\n    R_{34}(\\theta_{34}) \\; R_{24}(\\theta_{24}) \\;\n\n    R_{23}(\\theta_{23} ,\\, \\delta_3) \\;\n\n    R_{14}(\\theta_{14}) \\; R_{13}(\\theta_{13} ,\\, \\delta_2) \\; \n\n    R_{12}(\\theta_{12} ,\\, \\delta_1) \\,.\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:pee}\n\n    P_{ee} = 1 - \\sin^2 2 \\theta_{14} \\sin^2 \\frac{\\Delta_\\Sbl L }{2} \n\n    - c^4_{14} \\sin^2 2 \\theta_{13} \\sin^2 \\frac{\\Delta_\\Atm L}{2} \\, ,\n\n\\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    P(\\nu_\\alpha \\to \\nu_\\beta) &= \\delta_{\\alpha \\beta}\n\n    - 4 \\Re \\left[ U_{\\alpha 3} U^\\star_{\\beta 3} \\left(\n\n    \\delta_{\\alpha\\beta} - U^\\star_{\\alpha 3} U_{\\beta 3}\n\n    - U^\\star_{\\alpha 4} U_{\\beta 4} \\right) \\right]\n\n    \\sin^2 \\frac{\\Delta_{23} L }{2}\n\n    \\\\\n\n    &- 2 \\Re \\left[ \\delta_{\\alpha\\beta} U_{\\alpha 4} U^\\star_{\\beta 4}\n\n    - |U_{\\alpha 4}|^2 |U_{\\beta 4}|^2 \\right]\n\n    \\pm 2 \\Im \\left[ U_{\\alpha 3} U^\\star_{\\beta 3}\n\n    U^\\star_{\\alpha 4} U_{\\beta 4} \\right] \\sin \\Delta_{23} L \\,,\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    P_{\\mu\\mu} &= 1 - 2 c^2_{14} s^2_{24} (1 - c^2_{14} s^2_{24})\n\n    - 4 \\big\\lbrace s^2_{23} c^2_{24} \\left[ c^2_{24} (c^2_{23}\n\n    - s^2_{13}) - s^2_{14} s^2_{24} \\right]\n\n    \\\\    \n\n    &- 2 c^3_{24} s_{23} (1 - 2 s^2_{23}) s_{13} s_{14} s_{24}\n\n    \\cos(\\delta_2 - \\delta_3) \\big\\rbrace\n\n    \\sin^2 \\frac{\\Delta_\\Atm L }{2} \\,.\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:pmumu}\n\n    P_{\\mu\\mu} = 1 - 2 s^2_{24}\n\n    - 4 s^2_{23} \\left[ c^2_{23} (1 - 2 s^2_{24}) - s^2_{13} \\right]\n\n    \\sin^2 \\frac{\\Delta_\\Atm L }{2} \\,.\n\n\\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    \\label{eq:pmus}\n\n    P_{\\mu s} &= 2 c^2_{34} s^2_{24} + \\big\\lbrace\n\n    \\sin^2 2 \\theta_{23} (c^4_{13} c^2_{24} s^2_{34}\n\n    - c^2_{34} s^2_{24})\n\n    \\\\\n\n    &+ 2  c_{34} \\sin 2 \\theta_{23} s_{34} \\left[\n\n    s_{24} (1 -2 s^2_{23}) \\cos \\delta_3\n\n    + 2 s_{23} s_{13} s_{14} \\cos \\delta_2 \\right] \\big\\rbrace\n\n    \\sin^2 \\frac{\\Delta_\\Atm L }{2}\n\n    \\\\\n\n    &\\pm c_{34} \\sin 2 \\theta_{23} s_{24} s_{34}\n\n    \\sin\\delta_3 \\sin\\Delta_\\Atm L \\,.\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    \\label{eq:pmue}\n\n    P_{\\mu e} &= 4 \\left\\lbrace s^2_{23} s^2_{13}\n\n    [1 - s^2_{13} - s^2_{14} - s^2_{24}] \n\n    + s_{23} s_{13} s_{14} s_{24} \\cos(\\delta_2 - \\delta_3) \\right\\rbrace\n\n    \\sin^2 \\frac{\\Delta_\\Atm L}{2}\n\n    \\\\\n\n    & \\pm 2 s_{23} s_{13} s_{14} s_{24} \\sin(\\delta_2 - \\delta_3)\n\n    \\sin\\Delta_\\Atm L + 2 s^2_{14} s^2_{24} \\,.\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    \\label{eq:pmutau}\n\n    P_{\\mu \\tau} &= 2 s^2_{24} s^2_{34} + \\big\\lbrace \\sin^2 2 \\theta_{23}\n\n    [c^4_{13} c^2_{24} c^2_{34} - s^2_{24} s^2_{34}]\n\n    \\\\\n\n    &- 4 \\sin 2 \\theta_{23} s_{13} s_{14} [s_{23} s_{34} \\cos\\delta_2\n\n    + c_{23} s_{24} \\cos(\\delta_2 - \\delta_3)]\n\n    \\\\\n\n    &+ 2 \\sin 2 \\theta_{23} s_{24} s_{34} c^2_{13} c^2_{24} c_{34}\n\n    [c^2_{14} - 2 c^2_{13} s^2_{23}] \\cos\\delta_3 \\big\\rbrace\n\n    \\sin^2 \\frac{\\Delta_\\Atm L}{2}\n\n    \\\\\n\n    &\\mp \\sin 2 \\theta_{23} s_{24} s_{34} c^2_{13} c^2_{14} c^2_{24} c_{34}\n\n    \\sin\\delta_3 \\sin\\Delta_\\Atm L \\,.\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    P^{3\\nu}_{\\mu\\tau} = P_{\\mu\\tau} (\\theta_{i4} = 0)\n\n    \\simeq c^4_{13} \\sin^2 2 \\theta_{23}\n\n    \\sin^2 \\frac{\\Delta_\\Atm L}{2} \\,.\n\n\\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    \\Delta P_{\\mu \\tau} &\\equiv P_{\\mu \\tau} - P^{3\\nu}_{\\mu\\tau}\n\n    \\\\\n\n    &= \\left\\lbrace - (s^2_{24} + s^2_{34}) \\sin^2 2 \\theta_{23} \n\n    + 2 \\sin 2 \\theta_{23} (1 - 2 s^2_{13}) s_{24} s_{34}\n\n    \\cos\\delta_3 \\right\\rbrace \\sin^2 \\frac{\\Delta_\\Atm L}{2}\n\n    \\\\\n\n    &\\mp \\sin 2 \\theta_{23} s_{24} s_{34} \\sin\\delta_3\n\n    \\sin \\Delta_\\Atm L + \\dots\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    \\label{eq:dpmue}\n\n    \\Delta P_{\\mu e} &\\equiv P_{\\mu e} - P_{\\mu e}(\\theta_{i4} = 0)\n\n    \\\\\n\n    &= s_{23} s_{13} s_{14} s_{24} \\cos(\\delta_2 - \\delta_3)\n\n    \\sin^2 \\frac{\\Delta_\\Atm L}{2}\n\n    \\\\\n\n    &\\pm 2 s_{23} s_{13} s_{14} s_{24}\n\n    \\sin(\\delta_2 - \\delta_3) \\sin \\Delta_\\Atm L + \\dots\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    N = \\int \\frac{d \\phi_{\\nu_\\alpha}(E)}{dE} \\,\n\n    \\sigma_{\\nu_\\alpha}(E) \\, dE \\,,\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:signaltau}\n\n    N_{\\mu \\tau} = A \\int \\frac{ d\\phi_{\\nu_{\\mu}}(E)}{dE} \\,\n\n    P_{\\mu \\tau}(E) \\sigma_{\\nu_\\tau}^{CC}(E) \\,\n\n    \\varepsilon_{\\mu \\tau} \\, dE \\,.\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:signale}\n\n    N_{\\mu e} = A \\int \\frac{ d\\phi_{\\nu_{\\mu}}(E)}{dE} \\,\n\n    P_{\\mu e}(E) \\sigma_{\\nu_{e}}^{CC}(E) \\,\n\n    \\varepsilon_{\\mu e}(E) \\, dE \\,,\n\n\\end{equation}"}
{"table/equation": "\\begin{table}[t] \\centering\n\n    \\begin{tabular}{|c|c|c|c|}\n\n\t\\hline\n\n\t\\hline\n\n\t$\\nu_{\\mu}$ &  $\\bar \\nu_{\\mu}$ & $\\nu_e$ &  $\\bar \\nu_e$\\\\\n\n\t\\hline\n\n\t669.0 &13.7 & 5.9& 0.3\\\\\n\n\t\\hline\\hline\n\n    \\end{tabular}\n\n    \\caption{\\label{tab:rates}\\sl%\n\n      Nominal performance of the CNGS reference beam~\\cite{fluxes}.\n\n      The total non-oscillated CC event rates are calculated assuming\n\n      $10^{19}$ pot and 1 Kton lead target mass.}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[t] \\centering\n\n    \\begin{tabular}{|c|c|c|c|c|c|}\n\n\t\\hline\\hline\n\n\t$(\\theta_{13}; \\theta_{14}; \\theta_{24}; \\theta_{34})$ & $N_\\tau$ & background & $(\\theta_{13}; \\theta_{14}; \\theta_{24}; \\theta_{34})$ & $N_\\tau$ & background \\\\\n\n\t\\hline\n\n\t$(5^\\circ;  5^\\circ;  5^\\circ; 20^\\circ)$    &     8.9    & 1.0 & $(10^\\circ; 5^\\circ;  5^\\circ; 20^\\circ)$    &     8.5    & 1.0 \\\\  \n\n\t$(5^\\circ;  5^\\circ;  5^\\circ; 30^\\circ)$    &     6.9    & 1.0 & $(10^\\circ; 5^\\circ;  5^\\circ; 30^\\circ)$    &     6.5    & 1.0 \\\\\n\n\t$(5^\\circ;  5^\\circ; 10^\\circ; 20^\\circ)$    &     8.3    & 1.0 & $(10^\\circ; 5^\\circ; 10^\\circ; 20^\\circ)$    &     7.9    & 1.0 \\\\\n\n\t$(5^\\circ;  5^\\circ; 10^\\circ; 30^\\circ)$    &    10.5    & 1.0 & $(10^\\circ; 5^\\circ; 10^\\circ; 30^\\circ)$    &    10.3    & 1.0 \\\\\n\n\t3 families                          &    15.1    & 1.0 &          3 families                          &    14.4    & 1.0 \\\\\n\n\t\\hline \\hline\n\n    \\end{tabular}\n\n    \\caption{\\label{tab:rates3}\\sl%\n\n      Event rates and expected background for the $\\nu_\\mu \\to\n\n      \\nu_\\tau$ channel in the OPERA detector, for different values of\n\n      $\\theta_{14}, \\theta_{24}$ and $\\theta_{34}$ in the (3+1)\n\n      scheme. The other unknown angle, $\\theta_{13}$ has been fixed\n\n      to: $\\theta_{13} = 5^\\circ, 10^\\circ$. The CP-violating phases\n\n      are: $\\delta_1 = \\delta_2 = 0$; $\\delta_3 = 90^\\circ$.  As a\n\n      reference, the expected value in the case of standard\n\n      three-family oscillation (\\textit{i.e.}, for $\\theta_{i4} = 0$) is shown\n\n      for maximal CP-violating phase $\\delta$. The rates are computed\n\n      according to Eq.~\\eqref{eq:signaltau}.}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[t] \\centering\n\n    \\begin{tabular}{|c|c|c|c|c|c|}\n\n\t\\hline\\hline\n\n\t$(\\theta_{13}; \\theta_{14}; \\theta_{24}; \\theta_{34})$ & $N_e$ & $\\nu_e^{CC}$ & $\\nu_\\mu^{NC}$ & $\\tau \\to e$ & $\\nu_\\mu^{CC}$  \\\\ \\hline\n\n\t$(5^\\circ; 5^\\circ; 5^\\circ; 20^\\circ)$   &  3.5 & 19.4 & 5.3 & 2.8 & 0.9 \\\\\n\n\t$(5^\\circ; 5^\\circ; 5^\\circ; 30^\\circ)$   &  3.5 & 19.4 & 5.3 & 2.1 & 0.9 \\\\\n\n\t$(5^\\circ; 5^\\circ; 10^\\circ; 20^\\circ)$  &  2.4 & 19.4 & 5.3 & 2.3 & 0.9 \\\\\n\n\t$(5^\\circ; 5^\\circ; 10^\\circ; 30^\\circ)$  &  2.4 & 19.4 & 5.3 & 2.4 & 0.9 \\\\\n\n\t3 families                        &  3.7 & 19.7 & 5.3 & 4.6 & 0.9 \\\\\n\n\t\\hline\n\n\t$(10^\\circ; 5^\\circ; 5^\\circ; 20^\\circ)$  & 10.6 & 19.4 & 5.3 & 2.7 & 0.9 \\\\\n\n\t$(10^\\circ; 5^\\circ; 5^\\circ; 30^\\circ)$  & 10.4 & 19.4 & 5.3 & 2.0 & 0.9 \\\\\n\n\t$(10^\\circ; 5^\\circ; 10^\\circ; 20^\\circ)$ &  8.8 & 19.4 & 5.3 & 2.2 & 0.9 \\\\\n\n\t$(10^\\circ; 5^\\circ; 10^\\circ; 30^\\circ)$ &  8.6 & 19.4 & 5.3 & 2.4 & 0.9 \\\\\n\n\t3 families                        & 15.1 & 19.7 & 5.3 & 4.8 & 0.9 \\\\\n\n\t\\hline \\hline\n\n    \\end{tabular}\n\n    \\caption{\\label{tab:rates2}\\sl%\n\n      Event rates and expected background for the $\\nu_\\mu \\to \\nu_e$\n\n      channel in the OPERA detector, for different values of \n\n      $\\theta_{14}, \\theta_{24}$ and $\\theta_{34}$ in the (3+1)\n\n      scheme. The other unknown angle, $\\theta_{13}$, has been fixed\n\n      to: $\\theta_{13} = 5^\\circ, 10^\\circ$. The CP-violating phases\n\n      are: $\\delta_1 = \\delta_2 = 0$; $\\delta_3 = 90^\\circ$.  As a\n\n      reference, the expected value in the case of standard\n\n      three-family oscillation(\\textit{i.e.}, for $\\theta_{i4} = 0$) is shown\n\n      for maximal CP-violating phase $\\delta$.  The rates are computed\n\n      according to Eq.~\\eqref{eq:signale}. Backgrounds have been\n\n      computed following Ref.~\\cite{Komatsu:2002sz}.}\n\n\\end{table}"}
{"table/equation": "\\begin{equation}\n\n    \\eta_s = \\sum_{i \\,\\in\\, \\text{sol}} |U_{s i}|^2\n\n    \\qquad\\text{and}\\qquad\n\n    c_s = \\sum_{j \\,\\in\\, \\text{atm}} |U_{s j}|^2\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:3+1param}\n\n    U_\\text{SBL} =\n\n    R_{14}(\\theta_{14}) \\; R_{24}(\\theta_{24}) \\; R_{34}(\\theta_{34}) \\; \n\n    R_{23}(\\theta_{23} ,\\, \\delta_3) \\; R_{13}(\\theta_{13} ,\\, \\delta_2) \\;\n\n    R_{12}(\\theta_{12} ,\\, \\delta_1) \\,.\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:3+1param2}\n\n    U_\\text{atm} =\n\n    R_{34}(\\theta_{34}) \\; R_{24}(\\theta_{24}) \\;\n\n    R_{23}(\\theta_{23} ,\\, \\delta_3) \\;\n\n    R_{14}(\\theta_{14}) \\; R_{13}(\\theta_{13} ,\\, \\delta_2) \\; \n\n    R_{12}(\\theta_{12} ,\\, \\delta_1) \\,.\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:pee}\n\n    P_{ee} = 1 - \\sin^2 2 \\theta_{14} \\sin^2 \\frac{\\Delta_\\Sbl L }{2} \n\n    - c^4_{14} \\sin^2 2 \\theta_{13} \\sin^2 \\frac{\\Delta_\\Atm L}{2} \\, ,\n\n\\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    P(\\nu_\\alpha \\to \\nu_\\beta) &= \\delta_{\\alpha \\beta}\n\n    - 4 \\Re \\left[ U_{\\alpha 3} U^\\star_{\\beta 3} \\left(\n\n    \\delta_{\\alpha\\beta} - U^\\star_{\\alpha 3} U_{\\beta 3}\n\n    - U^\\star_{\\alpha 4} U_{\\beta 4} \\right) \\right]\n\n    \\sin^2 \\frac{\\Delta_{23} L }{2}\n\n    \\\\\n\n    &- 2 \\Re \\left[ \\delta_{\\alpha\\beta} U_{\\alpha 4} U^\\star_{\\beta 4}\n\n    - |U_{\\alpha 4}|^2 |U_{\\beta 4}|^2 \\right]\n\n    \\pm 2 \\Im \\left[ U_{\\alpha 3} U^\\star_{\\beta 3}\n\n    U^\\star_{\\alpha 4} U_{\\beta 4} \\right] \\sin \\Delta_{23} L \\,,\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    P_{\\mu\\mu} &= 1 - 2 c^2_{14} s^2_{24} (1 - c^2_{14} s^2_{24})\n\n    - 4 \\big\\lbrace s^2_{23} c^2_{24} \\left[ c^2_{24} (c^2_{23}\n\n    - s^2_{13}) - s^2_{14} s^2_{24} \\right]\n\n    \\\\    \n\n    &- 2 c^3_{24} s_{23} (1 - 2 s^2_{23}) s_{13} s_{14} s_{24}\n\n    \\cos(\\delta_2 - \\delta_3) \\big\\rbrace\n\n    \\sin^2 \\frac{\\Delta_\\Atm L }{2} \\,.\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:pmumu}\n\n    P_{\\mu\\mu} = 1 - 2 s^2_{24}\n\n    - 4 s^2_{23} \\left[ c^2_{23} (1 - 2 s^2_{24}) - s^2_{13} \\right]\n\n    \\sin^2 \\frac{\\Delta_\\Atm L }{2} \\,.\n\n\\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    \\label{eq:pmus}\n\n    P_{\\mu s} &= 2 c^2_{34} s^2_{24} + \\big\\lbrace\n\n    \\sin^2 2 \\theta_{23} (c^4_{13} c^2_{24} s^2_{34}\n\n    - c^2_{34} s^2_{24})\n\n    \\\\\n\n    &+ 2  c_{34} \\sin 2 \\theta_{23} s_{34} \\left[\n\n    s_{24} (1 -2 s^2_{23}) \\cos \\delta_3\n\n    + 2 s_{23} s_{13} s_{14} \\cos \\delta_2 \\right] \\big\\rbrace\n\n    \\sin^2 \\frac{\\Delta_\\Atm L }{2}\n\n    \\\\\n\n    &\\pm c_{34} \\sin 2 \\theta_{23} s_{24} s_{34}\n\n    \\sin\\delta_3 \\sin\\Delta_\\Atm L \\,.\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    \\label{eq:pmue}\n\n    P_{\\mu e} &= 4 \\left\\lbrace s^2_{23} s^2_{13}\n\n    [1 - s^2_{13} - s^2_{14} - s^2_{24}] \n\n    + s_{23} s_{13} s_{14} s_{24} \\cos(\\delta_2 - \\delta_3) \\right\\rbrace\n\n    \\sin^2 \\frac{\\Delta_\\Atm L}{2}\n\n    \\\\\n\n    & \\pm 2 s_{23} s_{13} s_{14} s_{24} \\sin(\\delta_2 - \\delta_3)\n\n    \\sin\\Delta_\\Atm L + 2 s^2_{14} s^2_{24} \\,.\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    \\label{eq:pmutau}\n\n    P_{\\mu \\tau} &= 2 s^2_{24} s^2_{34} + \\big\\lbrace \\sin^2 2 \\theta_{23}\n\n    [c^4_{13} c^2_{24} c^2_{34} - s^2_{24} s^2_{34}]\n\n    \\\\\n\n    &- 4 \\sin 2 \\theta_{23} s_{13} s_{14} [s_{23} s_{34} \\cos\\delta_2\n\n    + c_{23} s_{24} \\cos(\\delta_2 - \\delta_3)]\n\n    \\\\\n\n    &+ 2 \\sin 2 \\theta_{23} s_{24} s_{34} c^2_{13} c^2_{24} c_{34}\n\n    [c^2_{14} - 2 c^2_{13} s^2_{23}] \\cos\\delta_3 \\big\\rbrace\n\n    \\sin^2 \\frac{\\Delta_\\Atm L}{2}\n\n    \\\\\n\n    &\\mp \\sin 2 \\theta_{23} s_{24} s_{34} c^2_{13} c^2_{14} c^2_{24} c_{34}\n\n    \\sin\\delta_3 \\sin\\Delta_\\Atm L \\,.\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    P^{3\\nu}_{\\mu\\tau} = P_{\\mu\\tau} (\\theta_{i4} = 0)\n\n    \\simeq c^4_{13} \\sin^2 2 \\theta_{23}\n\n    \\sin^2 \\frac{\\Delta_\\Atm L}{2} \\,.\n\n\\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    \\Delta P_{\\mu \\tau} &\\equiv P_{\\mu \\tau} - P^{3\\nu}_{\\mu\\tau}\n\n    \\\\\n\n    &= \\left\\lbrace - (s^2_{24} + s^2_{34}) \\sin^2 2 \\theta_{23} \n\n    + 2 \\sin 2 \\theta_{23} (1 - 2 s^2_{13}) s_{24} s_{34}\n\n    \\cos\\delta_3 \\right\\rbrace \\sin^2 \\frac{\\Delta_\\Atm L}{2}\n\n    \\\\\n\n    &\\mp \\sin 2 \\theta_{23} s_{24} s_{34} \\sin\\delta_3\n\n    \\sin \\Delta_\\Atm L + \\dots\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    \\label{eq:dpmue}\n\n    \\Delta P_{\\mu e} &\\equiv P_{\\mu e} - P_{\\mu e}(\\theta_{i4} = 0)\n\n    \\\\\n\n    &= s_{23} s_{13} s_{14} s_{24} \\cos(\\delta_2 - \\delta_3)\n\n    \\sin^2 \\frac{\\Delta_\\Atm L}{2}\n\n    \\\\\n\n    &\\pm 2 s_{23} s_{13} s_{14} s_{24}\n\n    \\sin(\\delta_2 - \\delta_3) \\sin \\Delta_\\Atm L + \\dots\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    N = \\int \\frac{d \\phi_{\\nu_\\alpha}(E)}{dE} \\,\n\n    \\sigma_{\\nu_\\alpha}(E) \\, dE \\,,\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:signaltau}\n\n    N_{\\mu \\tau} = A \\int \\frac{ d\\phi_{\\nu_{\\mu}}(E)}{dE} \\,\n\n    P_{\\mu \\tau}(E) \\sigma_{\\nu_\\tau}^{CC}(E) \\,\n\n    \\varepsilon_{\\mu \\tau} \\, dE \\,.\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:signale}\n\n    N_{\\mu e} = A \\int \\frac{ d\\phi_{\\nu_{\\mu}}(E)}{dE} \\,\n\n    P_{\\mu e}(E) \\sigma_{\\nu_{e}}^{CC}(E) \\,\n\n    \\varepsilon_{\\mu e}(E) \\, dE \\,,\n\n\\end{equation}"}
{"table/equation": "\\begin{table}[t] \\centering\n\n    \\begin{tabular}{|c|c|c|c|}\n\n\t\\hline\n\n\t\\hline\n\n\t$\\nu_{\\mu}$ &  $\\bar \\nu_{\\mu}$ & $\\nu_e$ &  $\\bar \\nu_e$\\\\\n\n\t\\hline\n\n\t669.0 &13.7 & 5.9& 0.3\\\\\n\n\t\\hline\\hline\n\n    \\end{tabular}\n\n    \\caption{\\label{tab:rates}\\sl%\n\n      Nominal performance of the CNGS reference beam~\\cite{fluxes}.\n\n      The total non-oscillated CC event rates are calculated assuming\n\n      $10^{19}$ pot and 1 Kton lead target mass.}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[t] \\centering\n\n    \\begin{tabular}{|c|c|c|c|c|c|}\n\n\t\\hline\\hline\n\n\t$(\\theta_{13}; \\theta_{14}; \\theta_{24}; \\theta_{34})$ & $N_\\tau$ & background & $(\\theta_{13}; \\theta_{14}; \\theta_{24}; \\theta_{34})$ & $N_\\tau$ & background \\\\\n\n\t\\hline\n\n\t$(5^\\circ;  5^\\circ;  5^\\circ; 20^\\circ)$    &     8.9    & 1.0 & $(10^\\circ; 5^\\circ;  5^\\circ; 20^\\circ)$    &     8.5    & 1.0 \\\\  \n\n\t$(5^\\circ;  5^\\circ;  5^\\circ; 30^\\circ)$    &     6.9    & 1.0 & $(10^\\circ; 5^\\circ;  5^\\circ; 30^\\circ)$    &     6.5    & 1.0 \\\\\n\n\t$(5^\\circ;  5^\\circ; 10^\\circ; 20^\\circ)$    &     8.3    & 1.0 & $(10^\\circ; 5^\\circ; 10^\\circ; 20^\\circ)$    &     7.9    & 1.0 \\\\\n\n\t$(5^\\circ;  5^\\circ; 10^\\circ; 30^\\circ)$    &    10.5    & 1.0 & $(10^\\circ; 5^\\circ; 10^\\circ; 30^\\circ)$    &    10.3    & 1.0 \\\\\n\n\t3 families                          &    15.1    & 1.0 &          3 families                          &    14.4    & 1.0 \\\\\n\n\t\\hline \\hline\n\n    \\end{tabular}\n\n    \\caption{\\label{tab:rates3}\\sl%\n\n      Event rates and expected background for the $\\nu_\\mu \\to\n\n      \\nu_\\tau$ channel in the OPERA detector, for different values of\n\n      $\\theta_{14}, \\theta_{24}$ and $\\theta_{34}$ in the (3+1)\n\n      scheme. The other unknown angle, $\\theta_{13}$ has been fixed\n\n      to: $\\theta_{13} = 5^\\circ, 10^\\circ$. The CP-violating phases\n\n      are: $\\delta_1 = \\delta_2 = 0$; $\\delta_3 = 90^\\circ$.  As a\n\n      reference, the expected value in the case of standard\n\n      three-family oscillation (\\textit{i.e.}, for $\\theta_{i4} = 0$) is shown\n\n      for maximal CP-violating phase $\\delta$. The rates are computed\n\n      according to Eq.~\\eqref{eq:signaltau}.}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[t] \\centering\n\n    \\begin{tabular}{|c|c|c|c|c|c|}\n\n\t\\hline\\hline\n\n\t$(\\theta_{13}; \\theta_{14}; \\theta_{24}; \\theta_{34})$ & $N_e$ & $\\nu_e^{CC}$ & $\\nu_\\mu^{NC}$ & $\\tau \\to e$ & $\\nu_\\mu^{CC}$  \\\\ \\hline\n\n\t$(5^\\circ; 5^\\circ; 5^\\circ; 20^\\circ)$   &  3.5 & 19.4 & 5.3 & 2.8 & 0.9 \\\\\n\n\t$(5^\\circ; 5^\\circ; 5^\\circ; 30^\\circ)$   &  3.5 & 19.4 & 5.3 & 2.1 & 0.9 \\\\\n\n\t$(5^\\circ; 5^\\circ; 10^\\circ; 20^\\circ)$  &  2.4 & 19.4 & 5.3 & 2.3 & 0.9 \\\\\n\n\t$(5^\\circ; 5^\\circ; 10^\\circ; 30^\\circ)$  &  2.4 & 19.4 & 5.3 & 2.4 & 0.9 \\\\\n\n\t3 families                        &  3.7 & 19.7 & 5.3 & 4.6 & 0.9 \\\\\n\n\t\\hline\n\n\t$(10^\\circ; 5^\\circ; 5^\\circ; 20^\\circ)$  & 10.6 & 19.4 & 5.3 & 2.7 & 0.9 \\\\\n\n\t$(10^\\circ; 5^\\circ; 5^\\circ; 30^\\circ)$  & 10.4 & 19.4 & 5.3 & 2.0 & 0.9 \\\\\n\n\t$(10^\\circ; 5^\\circ; 10^\\circ; 20^\\circ)$ &  8.8 & 19.4 & 5.3 & 2.2 & 0.9 \\\\\n\n\t$(10^\\circ; 5^\\circ; 10^\\circ; 30^\\circ)$ &  8.6 & 19.4 & 5.3 & 2.4 & 0.9 \\\\\n\n\t3 families                        & 15.1 & 19.7 & 5.3 & 4.8 & 0.9 \\\\\n\n\t\\hline \\hline\n\n    \\end{tabular}\n\n    \\caption{\\label{tab:rates2}\\sl%\n\n      Event rates and expected background for the $\\nu_\\mu \\to \\nu_e$\n\n      channel in the OPERA detector, for different values of \n\n      $\\theta_{14}, \\theta_{24}$ and $\\theta_{34}$ in the (3+1)\n\n      scheme. The other unknown angle, $\\theta_{13}$, has been fixed\n\n      to: $\\theta_{13} = 5^\\circ, 10^\\circ$. The CP-violating phases\n\n      are: $\\delta_1 = \\delta_2 = 0$; $\\delta_3 = 90^\\circ$.  As a\n\n      reference, the expected value in the case of standard\n\n      three-family oscillation(\\textit{i.e.}, for $\\theta_{i4} = 0$) is shown\n\n      for maximal CP-violating phase $\\delta$.  The rates are computed\n\n      according to Eq.~\\eqref{eq:signale}. Backgrounds have been\n\n      computed following Ref.~\\cite{Komatsu:2002sz}.}\n\n\\end{table}"}
{"table/equation": "\\begin{equation}\n\n    \\eta_s = \\sum_{i \\,\\in\\, \\text{sol}} |U_{s i}|^2\n\n    \\qquad\\text{and}\\qquad\n\n    c_s = \\sum_{j \\,\\in\\, \\text{atm}} |U_{s j}|^2\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:3+1param}\n\n    U_\\text{SBL} =\n\n    R_{14}(\\theta_{14}) \\; R_{24}(\\theta_{24}) \\; R_{34}(\\theta_{34}) \\; \n\n    R_{23}(\\theta_{23} ,\\, \\delta_3) \\; R_{13}(\\theta_{13} ,\\, \\delta_2) \\;\n\n    R_{12}(\\theta_{12} ,\\, \\delta_1) \\,.\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:3+1param2}\n\n    U_\\text{atm} =\n\n    R_{34}(\\theta_{34}) \\; R_{24}(\\theta_{24}) \\;\n\n    R_{23}(\\theta_{23} ,\\, \\delta_3) \\;\n\n    R_{14}(\\theta_{14}) \\; R_{13}(\\theta_{13} ,\\, \\delta_2) \\; \n\n    R_{12}(\\theta_{12} ,\\, \\delta_1) \\,.\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:pee}\n\n    P_{ee} = 1 - \\sin^2 2 \\theta_{14} \\sin^2 \\frac{\\Delta_\\Sbl L }{2} \n\n    - c^4_{14} \\sin^2 2 \\theta_{13} \\sin^2 \\frac{\\Delta_\\Atm L}{2} \\, ,\n\n\\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    P(\\nu_\\alpha \\to \\nu_\\beta) &= \\delta_{\\alpha \\beta}\n\n    - 4 \\Re \\left[ U_{\\alpha 3} U^\\star_{\\beta 3} \\left(\n\n    \\delta_{\\alpha\\beta} - U^\\star_{\\alpha 3} U_{\\beta 3}\n\n    - U^\\star_{\\alpha 4} U_{\\beta 4} \\right) \\right]\n\n    \\sin^2 \\frac{\\Delta_{23} L }{2}\n\n    \\\\\n\n    &- 2 \\Re \\left[ \\delta_{\\alpha\\beta} U_{\\alpha 4} U^\\star_{\\beta 4}\n\n    - |U_{\\alpha 4}|^2 |U_{\\beta 4}|^2 \\right]\n\n    \\pm 2 \\Im \\left[ U_{\\alpha 3} U^\\star_{\\beta 3}\n\n    U^\\star_{\\alpha 4} U_{\\beta 4} \\right] \\sin \\Delta_{23} L \\,,\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    P_{\\mu\\mu} &= 1 - 2 c^2_{14} s^2_{24} (1 - c^2_{14} s^2_{24})\n\n    - 4 \\big\\lbrace s^2_{23} c^2_{24} \\left[ c^2_{24} (c^2_{23}\n\n    - s^2_{13}) - s^2_{14} s^2_{24} \\right]\n\n    \\\\    \n\n    &- 2 c^3_{24} s_{23} (1 - 2 s^2_{23}) s_{13} s_{14} s_{24}\n\n    \\cos(\\delta_2 - \\delta_3) \\big\\rbrace\n\n    \\sin^2 \\frac{\\Delta_\\Atm L }{2} \\,.\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:pmumu}\n\n    P_{\\mu\\mu} = 1 - 2 s^2_{24}\n\n    - 4 s^2_{23} \\left[ c^2_{23} (1 - 2 s^2_{24}) - s^2_{13} \\right]\n\n    \\sin^2 \\frac{\\Delta_\\Atm L }{2} \\,.\n\n\\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    \\label{eq:pmus}\n\n    P_{\\mu s} &= 2 c^2_{34} s^2_{24} + \\big\\lbrace\n\n    \\sin^2 2 \\theta_{23} (c^4_{13} c^2_{24} s^2_{34}\n\n    - c^2_{34} s^2_{24})\n\n    \\\\\n\n    &+ 2  c_{34} \\sin 2 \\theta_{23} s_{34} \\left[\n\n    s_{24} (1 -2 s^2_{23}) \\cos \\delta_3\n\n    + 2 s_{23} s_{13} s_{14} \\cos \\delta_2 \\right] \\big\\rbrace\n\n    \\sin^2 \\frac{\\Delta_\\Atm L }{2}\n\n    \\\\\n\n    &\\pm c_{34} \\sin 2 \\theta_{23} s_{24} s_{34}\n\n    \\sin\\delta_3 \\sin\\Delta_\\Atm L \\,.\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    \\label{eq:pmue}\n\n    P_{\\mu e} &= 4 \\left\\lbrace s^2_{23} s^2_{13}\n\n    [1 - s^2_{13} - s^2_{14} - s^2_{24}] \n\n    + s_{23} s_{13} s_{14} s_{24} \\cos(\\delta_2 - \\delta_3) \\right\\rbrace\n\n    \\sin^2 \\frac{\\Delta_\\Atm L}{2}\n\n    \\\\\n\n    & \\pm 2 s_{23} s_{13} s_{14} s_{24} \\sin(\\delta_2 - \\delta_3)\n\n    \\sin\\Delta_\\Atm L + 2 s^2_{14} s^2_{24} \\,.\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    \\label{eq:pmutau}\n\n    P_{\\mu \\tau} &= 2 s^2_{24} s^2_{34} + \\big\\lbrace \\sin^2 2 \\theta_{23}\n\n    [c^4_{13} c^2_{24} c^2_{34} - s^2_{24} s^2_{34}]\n\n    \\\\\n\n    &- 4 \\sin 2 \\theta_{23} s_{13} s_{14} [s_{23} s_{34} \\cos\\delta_2\n\n    + c_{23} s_{24} \\cos(\\delta_2 - \\delta_3)]\n\n    \\\\\n\n    &+ 2 \\sin 2 \\theta_{23} s_{24} s_{34} c^2_{13} c^2_{24} c_{34}\n\n    [c^2_{14} - 2 c^2_{13} s^2_{23}] \\cos\\delta_3 \\big\\rbrace\n\n    \\sin^2 \\frac{\\Delta_\\Atm L}{2}\n\n    \\\\\n\n    &\\mp \\sin 2 \\theta_{23} s_{24} s_{34} c^2_{13} c^2_{14} c^2_{24} c_{34}\n\n    \\sin\\delta_3 \\sin\\Delta_\\Atm L \\,.\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    P^{3\\nu}_{\\mu\\tau} = P_{\\mu\\tau} (\\theta_{i4} = 0)\n\n    \\simeq c^4_{13} \\sin^2 2 \\theta_{23}\n\n    \\sin^2 \\frac{\\Delta_\\Atm L}{2} \\,.\n\n\\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    \\Delta P_{\\mu \\tau} &\\equiv P_{\\mu \\tau} - P^{3\\nu}_{\\mu\\tau}\n\n    \\\\\n\n    &= \\left\\lbrace - (s^2_{24} + s^2_{34}) \\sin^2 2 \\theta_{23} \n\n    + 2 \\sin 2 \\theta_{23} (1 - 2 s^2_{13}) s_{24} s_{34}\n\n    \\cos\\delta_3 \\right\\rbrace \\sin^2 \\frac{\\Delta_\\Atm L}{2}\n\n    \\\\\n\n    &\\mp \\sin 2 \\theta_{23} s_{24} s_{34} \\sin\\delta_3\n\n    \\sin \\Delta_\\Atm L + \\dots\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    \\label{eq:dpmue}\n\n    \\Delta P_{\\mu e} &\\equiv P_{\\mu e} - P_{\\mu e}(\\theta_{i4} = 0)\n\n    \\\\\n\n    &= s_{23} s_{13} s_{14} s_{24} \\cos(\\delta_2 - \\delta_3)\n\n    \\sin^2 \\frac{\\Delta_\\Atm L}{2}\n\n    \\\\\n\n    &\\pm 2 s_{23} s_{13} s_{14} s_{24}\n\n    \\sin(\\delta_2 - \\delta_3) \\sin \\Delta_\\Atm L + \\dots\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    N = \\int \\frac{d \\phi_{\\nu_\\alpha}(E)}{dE} \\,\n\n    \\sigma_{\\nu_\\alpha}(E) \\, dE \\,,\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:signaltau}\n\n    N_{\\mu \\tau} = A \\int \\frac{ d\\phi_{\\nu_{\\mu}}(E)}{dE} \\,\n\n    P_{\\mu \\tau}(E) \\sigma_{\\nu_\\tau}^{CC}(E) \\,\n\n    \\varepsilon_{\\mu \\tau} \\, dE \\,.\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:signale}\n\n    N_{\\mu e} = A \\int \\frac{ d\\phi_{\\nu_{\\mu}}(E)}{dE} \\,\n\n    P_{\\mu e}(E) \\sigma_{\\nu_{e}}^{CC}(E) \\,\n\n    \\varepsilon_{\\mu e}(E) \\, dE \\,,\n\n\\end{equation}"}
{"table/equation": "\\begin{table}[th]\n\n    \\centering\n\n    \\caption{Architecture for synthetic dataset models used in the experiments.}\n\n    \\resizebox{0.75\\linewidth}{!}{ \n\n        \\begin{tabular}{lllllll} \n\n            \\toprule \n\n            \\cellcolor{lightblue}\\textbf{Layer} & Input & FC1 & FC2 & FC3 & FC4 & FC5 \\\\ \\midrule \n\n            \\cellcolor{lightblue}\\textbf{Dimensions} & [5] & [5, 32] & [32, 64] & [64, 128] & [128, 32] & [32, 4] \\\\ \\bottomrule\n\n        \\end{tabular}\n\n    }\n\n    \\label{tab: synthetic-arch}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[H]\n\n    \\centering\n\n    \\caption{Architecture for CIFAR-MNIST dataset models. Every convolutional layer is followed by a max pooling layer with kernel size 2 and stride 2. }\n\n    \\resizebox{\\linewidth}{!}{ \n\n        \\begin{tabular}{lllllll}\n\n            \\toprule\n\n             \\cellcolor{lightblue}\\textbf{Layer} & Input & Conv1 & Conv2 & Conv3 & Flatten & FC \\\\ \\midrule \n\n             \\cellcolor{lightblue}\\textbf{Dimensions} & [3, 32, 32] & [3, 32, 3, 3] & [32, 64, 3, 3] & [64, 128, 3, 3] & [2048] & [2048, 20] \\\\ \\bottomrule \n\n        \\end{tabular}\n\n    }\n\n    \\label{tab: conv-arch}\n\n\\end{table}"}
{"table/equation": "\\begin{equation}\n\n    \\begin{split}\n\n        x^{\\star} \\leftarrow  \\argmin_{x \\in \\mathbb{R}^d} \\Bigg[ f(x) \\coloneqq \\frac{1}{N} \\sum_{i=1}^N f_i(x) \\Bigg], \n\n        \\label{eq: optimization-problem}\n\n    \\end{split}\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    x_{i}^{t,k} = x_{i}^{t,k-1} - \\eta_{\\ell} \\nabla f_i\\big(x_i^{t,k-1}, \\xi_i^{t,k-1}\\big), \\quad x_{i}^{t} = x_{i}^{t,K}, \\text{ and  } x_g^{t} = x_g^{t-1} - \\eta_g (x_g^{t-1} - \\frac{1}{N} \\sum_{i=1}^N x_{i}^{t}) , \n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq: mask learning}\n\n    \\min_{x, \\{m_i\\}_{i\\in[N]}} \\frac{1}{N} \\sum_{i=1}^N f_i(x \\odot m_i),\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    x_{i}^{t,k} = x_{i}^{t,k-1} - \\eta_{\\ell} \\nabla f_i\\big(x_i^{t,k-1} \\odot m_i^t, \\xi_i^{t,k-1}\\big) \\; \\text{and} \\; x_g^{t} = x_g^{t-1} - \\eta_g (x_g^{t-1} - \\frac{\\sum_{i \\in [N]}x_{i}^{t} \\odot m_i^t}{\\sum_{i \\in [N]}m_i^t}).\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq: mask score learning}\n\n    \\min_{x, \\{s_i\\}_{i\\in[N]}} \\frac{1}{N} \\sum_{i=1}^N f_i(x \\odot \\mathcal{G}(s_i)).\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\mathcal{L}_s = f_i\\Big(x_i^{t,k} \\odot \\mathcal{G}\\left(s_i^{t,k}\\right), \\xi_i^{t,k-1}\\Big) - \\lambda \\|\\sigma\\left(s_{i}^{t,k}\\right) - \\theta_{g \\backslash \\{i\\}}^{t}\\|_2^2; \\qquad s_{i}^{t,k+1} \\gets s_{i}^{t,k} - \\eta_s \\nabla_{s} \\mathcal{L}_s, \n\n    \\label{eq: mask-objective-func}\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\|\\sigma\\left(s_{i}^{t,k}\\right) - \\theta_{g \\backslash \\{i\\}}^{t}\\|_2^2 \\leq h. \n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\mathcal{L}_x = f_i\\Big(x_i^{t,k} \\odot \\mathcal{G}\\left(s_i^{t,k}\\right), \\xi_i^{t,k-1}\\Big); \\qquad x_{i}^{t,k+1} \\gets x_{i}^{t,k} - \\eta_{\\ell} \\nabla_{x} \\mathcal{L}_x,\n\n\\end{equation}"}
{"table/equation": "\\begin{table}[t]\n\n    \\centering\n\n    \\caption{Ablation study of the parameters $\\lambda$ and $\\tau$ on the CIFAR-MNIST dataset with $N=2$ participants over three independent runs. The first column ($\\tau=0.0$) corresponds to the FedAvg algorithm. The last row $(-)$ presents results for the \\texttt{FedPeWS-Fixed} algorithm.} \n\n    \\resizebox{\\linewidth}{!}{ \n\n    \\begin{tabular}{r|cccccc}\n\n        \\toprule\n\n        & \\multicolumn{6}{c}{\\cellcolor{lightblue}\\textbf{Proportion of warmup rounds $\\tau = W/T$}} \\\\ \\cmidrule{2-7} \n\n        $\\mathbf{\\lambda} (\\downarrow) , \\tau (\\rightarrow)$ & $\\tau=0.0$ (FedAvg) & $\\tau=0.2$ & $\\tau=0.4$ & $\\tau=0.5$ & $\\tau=0.6$ & $\\tau=0.8$ \\\\ \\midrule \n\n        0.0 & \\multirow{12}{*}{$71.23 \\pm 0.71$} & $68.01 \\pm 0.88$ & $66.01\\pm0.48$ & $65.96\\pm1.03$ & $65.40\\pm1.95$ & $65.77\\pm0.41$ \\\\ \\cmidrule{3-7}\n\n        0.1 & & $68.77\\pm1.09$ & $70.39\\pm1.00$ & $70.61\\pm1.17$ & $69.48\\pm0.37$ & $70.36\\pm2.06$ \\\\\n\n        0.3 & & $70.86\\pm0.23$ & $73.00\\pm0.65$ & $73.91\\pm0.71$ & $73.26\\pm0.46$ & $73.02\\pm1.05$ \\\\\n\n        0.5 & & $71.43\\pm0.56$ & $74.17\\pm0.91$ & $73.84\\pm0.12$ & $73.66\\pm0.88$ & $75.05\\pm0.45$ \\\\\n\n        1.0 & & $72.26\\pm0.54$ & $74.46\\pm0.44$ & $74.54\\pm0.76$ & $74.91\\pm0.42$ & $73.81\\pm0.87$ \\\\\n\n        2.0 & & $72.61\\pm0.79$ & $73.68\\pm0.17$ & \\cellcolor{lightgreen} $\\mathbf{75.35\\pm0.50}$ & $74.76\\pm0.54$ & $74.46\\pm0.56$ \\\\\n\n        5.0 & & $72.60\\pm0.45$ & \\cellcolor{lightgreen} $\\mathbf{75.22\\pm0.33}$ & \\cellcolor{lightgreen} $\\mathbf{75.00\\pm0.74}$ & $75.01\\pm0.71$ & $73.96\\pm1.60$ \\\\\n\n        10.0 & & $72.29\\pm0.48$ & \\cellcolor{lightgreen} $\\mathbf{74.97\\pm0.65}$ & $74.31\\pm0.95$ & $74.03\\pm0.30$ & $71.91\\pm2.69$ \\\\\n\n        100.0 & & $71.64\\pm0.47$ & $72.92\\pm0.39$ & $73.96\\pm0.65$ & $73.13\\pm0.73$ & $72.43\\pm3.55$ \\\\\n\n        1000.0 & & $71.58\\pm0.53$ & $73.18\\pm0.73$ & $73.32\\pm1.70$ & $73.87\\pm1.16$ & $72.52\\pm1.92$ \\\\ \\cmidrule{3-7} \n\n        $-$ & & $72.72\\pm0.44$ & \\cellcolor{lightgreen} $\\mathbf{75.22\\pm0.19}$ & \\cellcolor{lightgreen} $\\mathbf{75.05\\pm0.42}$ & $72.51\\pm3.89$ & $73.77\\pm0.20$ \\\\ \n\n        \\bottomrule\n\n    \\end{tabular}\n\n    }\n\n    \\label{tab: cifar-mnist-ablation}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[H]\n\n    \\centering\n\n    \\caption{Comparison of approaches for handling data heterogeneity in federated learning.} \n\n    \\resizebox{0.8\\linewidth}{!}{ \n\n        \\begin{tabular}{lcccc}\n\n            \\toprule\n\n             & \\multirow{2}{*}{\\parbox{2.1cm}{\\centering \\textbf{Shared \\\\ Global Model}}} & \\multirow{2}{*}{\\parbox{2.5cm}{\\centering \\textbf{Level of Mask \\\\ Personalization}}} & \\multirow{2}{*}{\\parbox{2cm}{\\centering {\\textbf{Learnable \\\\ Mask}}}} & \\multirow{2}{*}{\\parbox{2cm}{\\centering {\\textbf{Learnable \\\\ Parameters}}}} \\\\ \n\n             & & & & \\\\ \\midrule \n\n             FedPM \\citep{isik2022sparse} & \\checkmark & parameter-level & \\checkmark & $\\bm{\\times}$ \\\\ \n\n             IST \\citep{yuan2022distributed} & \\checkmark & neuron-level & $\\bm{\\times}$ (random) & \\checkmark \\\\ \n\n             LTH \\citep{frankle2018lottery} & $\\bm{\\times}$ & parameter-level & $\\bm{\\times}$ (pruned) & \\checkmark \\\\ \n\n             FedWeIT \\citep{yoon2021fedweit} & $\\bm{\\times}$ & parameter-level & \\checkmark & \\checkmark \\\\ \n\n             FjORD \\citep{horvath2021fjord} & \\checkmark & parameter-level & $\\bm{\\times}$ (slimmed) & \\checkmark \\\\ \\midrule \n\n             \\rowcolor{lightgreen} \\texttt{FedPeWS} (ours) & \\checkmark & neuron-level & \\checkmark & \\checkmark \\\\ \\bottomrule \n\n        \\end{tabular}\n\n    }\n\n    \\label{tab: methods-comparison}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[b]\n\n    \\centering\n\n    \\caption{Ablation study of the parameters $\\lambda$ and $\\tau$ on the combination of \\{Path-OCT-Tissue\\}MNIST datasets with $N=3$ participants over three independent runs. The first column ($\\tau=0.0$) corresponds to the FedAvg algorithm. The last row $(-)$ presents results for the \\texttt{FedPeWS-Fixed} algorithm.} \n\n    \\resizebox{\\linewidth}{!}{ \n\n    \\begin{tabular}{r|cccccc}\n\n        \\toprule\n\n        & \\multicolumn{6}{c}{\\cellcolor{lightblue}\\textbf{Proportion of warmup rounds $\\tau = W/T$}} \\\\ \\cmidrule{2-7} \n\n        ${\\lambda} (\\downarrow) , \\tau (\\rightarrow)$ & $\\tau=0.0$ (FedAvg) & $\\tau=0.2$ & $\\tau=0.4$ & $\\tau=0.5$ & $\\tau=0.6$ & $\\tau=0.8$ \\\\ \\midrule \n\n        0.0 & \\multirow{12}{*}{$52.25\\pm0.57$} & $53.04\\pm2.08$ & $50.35\\pm0.61$ & $47.83\\pm1.04$ & $47.89\\pm1.17$ & $46.80\\pm1.76$ \\\\ \\cmidrule{3-7}\n\n        0.1 & & $52.52\\pm1.79$ & $51.72\\pm0.63$ & $49.83\\pm1.83$ & $48.83\\pm2.51$ & $47.50\\pm1.07$ \\\\\n\n        0.3 & & $52.74\\pm1.75$ & $54.36\\pm0.80$ & $51.07\\pm0.25$ & $50.94\\pm2.22$ & $50.62\\pm1.91$ \\\\\n\n        0.5 & & $54.30\\pm2.08$ & $54.42\\pm1.43$ & $52.02\\pm2.53$ & $51.43\\pm1.60$ & $48.97\\pm0.80$ \\\\\n\n        1.0 & & \\cellcolor{lightgreen} $\\mathbf{54.89\\pm0.72}$ & $53.59\\pm1.40$ & $51.49\\pm1.19$ & $50.29\\pm2.70$ & $52.21\\pm1.02$ \\\\\n\n        2.0 & & \\cellcolor{lightgreen} $\\mathbf{54.75\\pm1.12}$ & \\cellcolor{lightgreen} $\\mathbf{54.45\\pm1.78}$ & $53.02\\pm0.45$ & $52.26\\pm1.81$ & $52.49\\pm0.49$ \\\\ \n\n        5.0 & & \\cellcolor{lightgreen} $\\mathbf{54.91\\pm0.91}$ & \\cellcolor{lightgreen} $\\mathbf{54.99\\pm0.90}$ & $52.41\\pm0.45$ & $52.95\\pm1.49$ & $52.53\\pm1.35$ \\\\ \n\n        10.0 & & \\cellcolor{lightgreen} $\\mathbf{55.12\\pm1.16}$ & \\cellcolor{lightgreen} $\\mathbf{55.03\\pm1.39}$ & $52.85\\pm0.95$ & $50.82\\pm3.03$ & $52.27\\pm1.11$ \\\\ \n\n        100.0 & & $54.22\\pm1.74$ & $52.31\\pm2.55$ & $52.10\\pm0.87$ & $49.52\\pm4.34$ & $51.46\\pm2.92$ \\\\ \n\n        1000.0 & & $53.45\\pm1.65$ & $53.82\\pm2.16$ & $51.16\\pm1.92$ & $51.30\\pm2.02$ & $51.19\\pm1.05$ \\\\ \\cmidrule{3-7} \n\n        $-$ & & $53.69\\pm0.77$ & $51.78\\pm0.44$ & $50.24\\pm1.88$ & $51.12\\pm0.72$ & $49.87\\pm1.23$ \\\\ \n\n        \\bottomrule\n\n    \\end{tabular}\n\n    }\n\n    \\label{tab: pot-ablation}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[t]\n\n    \\centering\n\n    \\caption{The required number of collaboration rounds to reach target accuracy $\\upsilon$ $\\%$ and the final accuracy after $T$ rounds. The results are averaged over 3 seeds. \n\n    {\\color{red} $\\times$} indicates that the algorithm cannot reach target accuracy $\\upsilon$ within $T$ rounds and {\\color{red} NA} means that it reaches $\\upsilon$ only in one random seed. } \n\n    \\resizebox{\\linewidth}{!}{ \n\n        \\begin{tabular}{cr|ccccc} \n\n        \\toprule\n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Dataset $/$ Batch size}} & \\multicolumn{3}{c}{Synthetic-32K, $32$} & \\multicolumn{1}{c}{Synthetic-3.2K, $8$} \n\n        \\\\ \\midrule \n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Parameters} $\\{\\eta_g / \\lambda / \\tau\\}$} & $\\{1.0 / 5.0 / 0.125\\}$ & $\\{0.5/2.0/0.2\\}$ & $\\{0.25/1.0/0.1875\\}$ & \\multicolumn{1}{c}{$\\{0.1/2.0/0.1\\}$} \\\\ \\midrule\n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Target accuracy $\\upsilon (\\%)$}} & $99$ & $90$ & $75$ & $99$ \n\n        \\\\ \\midrule \n\n        \\multirow{2}{*}{\\parbox{3.5cm}{\\centering \\textbf{No. of rounds to \\\\ reach target accuracy}}} & FedAvg & $148\\pm3.79$ & \\color{red} \\cellcolor{lightred} $199\\pm \\text{NA}$ & \\color{red} \\cellcolor{lightred} $\\times$ & \\color{red} \\cellcolor{lightred} $371\\pm\\text{NA}$ \n\n        \\\\ \\cmidrule{2-6}\n\n         & \\texttt{FedPeWS} & \\cellcolor{lightgreen} $\\mathbf{115\\pm7.21}$ & \\cellcolor{lightgreen} $\\mathbf{182\\pm6.81}$ & \\cellcolor{lightgreen} $\\mathbf{286\\pm7.93}$ & \\cellcolor{lightgreen} $\\mathbf{301\\pm10.59}$ \n\n        \\\\ \\midrule \n\n        \\multirow{2}{*}{\\parbox{3.75cm}{\\centering \\textbf{Final accuracy after \\\\ $T$ collaboration rounds}}} & FedAvg & \\multicolumn{1}{c}{$99.94\\pm0.05$} & $91.40\\pm7.25$ & $67.64\\pm0.90$ & $97.33\\pm3.89$ \n\n        \\\\ \\cmidrule{2-6}\n\n        & \\texttt{FedPeWS} & \\multicolumn{1}{c}{\\cellcolor{lightgreen} $\\mathbf{99.96\\pm0.01}$} & \\cellcolor{lightgreen} $\\mathbf{99.49\\pm0.60}$ & \\cellcolor{lightgreen} $\\mathbf{83.50\\pm3.52}$ & \\cellcolor{lightgreen} $\\mathbf{99.66\\pm0.19}$ \n\n        \\\\ \\bottomrule \n\n        \\end{tabular}\n\n    }\n\n    \\label{table: convergence-table}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[t]\n\n    \\centering\n\n    \\caption{The required number of collaboration rounds to reach target accuracy $\\upsilon$ $\\%$ using FedProx algorithm and the final accuracy after $T$ rounds. The results are averaged over 3 seeds. \n\n    {\\color{red} $\\times$} indicates that the algorithm cannot reach target accuracy $\\upsilon$ within $T$ rounds.} \n\n    \\resizebox{\\linewidth}{!}{ \n\n        \\begin{tabular}{cr|ccccc} \n\n        \\toprule\n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Dataset $/$ Batch size}} & \\multicolumn{3}{c}{Synthetic-32K, $32$} & \\multicolumn{1}{c}{Synthetic-3.2K, $8$} \n\n        \\\\ \\midrule \n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Parameters} $\\{\\eta_g / \\lambda / \\tau\\}$} & $\\{1.0 / 0.1 / 0.125\\}$ & $\\{0.5/0.1/0.2\\}$ & $\\{0.25/1.0/0.1875\\}$ & \\multicolumn{1}{c}{$\\{0.1/1.0/0.1\\}$} \\\\ \\midrule\n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Target accuracy $\\upsilon (\\%)$}} & $99$ & $90$ & $75$ & $99$ \n\n        \\\\ \\midrule \n\n        \\multirow{2}{*}{\\parbox{3.5cm}{\\centering \\textbf{No. of rounds to \\\\ reach target accuracy}}} & FedProx & $138\\pm13.22$ & \\cellcolor{lightred} \\color{red} $\\times$ & \\cellcolor{lightred} \\color{red} $\\times$ & $362\\pm20.00$ \n\n        \\\\ \\cmidrule{2-6} \n\n         & \\texttt{FedPeWS} & \\cellcolor{lightgreen} $\\mathbf{115\\pm5.29}$ & \\cellcolor{lightgreen} $\\mathbf{211\\pm16.52}$ & \\cellcolor{lightgreen} $\\mathbf{314\\pm27.83}$ & \\cellcolor{lightgreen} $\\mathbf{344\\pm27.30}$ \n\n        \\\\ \\midrule \n\n        \\multirow{2}{*}{\\parbox{3.75cm}{\\centering \\textbf{Final accuracy after \\\\ $T$ collaboration rounds}}} & FedProx & \\multicolumn{1}{c}{$99.95\\pm0.02$} & $82.43\\pm7.98$ & $69.26\\pm6.03$ &\\cellcolor{lightgreen}  $\\mathbf{99.92\\pm0.06}$ \n\n        \\\\ \\cmidrule{2-6}\n\n        & \\texttt{FedPeWS} & \\multicolumn{1}{c}{\\cellcolor{lightgreen} $\\mathbf{99.96\\pm0.01}$} & \\cellcolor{lightgreen} $\\mathbf{98.40\\pm1.84}$ & \\cellcolor{lightgreen} $\\mathbf{90.40\\pm3.91}$ & \\cellcolor{lightgreen} $\\mathbf{99.92\\pm0.07}$ \n\n        \\\\ \\bottomrule \n\n        \\end{tabular}\n\n    }\n\n    \\label{table: convergence-table-fedprox}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[th]\n\n    \\centering\n\n    \\caption{Architecture for synthetic dataset models used in the experiments.}\n\n    \\resizebox{0.75\\linewidth}{!}{ \n\n        \\begin{tabular}{lllllll} \n\n            \\toprule \n\n            \\cellcolor{lightblue}\\textbf{Layer} & Input & FC1 & FC2 & FC3 & FC4 & FC5 \\\\ \\midrule \n\n            \\cellcolor{lightblue}\\textbf{Dimensions} & [5] & [5, 32] & [32, 64] & [64, 128] & [128, 32] & [32, 4] \\\\ \\bottomrule\n\n        \\end{tabular}\n\n    }\n\n    \\label{tab: synthetic-arch}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[H]\n\n    \\centering\n\n    \\caption{Architecture for CIFAR-MNIST dataset models. Every convolutional layer is followed by a max pooling layer with kernel size 2 and stride 2. }\n\n    \\resizebox{\\linewidth}{!}{ \n\n        \\begin{tabular}{lllllll}\n\n            \\toprule\n\n             \\cellcolor{lightblue}\\textbf{Layer} & Input & Conv1 & Conv2 & Conv3 & Flatten & FC \\\\ \\midrule \n\n             \\cellcolor{lightblue}\\textbf{Dimensions} & [3, 32, 32] & [3, 32, 3, 3] & [32, 64, 3, 3] & [64, 128, 3, 3] & [2048] & [2048, 20] \\\\ \\bottomrule \n\n        \\end{tabular}\n\n    }\n\n    \\label{tab: conv-arch}\n\n\\end{table}"}
{"table/equation": "\\begin{equation}\n\n    \\begin{split}\n\n        x^{\\star} \\leftarrow  \\argmin_{x \\in \\mathbb{R}^d} \\Bigg[ f(x) \\coloneqq \\frac{1}{N} \\sum_{i=1}^N f_i(x) \\Bigg], \n\n        \\label{eq: optimization-problem}\n\n    \\end{split}\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    x_{i}^{t,k} = x_{i}^{t,k-1} - \\eta_{\\ell} \\nabla f_i\\big(x_i^{t,k-1}, \\xi_i^{t,k-1}\\big), \\quad x_{i}^{t} = x_{i}^{t,K}, \\text{ and  } x_g^{t} = x_g^{t-1} - \\eta_g (x_g^{t-1} - \\frac{1}{N} \\sum_{i=1}^N x_{i}^{t}) , \n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq: mask learning}\n\n    \\min_{x, \\{m_i\\}_{i\\in[N]}} \\frac{1}{N} \\sum_{i=1}^N f_i(x \\odot m_i),\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    x_{i}^{t,k} = x_{i}^{t,k-1} - \\eta_{\\ell} \\nabla f_i\\big(x_i^{t,k-1} \\odot m_i^t, \\xi_i^{t,k-1}\\big) \\; \\text{and} \\; x_g^{t} = x_g^{t-1} - \\eta_g (x_g^{t-1} - \\frac{\\sum_{i \\in [N]}x_{i}^{t} \\odot m_i^t}{\\sum_{i \\in [N]}m_i^t}).\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq: mask score learning}\n\n    \\min_{x, \\{s_i\\}_{i\\in[N]}} \\frac{1}{N} \\sum_{i=1}^N f_i(x \\odot \\mathcal{G}(s_i)).\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\mathcal{L}_s = f_i\\Big(x_i^{t,k} \\odot \\mathcal{G}\\left(s_i^{t,k}\\right), \\xi_i^{t,k-1}\\Big) - \\lambda \\|\\sigma\\left(s_{i}^{t,k}\\right) - \\theta_{g \\backslash \\{i\\}}^{t}\\|_2^2; \\qquad s_{i}^{t,k+1} \\gets s_{i}^{t,k} - \\eta_s \\nabla_{s} \\mathcal{L}_s, \n\n    \\label{eq: mask-objective-func}\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\|\\sigma\\left(s_{i}^{t,k}\\right) - \\theta_{g \\backslash \\{i\\}}^{t}\\|_2^2 \\leq h. \n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\mathcal{L}_x = f_i\\Big(x_i^{t,k} \\odot \\mathcal{G}\\left(s_i^{t,k}\\right), \\xi_i^{t,k-1}\\Big); \\qquad x_{i}^{t,k+1} \\gets x_{i}^{t,k} - \\eta_{\\ell} \\nabla_{x} \\mathcal{L}_x,\n\n\\end{equation}"}
{"table/equation": "\\begin{table}[t]\n\n    \\centering\n\n    \\caption{Ablation study of the parameters $\\lambda$ and $\\tau$ on the CIFAR-MNIST dataset with $N=2$ participants over three independent runs. The first column ($\\tau=0.0$) corresponds to the FedAvg algorithm. The last row $(-)$ presents results for the \\texttt{FedPeWS-Fixed} algorithm.} \n\n    \\resizebox{\\linewidth}{!}{ \n\n    \\begin{tabular}{r|cccccc}\n\n        \\toprule\n\n        & \\multicolumn{6}{c}{\\cellcolor{lightblue}\\textbf{Proportion of warmup rounds $\\tau = W/T$}} \\\\ \\cmidrule{2-7} \n\n        $\\mathbf{\\lambda} (\\downarrow) , \\tau (\\rightarrow)$ & $\\tau=0.0$ (FedAvg) & $\\tau=0.2$ & $\\tau=0.4$ & $\\tau=0.5$ & $\\tau=0.6$ & $\\tau=0.8$ \\\\ \\midrule \n\n        0.0 & \\multirow{12}{*}{$71.23 \\pm 0.71$} & $68.01 \\pm 0.88$ & $66.01\\pm0.48$ & $65.96\\pm1.03$ & $65.40\\pm1.95$ & $65.77\\pm0.41$ \\\\ \\cmidrule{3-7}\n\n        0.1 & & $68.77\\pm1.09$ & $70.39\\pm1.00$ & $70.61\\pm1.17$ & $69.48\\pm0.37$ & $70.36\\pm2.06$ \\\\\n\n        0.3 & & $70.86\\pm0.23$ & $73.00\\pm0.65$ & $73.91\\pm0.71$ & $73.26\\pm0.46$ & $73.02\\pm1.05$ \\\\\n\n        0.5 & & $71.43\\pm0.56$ & $74.17\\pm0.91$ & $73.84\\pm0.12$ & $73.66\\pm0.88$ & $75.05\\pm0.45$ \\\\\n\n        1.0 & & $72.26\\pm0.54$ & $74.46\\pm0.44$ & $74.54\\pm0.76$ & $74.91\\pm0.42$ & $73.81\\pm0.87$ \\\\\n\n        2.0 & & $72.61\\pm0.79$ & $73.68\\pm0.17$ & \\cellcolor{lightgreen} $\\mathbf{75.35\\pm0.50}$ & $74.76\\pm0.54$ & $74.46\\pm0.56$ \\\\\n\n        5.0 & & $72.60\\pm0.45$ & \\cellcolor{lightgreen} $\\mathbf{75.22\\pm0.33}$ & \\cellcolor{lightgreen} $\\mathbf{75.00\\pm0.74}$ & $75.01\\pm0.71$ & $73.96\\pm1.60$ \\\\\n\n        10.0 & & $72.29\\pm0.48$ & \\cellcolor{lightgreen} $\\mathbf{74.97\\pm0.65}$ & $74.31\\pm0.95$ & $74.03\\pm0.30$ & $71.91\\pm2.69$ \\\\\n\n        100.0 & & $71.64\\pm0.47$ & $72.92\\pm0.39$ & $73.96\\pm0.65$ & $73.13\\pm0.73$ & $72.43\\pm3.55$ \\\\\n\n        1000.0 & & $71.58\\pm0.53$ & $73.18\\pm0.73$ & $73.32\\pm1.70$ & $73.87\\pm1.16$ & $72.52\\pm1.92$ \\\\ \\cmidrule{3-7} \n\n        $-$ & & $72.72\\pm0.44$ & \\cellcolor{lightgreen} $\\mathbf{75.22\\pm0.19}$ & \\cellcolor{lightgreen} $\\mathbf{75.05\\pm0.42}$ & $72.51\\pm3.89$ & $73.77\\pm0.20$ \\\\ \n\n        \\bottomrule\n\n    \\end{tabular}\n\n    }\n\n    \\label{tab: cifar-mnist-ablation}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[H]\n\n    \\centering\n\n    \\caption{Comparison of approaches for handling data heterogeneity in federated learning.} \n\n    \\resizebox{0.8\\linewidth}{!}{ \n\n        \\begin{tabular}{lcccc}\n\n            \\toprule\n\n             & \\multirow{2}{*}{\\parbox{2.1cm}{\\centering \\textbf{Shared \\\\ Global Model}}} & \\multirow{2}{*}{\\parbox{2.5cm}{\\centering \\textbf{Level of Mask \\\\ Personalization}}} & \\multirow{2}{*}{\\parbox{2cm}{\\centering {\\textbf{Learnable \\\\ Mask}}}} & \\multirow{2}{*}{\\parbox{2cm}{\\centering {\\textbf{Learnable \\\\ Parameters}}}} \\\\ \n\n             & & & & \\\\ \\midrule \n\n             FedPM \\citep{isik2022sparse} & \\checkmark & parameter-level & \\checkmark & $\\bm{\\times}$ \\\\ \n\n             IST \\citep{yuan2022distributed} & \\checkmark & neuron-level & $\\bm{\\times}$ (random) & \\checkmark \\\\ \n\n             LTH \\citep{frankle2018lottery} & $\\bm{\\times}$ & parameter-level & $\\bm{\\times}$ (pruned) & \\checkmark \\\\ \n\n             FedWeIT \\citep{yoon2021fedweit} & $\\bm{\\times}$ & parameter-level & \\checkmark & \\checkmark \\\\ \n\n             FjORD \\citep{horvath2021fjord} & \\checkmark & parameter-level & $\\bm{\\times}$ (slimmed) & \\checkmark \\\\ \\midrule \n\n             \\rowcolor{lightgreen} \\texttt{FedPeWS} (ours) & \\checkmark & neuron-level & \\checkmark & \\checkmark \\\\ \\bottomrule \n\n        \\end{tabular}\n\n    }\n\n    \\label{tab: methods-comparison}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[b]\n\n    \\centering\n\n    \\caption{Ablation study of the parameters $\\lambda$ and $\\tau$ on the combination of \\{Path-OCT-Tissue\\}MNIST datasets with $N=3$ participants over three independent runs. The first column ($\\tau=0.0$) corresponds to the FedAvg algorithm. The last row $(-)$ presents results for the \\texttt{FedPeWS-Fixed} algorithm.} \n\n    \\resizebox{\\linewidth}{!}{ \n\n    \\begin{tabular}{r|cccccc}\n\n        \\toprule\n\n        & \\multicolumn{6}{c}{\\cellcolor{lightblue}\\textbf{Proportion of warmup rounds $\\tau = W/T$}} \\\\ \\cmidrule{2-7} \n\n        ${\\lambda} (\\downarrow) , \\tau (\\rightarrow)$ & $\\tau=0.0$ (FedAvg) & $\\tau=0.2$ & $\\tau=0.4$ & $\\tau=0.5$ & $\\tau=0.6$ & $\\tau=0.8$ \\\\ \\midrule \n\n        0.0 & \\multirow{12}{*}{$52.25\\pm0.57$} & $53.04\\pm2.08$ & $50.35\\pm0.61$ & $47.83\\pm1.04$ & $47.89\\pm1.17$ & $46.80\\pm1.76$ \\\\ \\cmidrule{3-7}\n\n        0.1 & & $52.52\\pm1.79$ & $51.72\\pm0.63$ & $49.83\\pm1.83$ & $48.83\\pm2.51$ & $47.50\\pm1.07$ \\\\\n\n        0.3 & & $52.74\\pm1.75$ & $54.36\\pm0.80$ & $51.07\\pm0.25$ & $50.94\\pm2.22$ & $50.62\\pm1.91$ \\\\\n\n        0.5 & & $54.30\\pm2.08$ & $54.42\\pm1.43$ & $52.02\\pm2.53$ & $51.43\\pm1.60$ & $48.97\\pm0.80$ \\\\\n\n        1.0 & & \\cellcolor{lightgreen} $\\mathbf{54.89\\pm0.72}$ & $53.59\\pm1.40$ & $51.49\\pm1.19$ & $50.29\\pm2.70$ & $52.21\\pm1.02$ \\\\\n\n        2.0 & & \\cellcolor{lightgreen} $\\mathbf{54.75\\pm1.12}$ & \\cellcolor{lightgreen} $\\mathbf{54.45\\pm1.78}$ & $53.02\\pm0.45$ & $52.26\\pm1.81$ & $52.49\\pm0.49$ \\\\ \n\n        5.0 & & \\cellcolor{lightgreen} $\\mathbf{54.91\\pm0.91}$ & \\cellcolor{lightgreen} $\\mathbf{54.99\\pm0.90}$ & $52.41\\pm0.45$ & $52.95\\pm1.49$ & $52.53\\pm1.35$ \\\\ \n\n        10.0 & & \\cellcolor{lightgreen} $\\mathbf{55.12\\pm1.16}$ & \\cellcolor{lightgreen} $\\mathbf{55.03\\pm1.39}$ & $52.85\\pm0.95$ & $50.82\\pm3.03$ & $52.27\\pm1.11$ \\\\ \n\n        100.0 & & $54.22\\pm1.74$ & $52.31\\pm2.55$ & $52.10\\pm0.87$ & $49.52\\pm4.34$ & $51.46\\pm2.92$ \\\\ \n\n        1000.0 & & $53.45\\pm1.65$ & $53.82\\pm2.16$ & $51.16\\pm1.92$ & $51.30\\pm2.02$ & $51.19\\pm1.05$ \\\\ \\cmidrule{3-7} \n\n        $-$ & & $53.69\\pm0.77$ & $51.78\\pm0.44$ & $50.24\\pm1.88$ & $51.12\\pm0.72$ & $49.87\\pm1.23$ \\\\ \n\n        \\bottomrule\n\n    \\end{tabular}\n\n    }\n\n    \\label{tab: pot-ablation}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[t]\n\n    \\centering\n\n    \\caption{The required number of collaboration rounds to reach target accuracy $\\upsilon$ $\\%$ and the final accuracy after $T$ rounds. The results are averaged over 3 seeds. \n\n    {\\color{red} $\\times$} indicates that the algorithm cannot reach target accuracy $\\upsilon$ within $T$ rounds and {\\color{red} NA} means that it reaches $\\upsilon$ only in one random seed. } \n\n    \\resizebox{\\linewidth}{!}{ \n\n        \\begin{tabular}{cr|ccccc} \n\n        \\toprule\n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Dataset $/$ Batch size}} & \\multicolumn{3}{c}{Synthetic-32K, $32$} & \\multicolumn{1}{c}{Synthetic-3.2K, $8$} \n\n        \\\\ \\midrule \n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Parameters} $\\{\\eta_g / \\lambda / \\tau\\}$} & $\\{1.0 / 5.0 / 0.125\\}$ & $\\{0.5/2.0/0.2\\}$ & $\\{0.25/1.0/0.1875\\}$ & \\multicolumn{1}{c}{$\\{0.1/2.0/0.1\\}$} \\\\ \\midrule\n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Target accuracy $\\upsilon (\\%)$}} & $99$ & $90$ & $75$ & $99$ \n\n        \\\\ \\midrule \n\n        \\multirow{2}{*}{\\parbox{3.5cm}{\\centering \\textbf{No. of rounds to \\\\ reach target accuracy}}} & FedAvg & $148\\pm3.79$ & \\color{red} \\cellcolor{lightred} $199\\pm \\text{NA}$ & \\color{red} \\cellcolor{lightred} $\\times$ & \\color{red} \\cellcolor{lightred} $371\\pm\\text{NA}$ \n\n        \\\\ \\cmidrule{2-6}\n\n         & \\texttt{FedPeWS} & \\cellcolor{lightgreen} $\\mathbf{115\\pm7.21}$ & \\cellcolor{lightgreen} $\\mathbf{182\\pm6.81}$ & \\cellcolor{lightgreen} $\\mathbf{286\\pm7.93}$ & \\cellcolor{lightgreen} $\\mathbf{301\\pm10.59}$ \n\n        \\\\ \\midrule \n\n        \\multirow{2}{*}{\\parbox{3.75cm}{\\centering \\textbf{Final accuracy after \\\\ $T$ collaboration rounds}}} & FedAvg & \\multicolumn{1}{c}{$99.94\\pm0.05$} & $91.40\\pm7.25$ & $67.64\\pm0.90$ & $97.33\\pm3.89$ \n\n        \\\\ \\cmidrule{2-6}\n\n        & \\texttt{FedPeWS} & \\multicolumn{1}{c}{\\cellcolor{lightgreen} $\\mathbf{99.96\\pm0.01}$} & \\cellcolor{lightgreen} $\\mathbf{99.49\\pm0.60}$ & \\cellcolor{lightgreen} $\\mathbf{83.50\\pm3.52}$ & \\cellcolor{lightgreen} $\\mathbf{99.66\\pm0.19}$ \n\n        \\\\ \\bottomrule \n\n        \\end{tabular}\n\n    }\n\n    \\label{table: convergence-table}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[t]\n\n    \\centering\n\n    \\caption{The required number of collaboration rounds to reach target accuracy $\\upsilon$ $\\%$ using FedProx algorithm and the final accuracy after $T$ rounds. The results are averaged over 3 seeds. \n\n    {\\color{red} $\\times$} indicates that the algorithm cannot reach target accuracy $\\upsilon$ within $T$ rounds.} \n\n    \\resizebox{\\linewidth}{!}{ \n\n        \\begin{tabular}{cr|ccccc} \n\n        \\toprule\n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Dataset $/$ Batch size}} & \\multicolumn{3}{c}{Synthetic-32K, $32$} & \\multicolumn{1}{c}{Synthetic-3.2K, $8$} \n\n        \\\\ \\midrule \n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Parameters} $\\{\\eta_g / \\lambda / \\tau\\}$} & $\\{1.0 / 0.1 / 0.125\\}$ & $\\{0.5/0.1/0.2\\}$ & $\\{0.25/1.0/0.1875\\}$ & \\multicolumn{1}{c}{$\\{0.1/1.0/0.1\\}$} \\\\ \\midrule\n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Target accuracy $\\upsilon (\\%)$}} & $99$ & $90$ & $75$ & $99$ \n\n        \\\\ \\midrule \n\n        \\multirow{2}{*}{\\parbox{3.5cm}{\\centering \\textbf{No. of rounds to \\\\ reach target accuracy}}} & FedProx & $138\\pm13.22$ & \\cellcolor{lightred} \\color{red} $\\times$ & \\cellcolor{lightred} \\color{red} $\\times$ & $362\\pm20.00$ \n\n        \\\\ \\cmidrule{2-6} \n\n         & \\texttt{FedPeWS} & \\cellcolor{lightgreen} $\\mathbf{115\\pm5.29}$ & \\cellcolor{lightgreen} $\\mathbf{211\\pm16.52}$ & \\cellcolor{lightgreen} $\\mathbf{314\\pm27.83}$ & \\cellcolor{lightgreen} $\\mathbf{344\\pm27.30}$ \n\n        \\\\ \\midrule \n\n        \\multirow{2}{*}{\\parbox{3.75cm}{\\centering \\textbf{Final accuracy after \\\\ $T$ collaboration rounds}}} & FedProx & \\multicolumn{1}{c}{$99.95\\pm0.02$} & $82.43\\pm7.98$ & $69.26\\pm6.03$ &\\cellcolor{lightgreen}  $\\mathbf{99.92\\pm0.06}$ \n\n        \\\\ \\cmidrule{2-6}\n\n        & \\texttt{FedPeWS} & \\multicolumn{1}{c}{\\cellcolor{lightgreen} $\\mathbf{99.96\\pm0.01}$} & \\cellcolor{lightgreen} $\\mathbf{98.40\\pm1.84}$ & \\cellcolor{lightgreen} $\\mathbf{90.40\\pm3.91}$ & \\cellcolor{lightgreen} $\\mathbf{99.92\\pm0.07}$ \n\n        \\\\ \\bottomrule \n\n        \\end{tabular}\n\n    }\n\n    \\label{table: convergence-table-fedprox}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[th]\n\n    \\centering\n\n    \\caption{Architecture for synthetic dataset models used in the experiments.}\n\n    \\resizebox{0.75\\linewidth}{!}{ \n\n        \\begin{tabular}{lllllll} \n\n            \\toprule \n\n            \\cellcolor{lightblue}\\textbf{Layer} & Input & FC1 & FC2 & FC3 & FC4 & FC5 \\\\ \\midrule \n\n            \\cellcolor{lightblue}\\textbf{Dimensions} & [5] & [5, 32] & [32, 64] & [64, 128] & [128, 32] & [32, 4] \\\\ \\bottomrule\n\n        \\end{tabular}\n\n    }\n\n    \\label{tab: synthetic-arch}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[H]\n\n    \\centering\n\n    \\caption{Architecture for CIFAR-MNIST dataset models. Every convolutional layer is followed by a max pooling layer with kernel size 2 and stride 2. }\n\n    \\resizebox{\\linewidth}{!}{ \n\n        \\begin{tabular}{lllllll}\n\n            \\toprule\n\n             \\cellcolor{lightblue}\\textbf{Layer} & Input & Conv1 & Conv2 & Conv3 & Flatten & FC \\\\ \\midrule \n\n             \\cellcolor{lightblue}\\textbf{Dimensions} & [3, 32, 32] & [3, 32, 3, 3] & [32, 64, 3, 3] & [64, 128, 3, 3] & [2048] & [2048, 20] \\\\ \\bottomrule \n\n        \\end{tabular}\n\n    }\n\n    \\label{tab: conv-arch}\n\n\\end{table}"}
{"table/equation": "\\begin{equation}\n\n    \\begin{split}\n\n        x^{\\star} \\leftarrow  \\argmin_{x \\in \\mathbb{R}^d} \\Bigg[ f(x) \\coloneqq \\frac{1}{N} \\sum_{i=1}^N f_i(x) \\Bigg], \n\n        \\label{eq: optimization-problem}\n\n    \\end{split}\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    x_{i}^{t,k} = x_{i}^{t,k-1} - \\eta_{\\ell} \\nabla f_i\\big(x_i^{t,k-1}, \\xi_i^{t,k-1}\\big), \\quad x_{i}^{t} = x_{i}^{t,K}, \\text{ and  } x_g^{t} = x_g^{t-1} - \\eta_g (x_g^{t-1} - \\frac{1}{N} \\sum_{i=1}^N x_{i}^{t}) , \n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq: mask learning}\n\n    \\min_{x, \\{m_i\\}_{i\\in[N]}} \\frac{1}{N} \\sum_{i=1}^N f_i(x \\odot m_i),\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    x_{i}^{t,k} = x_{i}^{t,k-1} - \\eta_{\\ell} \\nabla f_i\\big(x_i^{t,k-1} \\odot m_i^t, \\xi_i^{t,k-1}\\big) \\; \\text{and} \\; x_g^{t} = x_g^{t-1} - \\eta_g (x_g^{t-1} - \\frac{\\sum_{i \\in [N]}x_{i}^{t} \\odot m_i^t}{\\sum_{i \\in [N]}m_i^t}).\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq: mask score learning}\n\n    \\min_{x, \\{s_i\\}_{i\\in[N]}} \\frac{1}{N} \\sum_{i=1}^N f_i(x \\odot \\mathcal{G}(s_i)).\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\mathcal{L}_s = f_i\\Big(x_i^{t,k} \\odot \\mathcal{G}\\left(s_i^{t,k}\\right), \\xi_i^{t,k-1}\\Big) - \\lambda \\|\\sigma\\left(s_{i}^{t,k}\\right) - \\theta_{g \\backslash \\{i\\}}^{t}\\|_2^2; \\qquad s_{i}^{t,k+1} \\gets s_{i}^{t,k} - \\eta_s \\nabla_{s} \\mathcal{L}_s, \n\n    \\label{eq: mask-objective-func}\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\|\\sigma\\left(s_{i}^{t,k}\\right) - \\theta_{g \\backslash \\{i\\}}^{t}\\|_2^2 \\leq h. \n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\mathcal{L}_x = f_i\\Big(x_i^{t,k} \\odot \\mathcal{G}\\left(s_i^{t,k}\\right), \\xi_i^{t,k-1}\\Big); \\qquad x_{i}^{t,k+1} \\gets x_{i}^{t,k} - \\eta_{\\ell} \\nabla_{x} \\mathcal{L}_x,\n\n\\end{equation}"}
{"table/equation": "\\begin{table}[t]\n\n    \\centering\n\n    \\caption{Ablation study of the parameters $\\lambda$ and $\\tau$ on the CIFAR-MNIST dataset with $N=2$ participants over three independent runs. The first column ($\\tau=0.0$) corresponds to the FedAvg algorithm. The last row $(-)$ presents results for the \\texttt{FedPeWS-Fixed} algorithm.} \n\n    \\resizebox{\\linewidth}{!}{ \n\n    \\begin{tabular}{r|cccccc}\n\n        \\toprule\n\n        & \\multicolumn{6}{c}{\\cellcolor{lightblue}\\textbf{Proportion of warmup rounds $\\tau = W/T$}} \\\\ \\cmidrule{2-7} \n\n        $\\mathbf{\\lambda} (\\downarrow) , \\tau (\\rightarrow)$ & $\\tau=0.0$ (FedAvg) & $\\tau=0.2$ & $\\tau=0.4$ & $\\tau=0.5$ & $\\tau=0.6$ & $\\tau=0.8$ \\\\ \\midrule \n\n        0.0 & \\multirow{12}{*}{$71.23 \\pm 0.71$} & $68.01 \\pm 0.88$ & $66.01\\pm0.48$ & $65.96\\pm1.03$ & $65.40\\pm1.95$ & $65.77\\pm0.41$ \\\\ \\cmidrule{3-7}\n\n        0.1 & & $68.77\\pm1.09$ & $70.39\\pm1.00$ & $70.61\\pm1.17$ & $69.48\\pm0.37$ & $70.36\\pm2.06$ \\\\\n\n        0.3 & & $70.86\\pm0.23$ & $73.00\\pm0.65$ & $73.91\\pm0.71$ & $73.26\\pm0.46$ & $73.02\\pm1.05$ \\\\\n\n        0.5 & & $71.43\\pm0.56$ & $74.17\\pm0.91$ & $73.84\\pm0.12$ & $73.66\\pm0.88$ & $75.05\\pm0.45$ \\\\\n\n        1.0 & & $72.26\\pm0.54$ & $74.46\\pm0.44$ & $74.54\\pm0.76$ & $74.91\\pm0.42$ & $73.81\\pm0.87$ \\\\\n\n        2.0 & & $72.61\\pm0.79$ & $73.68\\pm0.17$ & \\cellcolor{lightgreen} $\\mathbf{75.35\\pm0.50}$ & $74.76\\pm0.54$ & $74.46\\pm0.56$ \\\\\n\n        5.0 & & $72.60\\pm0.45$ & \\cellcolor{lightgreen} $\\mathbf{75.22\\pm0.33}$ & \\cellcolor{lightgreen} $\\mathbf{75.00\\pm0.74}$ & $75.01\\pm0.71$ & $73.96\\pm1.60$ \\\\\n\n        10.0 & & $72.29\\pm0.48$ & \\cellcolor{lightgreen} $\\mathbf{74.97\\pm0.65}$ & $74.31\\pm0.95$ & $74.03\\pm0.30$ & $71.91\\pm2.69$ \\\\\n\n        100.0 & & $71.64\\pm0.47$ & $72.92\\pm0.39$ & $73.96\\pm0.65$ & $73.13\\pm0.73$ & $72.43\\pm3.55$ \\\\\n\n        1000.0 & & $71.58\\pm0.53$ & $73.18\\pm0.73$ & $73.32\\pm1.70$ & $73.87\\pm1.16$ & $72.52\\pm1.92$ \\\\ \\cmidrule{3-7} \n\n        $-$ & & $72.72\\pm0.44$ & \\cellcolor{lightgreen} $\\mathbf{75.22\\pm0.19}$ & \\cellcolor{lightgreen} $\\mathbf{75.05\\pm0.42}$ & $72.51\\pm3.89$ & $73.77\\pm0.20$ \\\\ \n\n        \\bottomrule\n\n    \\end{tabular}\n\n    }\n\n    \\label{tab: cifar-mnist-ablation}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[H]\n\n    \\centering\n\n    \\caption{Comparison of approaches for handling data heterogeneity in federated learning.} \n\n    \\resizebox{0.8\\linewidth}{!}{ \n\n        \\begin{tabular}{lcccc}\n\n            \\toprule\n\n             & \\multirow{2}{*}{\\parbox{2.1cm}{\\centering \\textbf{Shared \\\\ Global Model}}} & \\multirow{2}{*}{\\parbox{2.5cm}{\\centering \\textbf{Level of Mask \\\\ Personalization}}} & \\multirow{2}{*}{\\parbox{2cm}{\\centering {\\textbf{Learnable \\\\ Mask}}}} & \\multirow{2}{*}{\\parbox{2cm}{\\centering {\\textbf{Learnable \\\\ Parameters}}}} \\\\ \n\n             & & & & \\\\ \\midrule \n\n             FedPM \\citep{isik2022sparse} & \\checkmark & parameter-level & \\checkmark & $\\bm{\\times}$ \\\\ \n\n             IST \\citep{yuan2022distributed} & \\checkmark & neuron-level & $\\bm{\\times}$ (random) & \\checkmark \\\\ \n\n             LTH \\citep{frankle2018lottery} & $\\bm{\\times}$ & parameter-level & $\\bm{\\times}$ (pruned) & \\checkmark \\\\ \n\n             FedWeIT \\citep{yoon2021fedweit} & $\\bm{\\times}$ & parameter-level & \\checkmark & \\checkmark \\\\ \n\n             FjORD \\citep{horvath2021fjord} & \\checkmark & parameter-level & $\\bm{\\times}$ (slimmed) & \\checkmark \\\\ \\midrule \n\n             \\rowcolor{lightgreen} \\texttt{FedPeWS} (ours) & \\checkmark & neuron-level & \\checkmark & \\checkmark \\\\ \\bottomrule \n\n        \\end{tabular}\n\n    }\n\n    \\label{tab: methods-comparison}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[b]\n\n    \\centering\n\n    \\caption{Ablation study of the parameters $\\lambda$ and $\\tau$ on the combination of \\{Path-OCT-Tissue\\}MNIST datasets with $N=3$ participants over three independent runs. The first column ($\\tau=0.0$) corresponds to the FedAvg algorithm. The last row $(-)$ presents results for the \\texttt{FedPeWS-Fixed} algorithm.} \n\n    \\resizebox{\\linewidth}{!}{ \n\n    \\begin{tabular}{r|cccccc}\n\n        \\toprule\n\n        & \\multicolumn{6}{c}{\\cellcolor{lightblue}\\textbf{Proportion of warmup rounds $\\tau = W/T$}} \\\\ \\cmidrule{2-7} \n\n        ${\\lambda} (\\downarrow) , \\tau (\\rightarrow)$ & $\\tau=0.0$ (FedAvg) & $\\tau=0.2$ & $\\tau=0.4$ & $\\tau=0.5$ & $\\tau=0.6$ & $\\tau=0.8$ \\\\ \\midrule \n\n        0.0 & \\multirow{12}{*}{$52.25\\pm0.57$} & $53.04\\pm2.08$ & $50.35\\pm0.61$ & $47.83\\pm1.04$ & $47.89\\pm1.17$ & $46.80\\pm1.76$ \\\\ \\cmidrule{3-7}\n\n        0.1 & & $52.52\\pm1.79$ & $51.72\\pm0.63$ & $49.83\\pm1.83$ & $48.83\\pm2.51$ & $47.50\\pm1.07$ \\\\\n\n        0.3 & & $52.74\\pm1.75$ & $54.36\\pm0.80$ & $51.07\\pm0.25$ & $50.94\\pm2.22$ & $50.62\\pm1.91$ \\\\\n\n        0.5 & & $54.30\\pm2.08$ & $54.42\\pm1.43$ & $52.02\\pm2.53$ & $51.43\\pm1.60$ & $48.97\\pm0.80$ \\\\\n\n        1.0 & & \\cellcolor{lightgreen} $\\mathbf{54.89\\pm0.72}$ & $53.59\\pm1.40$ & $51.49\\pm1.19$ & $50.29\\pm2.70$ & $52.21\\pm1.02$ \\\\\n\n        2.0 & & \\cellcolor{lightgreen} $\\mathbf{54.75\\pm1.12}$ & \\cellcolor{lightgreen} $\\mathbf{54.45\\pm1.78}$ & $53.02\\pm0.45$ & $52.26\\pm1.81$ & $52.49\\pm0.49$ \\\\ \n\n        5.0 & & \\cellcolor{lightgreen} $\\mathbf{54.91\\pm0.91}$ & \\cellcolor{lightgreen} $\\mathbf{54.99\\pm0.90}$ & $52.41\\pm0.45$ & $52.95\\pm1.49$ & $52.53\\pm1.35$ \\\\ \n\n        10.0 & & \\cellcolor{lightgreen} $\\mathbf{55.12\\pm1.16}$ & \\cellcolor{lightgreen} $\\mathbf{55.03\\pm1.39}$ & $52.85\\pm0.95$ & $50.82\\pm3.03$ & $52.27\\pm1.11$ \\\\ \n\n        100.0 & & $54.22\\pm1.74$ & $52.31\\pm2.55$ & $52.10\\pm0.87$ & $49.52\\pm4.34$ & $51.46\\pm2.92$ \\\\ \n\n        1000.0 & & $53.45\\pm1.65$ & $53.82\\pm2.16$ & $51.16\\pm1.92$ & $51.30\\pm2.02$ & $51.19\\pm1.05$ \\\\ \\cmidrule{3-7} \n\n        $-$ & & $53.69\\pm0.77$ & $51.78\\pm0.44$ & $50.24\\pm1.88$ & $51.12\\pm0.72$ & $49.87\\pm1.23$ \\\\ \n\n        \\bottomrule\n\n    \\end{tabular}\n\n    }\n\n    \\label{tab: pot-ablation}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[t]\n\n    \\centering\n\n    \\caption{The required number of collaboration rounds to reach target accuracy $\\upsilon$ $\\%$ and the final accuracy after $T$ rounds. The results are averaged over 3 seeds. \n\n    {\\color{red} $\\times$} indicates that the algorithm cannot reach target accuracy $\\upsilon$ within $T$ rounds and {\\color{red} NA} means that it reaches $\\upsilon$ only in one random seed. } \n\n    \\resizebox{\\linewidth}{!}{ \n\n        \\begin{tabular}{cr|ccccc} \n\n        \\toprule\n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Dataset $/$ Batch size}} & \\multicolumn{3}{c}{Synthetic-32K, $32$} & \\multicolumn{1}{c}{Synthetic-3.2K, $8$} \n\n        \\\\ \\midrule \n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Parameters} $\\{\\eta_g / \\lambda / \\tau\\}$} & $\\{1.0 / 5.0 / 0.125\\}$ & $\\{0.5/2.0/0.2\\}$ & $\\{0.25/1.0/0.1875\\}$ & \\multicolumn{1}{c}{$\\{0.1/2.0/0.1\\}$} \\\\ \\midrule\n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Target accuracy $\\upsilon (\\%)$}} & $99$ & $90$ & $75$ & $99$ \n\n        \\\\ \\midrule \n\n        \\multirow{2}{*}{\\parbox{3.5cm}{\\centering \\textbf{No. of rounds to \\\\ reach target accuracy}}} & FedAvg & $148\\pm3.79$ & \\color{red} \\cellcolor{lightred} $199\\pm \\text{NA}$ & \\color{red} \\cellcolor{lightred} $\\times$ & \\color{red} \\cellcolor{lightred} $371\\pm\\text{NA}$ \n\n        \\\\ \\cmidrule{2-6}\n\n         & \\texttt{FedPeWS} & \\cellcolor{lightgreen} $\\mathbf{115\\pm7.21}$ & \\cellcolor{lightgreen} $\\mathbf{182\\pm6.81}$ & \\cellcolor{lightgreen} $\\mathbf{286\\pm7.93}$ & \\cellcolor{lightgreen} $\\mathbf{301\\pm10.59}$ \n\n        \\\\ \\midrule \n\n        \\multirow{2}{*}{\\parbox{3.75cm}{\\centering \\textbf{Final accuracy after \\\\ $T$ collaboration rounds}}} & FedAvg & \\multicolumn{1}{c}{$99.94\\pm0.05$} & $91.40\\pm7.25$ & $67.64\\pm0.90$ & $97.33\\pm3.89$ \n\n        \\\\ \\cmidrule{2-6}\n\n        & \\texttt{FedPeWS} & \\multicolumn{1}{c}{\\cellcolor{lightgreen} $\\mathbf{99.96\\pm0.01}$} & \\cellcolor{lightgreen} $\\mathbf{99.49\\pm0.60}$ & \\cellcolor{lightgreen} $\\mathbf{83.50\\pm3.52}$ & \\cellcolor{lightgreen} $\\mathbf{99.66\\pm0.19}$ \n\n        \\\\ \\bottomrule \n\n        \\end{tabular}\n\n    }\n\n    \\label{table: convergence-table}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[t]\n\n    \\centering\n\n    \\caption{The required number of collaboration rounds to reach target accuracy $\\upsilon$ $\\%$ using FedProx algorithm and the final accuracy after $T$ rounds. The results are averaged over 3 seeds. \n\n    {\\color{red} $\\times$} indicates that the algorithm cannot reach target accuracy $\\upsilon$ within $T$ rounds.} \n\n    \\resizebox{\\linewidth}{!}{ \n\n        \\begin{tabular}{cr|ccccc} \n\n        \\toprule\n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Dataset $/$ Batch size}} & \\multicolumn{3}{c}{Synthetic-32K, $32$} & \\multicolumn{1}{c}{Synthetic-3.2K, $8$} \n\n        \\\\ \\midrule \n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Parameters} $\\{\\eta_g / \\lambda / \\tau\\}$} & $\\{1.0 / 0.1 / 0.125\\}$ & $\\{0.5/0.1/0.2\\}$ & $\\{0.25/1.0/0.1875\\}$ & \\multicolumn{1}{c}{$\\{0.1/1.0/0.1\\}$} \\\\ \\midrule\n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Target accuracy $\\upsilon (\\%)$}} & $99$ & $90$ & $75$ & $99$ \n\n        \\\\ \\midrule \n\n        \\multirow{2}{*}{\\parbox{3.5cm}{\\centering \\textbf{No. of rounds to \\\\ reach target accuracy}}} & FedProx & $138\\pm13.22$ & \\cellcolor{lightred} \\color{red} $\\times$ & \\cellcolor{lightred} \\color{red} $\\times$ & $362\\pm20.00$ \n\n        \\\\ \\cmidrule{2-6} \n\n         & \\texttt{FedPeWS} & \\cellcolor{lightgreen} $\\mathbf{115\\pm5.29}$ & \\cellcolor{lightgreen} $\\mathbf{211\\pm16.52}$ & \\cellcolor{lightgreen} $\\mathbf{314\\pm27.83}$ & \\cellcolor{lightgreen} $\\mathbf{344\\pm27.30}$ \n\n        \\\\ \\midrule \n\n        \\multirow{2}{*}{\\parbox{3.75cm}{\\centering \\textbf{Final accuracy after \\\\ $T$ collaboration rounds}}} & FedProx & \\multicolumn{1}{c}{$99.95\\pm0.02$} & $82.43\\pm7.98$ & $69.26\\pm6.03$ &\\cellcolor{lightgreen}  $\\mathbf{99.92\\pm0.06}$ \n\n        \\\\ \\cmidrule{2-6}\n\n        & \\texttt{FedPeWS} & \\multicolumn{1}{c}{\\cellcolor{lightgreen} $\\mathbf{99.96\\pm0.01}$} & \\cellcolor{lightgreen} $\\mathbf{98.40\\pm1.84}$ & \\cellcolor{lightgreen} $\\mathbf{90.40\\pm3.91}$ & \\cellcolor{lightgreen} $\\mathbf{99.92\\pm0.07}$ \n\n        \\\\ \\bottomrule \n\n        \\end{tabular}\n\n    }\n\n    \\label{table: convergence-table-fedprox}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[th]\n\n    \\centering\n\n    \\caption{Architecture for synthetic dataset models used in the experiments.}\n\n    \\resizebox{0.75\\linewidth}{!}{ \n\n        \\begin{tabular}{lllllll} \n\n            \\toprule \n\n            \\cellcolor{lightblue}\\textbf{Layer} & Input & FC1 & FC2 & FC3 & FC4 & FC5 \\\\ \\midrule \n\n            \\cellcolor{lightblue}\\textbf{Dimensions} & [5] & [5, 32] & [32, 64] & [64, 128] & [128, 32] & [32, 4] \\\\ \\bottomrule\n\n        \\end{tabular}\n\n    }\n\n    \\label{tab: synthetic-arch}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[H]\n\n    \\centering\n\n    \\caption{Architecture for CIFAR-MNIST dataset models. Every convolutional layer is followed by a max pooling layer with kernel size 2 and stride 2. }\n\n    \\resizebox{\\linewidth}{!}{ \n\n        \\begin{tabular}{lllllll}\n\n            \\toprule\n\n             \\cellcolor{lightblue}\\textbf{Layer} & Input & Conv1 & Conv2 & Conv3 & Flatten & FC \\\\ \\midrule \n\n             \\cellcolor{lightblue}\\textbf{Dimensions} & [3, 32, 32] & [3, 32, 3, 3] & [32, 64, 3, 3] & [64, 128, 3, 3] & [2048] & [2048, 20] \\\\ \\bottomrule \n\n        \\end{tabular}\n\n    }\n\n    \\label{tab: conv-arch}\n\n\\end{table}"}
{"table/equation": "\\begin{equation}\n\n    \\begin{split}\n\n        x^{\\star} \\leftarrow  \\argmin_{x \\in \\mathbb{R}^d} \\Bigg[ f(x) \\coloneqq \\frac{1}{N} \\sum_{i=1}^N f_i(x) \\Bigg], \n\n        \\label{eq: optimization-problem}\n\n    \\end{split}\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    x_{i}^{t,k} = x_{i}^{t,k-1} - \\eta_{\\ell} \\nabla f_i\\big(x_i^{t,k-1}, \\xi_i^{t,k-1}\\big), \\quad x_{i}^{t} = x_{i}^{t,K}, \\text{ and  } x_g^{t} = x_g^{t-1} - \\eta_g (x_g^{t-1} - \\frac{1}{N} \\sum_{i=1}^N x_{i}^{t}) , \n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq: mask learning}\n\n    \\min_{x, \\{m_i\\}_{i\\in[N]}} \\frac{1}{N} \\sum_{i=1}^N f_i(x \\odot m_i),\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    x_{i}^{t,k} = x_{i}^{t,k-1} - \\eta_{\\ell} \\nabla f_i\\big(x_i^{t,k-1} \\odot m_i^t, \\xi_i^{t,k-1}\\big) \\; \\text{and} \\; x_g^{t} = x_g^{t-1} - \\eta_g (x_g^{t-1} - \\frac{\\sum_{i \\in [N]}x_{i}^{t} \\odot m_i^t}{\\sum_{i \\in [N]}m_i^t}).\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq: mask score learning}\n\n    \\min_{x, \\{s_i\\}_{i\\in[N]}} \\frac{1}{N} \\sum_{i=1}^N f_i(x \\odot \\mathcal{G}(s_i)).\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\mathcal{L}_s = f_i\\Big(x_i^{t,k} \\odot \\mathcal{G}\\left(s_i^{t,k}\\right), \\xi_i^{t,k-1}\\Big) - \\lambda \\|\\sigma\\left(s_{i}^{t,k}\\right) - \\theta_{g \\backslash \\{i\\}}^{t}\\|_2^2; \\qquad s_{i}^{t,k+1} \\gets s_{i}^{t,k} - \\eta_s \\nabla_{s} \\mathcal{L}_s, \n\n    \\label{eq: mask-objective-func}\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\|\\sigma\\left(s_{i}^{t,k}\\right) - \\theta_{g \\backslash \\{i\\}}^{t}\\|_2^2 \\leq h. \n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\mathcal{L}_x = f_i\\Big(x_i^{t,k} \\odot \\mathcal{G}\\left(s_i^{t,k}\\right), \\xi_i^{t,k-1}\\Big); \\qquad x_{i}^{t,k+1} \\gets x_{i}^{t,k} - \\eta_{\\ell} \\nabla_{x} \\mathcal{L}_x,\n\n\\end{equation}"}
{"table/equation": "\\begin{table}[t]\n\n    \\centering\n\n    \\caption{Ablation study of the parameters $\\lambda$ and $\\tau$ on the CIFAR-MNIST dataset with $N=2$ participants over three independent runs. The first column ($\\tau=0.0$) corresponds to the FedAvg algorithm. The last row $(-)$ presents results for the \\texttt{FedPeWS-Fixed} algorithm.} \n\n    \\resizebox{\\linewidth}{!}{ \n\n    \\begin{tabular}{r|cccccc}\n\n        \\toprule\n\n        & \\multicolumn{6}{c}{\\cellcolor{lightblue}\\textbf{Proportion of warmup rounds $\\tau = W/T$}} \\\\ \\cmidrule{2-7} \n\n        $\\mathbf{\\lambda} (\\downarrow) , \\tau (\\rightarrow)$ & $\\tau=0.0$ (FedAvg) & $\\tau=0.2$ & $\\tau=0.4$ & $\\tau=0.5$ & $\\tau=0.6$ & $\\tau=0.8$ \\\\ \\midrule \n\n        0.0 & \\multirow{12}{*}{$71.23 \\pm 0.71$} & $68.01 \\pm 0.88$ & $66.01\\pm0.48$ & $65.96\\pm1.03$ & $65.40\\pm1.95$ & $65.77\\pm0.41$ \\\\ \\cmidrule{3-7}\n\n        0.1 & & $68.77\\pm1.09$ & $70.39\\pm1.00$ & $70.61\\pm1.17$ & $69.48\\pm0.37$ & $70.36\\pm2.06$ \\\\\n\n        0.3 & & $70.86\\pm0.23$ & $73.00\\pm0.65$ & $73.91\\pm0.71$ & $73.26\\pm0.46$ & $73.02\\pm1.05$ \\\\\n\n        0.5 & & $71.43\\pm0.56$ & $74.17\\pm0.91$ & $73.84\\pm0.12$ & $73.66\\pm0.88$ & $75.05\\pm0.45$ \\\\\n\n        1.0 & & $72.26\\pm0.54$ & $74.46\\pm0.44$ & $74.54\\pm0.76$ & $74.91\\pm0.42$ & $73.81\\pm0.87$ \\\\\n\n        2.0 & & $72.61\\pm0.79$ & $73.68\\pm0.17$ & \\cellcolor{lightgreen} $\\mathbf{75.35\\pm0.50}$ & $74.76\\pm0.54$ & $74.46\\pm0.56$ \\\\\n\n        5.0 & & $72.60\\pm0.45$ & \\cellcolor{lightgreen} $\\mathbf{75.22\\pm0.33}$ & \\cellcolor{lightgreen} $\\mathbf{75.00\\pm0.74}$ & $75.01\\pm0.71$ & $73.96\\pm1.60$ \\\\\n\n        10.0 & & $72.29\\pm0.48$ & \\cellcolor{lightgreen} $\\mathbf{74.97\\pm0.65}$ & $74.31\\pm0.95$ & $74.03\\pm0.30$ & $71.91\\pm2.69$ \\\\\n\n        100.0 & & $71.64\\pm0.47$ & $72.92\\pm0.39$ & $73.96\\pm0.65$ & $73.13\\pm0.73$ & $72.43\\pm3.55$ \\\\\n\n        1000.0 & & $71.58\\pm0.53$ & $73.18\\pm0.73$ & $73.32\\pm1.70$ & $73.87\\pm1.16$ & $72.52\\pm1.92$ \\\\ \\cmidrule{3-7} \n\n        $-$ & & $72.72\\pm0.44$ & \\cellcolor{lightgreen} $\\mathbf{75.22\\pm0.19}$ & \\cellcolor{lightgreen} $\\mathbf{75.05\\pm0.42}$ & $72.51\\pm3.89$ & $73.77\\pm0.20$ \\\\ \n\n        \\bottomrule\n\n    \\end{tabular}\n\n    }\n\n    \\label{tab: cifar-mnist-ablation}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[H]\n\n    \\centering\n\n    \\caption{Comparison of approaches for handling data heterogeneity in federated learning.} \n\n    \\resizebox{0.8\\linewidth}{!}{ \n\n        \\begin{tabular}{lcccc}\n\n            \\toprule\n\n             & \\multirow{2}{*}{\\parbox{2.1cm}{\\centering \\textbf{Shared \\\\ Global Model}}} & \\multirow{2}{*}{\\parbox{2.5cm}{\\centering \\textbf{Level of Mask \\\\ Personalization}}} & \\multirow{2}{*}{\\parbox{2cm}{\\centering {\\textbf{Learnable \\\\ Mask}}}} & \\multirow{2}{*}{\\parbox{2cm}{\\centering {\\textbf{Learnable \\\\ Parameters}}}} \\\\ \n\n             & & & & \\\\ \\midrule \n\n             FedPM \\citep{isik2022sparse} & \\checkmark & parameter-level & \\checkmark & $\\bm{\\times}$ \\\\ \n\n             IST \\citep{yuan2022distributed} & \\checkmark & neuron-level & $\\bm{\\times}$ (random) & \\checkmark \\\\ \n\n             LTH \\citep{frankle2018lottery} & $\\bm{\\times}$ & parameter-level & $\\bm{\\times}$ (pruned) & \\checkmark \\\\ \n\n             FedWeIT \\citep{yoon2021fedweit} & $\\bm{\\times}$ & parameter-level & \\checkmark & \\checkmark \\\\ \n\n             FjORD \\citep{horvath2021fjord} & \\checkmark & parameter-level & $\\bm{\\times}$ (slimmed) & \\checkmark \\\\ \\midrule \n\n             \\rowcolor{lightgreen} \\texttt{FedPeWS} (ours) & \\checkmark & neuron-level & \\checkmark & \\checkmark \\\\ \\bottomrule \n\n        \\end{tabular}\n\n    }\n\n    \\label{tab: methods-comparison}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[b]\n\n    \\centering\n\n    \\caption{Ablation study of the parameters $\\lambda$ and $\\tau$ on the combination of \\{Path-OCT-Tissue\\}MNIST datasets with $N=3$ participants over three independent runs. The first column ($\\tau=0.0$) corresponds to the FedAvg algorithm. The last row $(-)$ presents results for the \\texttt{FedPeWS-Fixed} algorithm.} \n\n    \\resizebox{\\linewidth}{!}{ \n\n    \\begin{tabular}{r|cccccc}\n\n        \\toprule\n\n        & \\multicolumn{6}{c}{\\cellcolor{lightblue}\\textbf{Proportion of warmup rounds $\\tau = W/T$}} \\\\ \\cmidrule{2-7} \n\n        ${\\lambda} (\\downarrow) , \\tau (\\rightarrow)$ & $\\tau=0.0$ (FedAvg) & $\\tau=0.2$ & $\\tau=0.4$ & $\\tau=0.5$ & $\\tau=0.6$ & $\\tau=0.8$ \\\\ \\midrule \n\n        0.0 & \\multirow{12}{*}{$52.25\\pm0.57$} & $53.04\\pm2.08$ & $50.35\\pm0.61$ & $47.83\\pm1.04$ & $47.89\\pm1.17$ & $46.80\\pm1.76$ \\\\ \\cmidrule{3-7}\n\n        0.1 & & $52.52\\pm1.79$ & $51.72\\pm0.63$ & $49.83\\pm1.83$ & $48.83\\pm2.51$ & $47.50\\pm1.07$ \\\\\n\n        0.3 & & $52.74\\pm1.75$ & $54.36\\pm0.80$ & $51.07\\pm0.25$ & $50.94\\pm2.22$ & $50.62\\pm1.91$ \\\\\n\n        0.5 & & $54.30\\pm2.08$ & $54.42\\pm1.43$ & $52.02\\pm2.53$ & $51.43\\pm1.60$ & $48.97\\pm0.80$ \\\\\n\n        1.0 & & \\cellcolor{lightgreen} $\\mathbf{54.89\\pm0.72}$ & $53.59\\pm1.40$ & $51.49\\pm1.19$ & $50.29\\pm2.70$ & $52.21\\pm1.02$ \\\\\n\n        2.0 & & \\cellcolor{lightgreen} $\\mathbf{54.75\\pm1.12}$ & \\cellcolor{lightgreen} $\\mathbf{54.45\\pm1.78}$ & $53.02\\pm0.45$ & $52.26\\pm1.81$ & $52.49\\pm0.49$ \\\\ \n\n        5.0 & & \\cellcolor{lightgreen} $\\mathbf{54.91\\pm0.91}$ & \\cellcolor{lightgreen} $\\mathbf{54.99\\pm0.90}$ & $52.41\\pm0.45$ & $52.95\\pm1.49$ & $52.53\\pm1.35$ \\\\ \n\n        10.0 & & \\cellcolor{lightgreen} $\\mathbf{55.12\\pm1.16}$ & \\cellcolor{lightgreen} $\\mathbf{55.03\\pm1.39}$ & $52.85\\pm0.95$ & $50.82\\pm3.03$ & $52.27\\pm1.11$ \\\\ \n\n        100.0 & & $54.22\\pm1.74$ & $52.31\\pm2.55$ & $52.10\\pm0.87$ & $49.52\\pm4.34$ & $51.46\\pm2.92$ \\\\ \n\n        1000.0 & & $53.45\\pm1.65$ & $53.82\\pm2.16$ & $51.16\\pm1.92$ & $51.30\\pm2.02$ & $51.19\\pm1.05$ \\\\ \\cmidrule{3-7} \n\n        $-$ & & $53.69\\pm0.77$ & $51.78\\pm0.44$ & $50.24\\pm1.88$ & $51.12\\pm0.72$ & $49.87\\pm1.23$ \\\\ \n\n        \\bottomrule\n\n    \\end{tabular}\n\n    }\n\n    \\label{tab: pot-ablation}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[t]\n\n    \\centering\n\n    \\caption{The required number of collaboration rounds to reach target accuracy $\\upsilon$ $\\%$ and the final accuracy after $T$ rounds. The results are averaged over 3 seeds. \n\n    {\\color{red} $\\times$} indicates that the algorithm cannot reach target accuracy $\\upsilon$ within $T$ rounds and {\\color{red} NA} means that it reaches $\\upsilon$ only in one random seed. } \n\n    \\resizebox{\\linewidth}{!}{ \n\n        \\begin{tabular}{cr|ccccc} \n\n        \\toprule\n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Dataset $/$ Batch size}} & \\multicolumn{3}{c}{Synthetic-32K, $32$} & \\multicolumn{1}{c}{Synthetic-3.2K, $8$} \n\n        \\\\ \\midrule \n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Parameters} $\\{\\eta_g / \\lambda / \\tau\\}$} & $\\{1.0 / 5.0 / 0.125\\}$ & $\\{0.5/2.0/0.2\\}$ & $\\{0.25/1.0/0.1875\\}$ & \\multicolumn{1}{c}{$\\{0.1/2.0/0.1\\}$} \\\\ \\midrule\n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Target accuracy $\\upsilon (\\%)$}} & $99$ & $90$ & $75$ & $99$ \n\n        \\\\ \\midrule \n\n        \\multirow{2}{*}{\\parbox{3.5cm}{\\centering \\textbf{No. of rounds to \\\\ reach target accuracy}}} & FedAvg & $148\\pm3.79$ & \\color{red} \\cellcolor{lightred} $199\\pm \\text{NA}$ & \\color{red} \\cellcolor{lightred} $\\times$ & \\color{red} \\cellcolor{lightred} $371\\pm\\text{NA}$ \n\n        \\\\ \\cmidrule{2-6}\n\n         & \\texttt{FedPeWS} & \\cellcolor{lightgreen} $\\mathbf{115\\pm7.21}$ & \\cellcolor{lightgreen} $\\mathbf{182\\pm6.81}$ & \\cellcolor{lightgreen} $\\mathbf{286\\pm7.93}$ & \\cellcolor{lightgreen} $\\mathbf{301\\pm10.59}$ \n\n        \\\\ \\midrule \n\n        \\multirow{2}{*}{\\parbox{3.75cm}{\\centering \\textbf{Final accuracy after \\\\ $T$ collaboration rounds}}} & FedAvg & \\multicolumn{1}{c}{$99.94\\pm0.05$} & $91.40\\pm7.25$ & $67.64\\pm0.90$ & $97.33\\pm3.89$ \n\n        \\\\ \\cmidrule{2-6}\n\n        & \\texttt{FedPeWS} & \\multicolumn{1}{c}{\\cellcolor{lightgreen} $\\mathbf{99.96\\pm0.01}$} & \\cellcolor{lightgreen} $\\mathbf{99.49\\pm0.60}$ & \\cellcolor{lightgreen} $\\mathbf{83.50\\pm3.52}$ & \\cellcolor{lightgreen} $\\mathbf{99.66\\pm0.19}$ \n\n        \\\\ \\bottomrule \n\n        \\end{tabular}\n\n    }\n\n    \\label{table: convergence-table}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[t]\n\n    \\centering\n\n    \\caption{The required number of collaboration rounds to reach target accuracy $\\upsilon$ $\\%$ using FedProx algorithm and the final accuracy after $T$ rounds. The results are averaged over 3 seeds. \n\n    {\\color{red} $\\times$} indicates that the algorithm cannot reach target accuracy $\\upsilon$ within $T$ rounds.} \n\n    \\resizebox{\\linewidth}{!}{ \n\n        \\begin{tabular}{cr|ccccc} \n\n        \\toprule\n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Dataset $/$ Batch size}} & \\multicolumn{3}{c}{Synthetic-32K, $32$} & \\multicolumn{1}{c}{Synthetic-3.2K, $8$} \n\n        \\\\ \\midrule \n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Parameters} $\\{\\eta_g / \\lambda / \\tau\\}$} & $\\{1.0 / 0.1 / 0.125\\}$ & $\\{0.5/0.1/0.2\\}$ & $\\{0.25/1.0/0.1875\\}$ & \\multicolumn{1}{c}{$\\{0.1/1.0/0.1\\}$} \\\\ \\midrule\n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Target accuracy $\\upsilon (\\%)$}} & $99$ & $90$ & $75$ & $99$ \n\n        \\\\ \\midrule \n\n        \\multirow{2}{*}{\\parbox{3.5cm}{\\centering \\textbf{No. of rounds to \\\\ reach target accuracy}}} & FedProx & $138\\pm13.22$ & \\cellcolor{lightred} \\color{red} $\\times$ & \\cellcolor{lightred} \\color{red} $\\times$ & $362\\pm20.00$ \n\n        \\\\ \\cmidrule{2-6} \n\n         & \\texttt{FedPeWS} & \\cellcolor{lightgreen} $\\mathbf{115\\pm5.29}$ & \\cellcolor{lightgreen} $\\mathbf{211\\pm16.52}$ & \\cellcolor{lightgreen} $\\mathbf{314\\pm27.83}$ & \\cellcolor{lightgreen} $\\mathbf{344\\pm27.30}$ \n\n        \\\\ \\midrule \n\n        \\multirow{2}{*}{\\parbox{3.75cm}{\\centering \\textbf{Final accuracy after \\\\ $T$ collaboration rounds}}} & FedProx & \\multicolumn{1}{c}{$99.95\\pm0.02$} & $82.43\\pm7.98$ & $69.26\\pm6.03$ &\\cellcolor{lightgreen}  $\\mathbf{99.92\\pm0.06}$ \n\n        \\\\ \\cmidrule{2-6}\n\n        & \\texttt{FedPeWS} & \\multicolumn{1}{c}{\\cellcolor{lightgreen} $\\mathbf{99.96\\pm0.01}$} & \\cellcolor{lightgreen} $\\mathbf{98.40\\pm1.84}$ & \\cellcolor{lightgreen} $\\mathbf{90.40\\pm3.91}$ & \\cellcolor{lightgreen} $\\mathbf{99.92\\pm0.07}$ \n\n        \\\\ \\bottomrule \n\n        \\end{tabular}\n\n    }\n\n    \\label{table: convergence-table-fedprox}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[h!]\n\n\\centering\n\n\\caption{\\label{baro-table-1}\n\nComparison of Young's moduli calculated with three distinct methods\n\nfor single-layer graphene (SLG) as well as carbon nanomembranes 1 \\& 2, \n\ncompare Figs.~\\xref{graphene} and \\xref{CNM}, for temperatures in the range of 50~K to 350~K.\n\nAll values are in GPa.} \n\n\\begin{tabular}{|l|c|c|c|} %  \n\n\\hline\n\n\\textbf{Method} & \\textbf{SLG} & \\textbf{CNM1} & \\textbf{CNM2} \\\\\n\n\\hline\n\nScaling           &     $1088\\pm11$        &    $240\\pm5$         &     $253\\pm3$  \\\\\n\n\\hline\n\nStress-Strain     &   $854\\pm5.2$          &     $35.835\\pm0.835$        & $35.71\\pm5.13$     \\\\\n\n\\hline\n\nBarostated dyn., PC in z &     $854\\pm23$        &     $33\\pm5$        &  $31\\pm3$   \\\\\n\n\\hline\n\nBarostated dyn., without PC in z&   $854\\pm19$          &  $32\\pm7$ & $28\\pm3$            \\\\\n\n\\hline\n\n\\end{tabular}\n\n\\label{table}\n\n\\end{table}"}
{"table/equation": "\\begin{equation}\n\n\\label{E-2-2}\n\nE_\\alpha = \\frac{1}{V_0} \\left(\\frac{\\partial^2 U}{\\partial \\alpha^2}\\right)_{\\alpha=1}\n\n\\ ,\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n\\label{E-2-2b}\n\n\\vec{x}_i \\rightarrow \\vec{x}_i + (\\alpha - 1) \\\n\n\\vec{e}_{\\alpha}\\cdot\\vec{x}_i \\ \\vec{e}_{\\alpha}\n\n\\ .\n\n\\end{equation}"}
{"table/equation": "\\begin{table}[h!]\n\n\\centering\n\n\\caption{\\label{baro-table-1}\n\nComparison of Young's moduli calculated with three distinct methods\n\nfor single-layer graphene (SLG) as well as carbon nanomembranes 1 \\& 2, \n\ncompare Figs.~\\xref{graphene} and \\xref{CNM}, for temperatures in the range of 50~K to 350~K.\n\nAll values are in GPa.} \n\n\\begin{tabular}{|l|c|c|c|} %  \n\n\\hline\n\n\\textbf{Method} & \\textbf{SLG} & \\textbf{CNM1} & \\textbf{CNM2} \\\\\n\n\\hline\n\nScaling           &     $1088\\pm11$        &    $240\\pm5$         &     $253\\pm3$  \\\\\n\n\\hline\n\nStress-Strain     &   $854\\pm5.2$          &     $35.835\\pm0.835$        & $35.71\\pm5.13$     \\\\\n\n\\hline\n\nBarostated dyn., PC in z &     $854\\pm23$        &     $33\\pm5$        &  $31\\pm3$   \\\\\n\n\\hline\n\nBarostated dyn., without PC in z&   $854\\pm19$          &  $32\\pm7$ & $28\\pm3$            \\\\\n\n\\hline\n\n\\end{tabular}\n\n\\label{table}\n\n\\end{table}"}
{"table/equation": "\\begin{equation}\n\n\\label{E-2-2}\n\nE_\\alpha = \\frac{1}{V_0} \\left(\\frac{\\partial^2 U}{\\partial \\alpha^2}\\right)_{\\alpha=1}\n\n\\ ,\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n\\label{E-2-2b}\n\n\\vec{x}_i \\rightarrow \\vec{x}_i + (\\alpha - 1) \\\n\n\\vec{e}_{\\alpha}\\cdot\\vec{x}_i \\ \\vec{e}_{\\alpha}\n\n\\ .\n\n\\end{equation}"}
{"table/equation": "\\begin{table}[h!]\n\n\\centering\n\n\\caption{\\label{baro-table-1}\n\nComparison of Young's moduli calculated with three distinct methods\n\nfor single-layer graphene (SLG) as well as carbon nanomembranes 1 \\& 2, \n\ncompare Figs.~\\xref{graphene} and \\xref{CNM}, for temperatures in the range of 50~K to 350~K.\n\nAll values are in GPa.} \n\n\\begin{tabular}{|l|c|c|c|} %  \n\n\\hline\n\n\\textbf{Method} & \\textbf{SLG} & \\textbf{CNM1} & \\textbf{CNM2} \\\\\n\n\\hline\n\nScaling           &     $1088\\pm11$        &    $240\\pm5$         &     $253\\pm3$  \\\\\n\n\\hline\n\nStress-Strain     &   $854\\pm5.2$          &     $35.835\\pm0.835$        & $35.71\\pm5.13$     \\\\\n\n\\hline\n\nBarostated dyn., PC in z &     $854\\pm23$        &     $33\\pm5$        &  $31\\pm3$   \\\\\n\n\\hline\n\nBarostated dyn., without PC in z&   $854\\pm19$          &  $32\\pm7$ & $28\\pm3$            \\\\\n\n\\hline\n\n\\end{tabular}\n\n\\label{table}\n\n\\end{table}"}
{"table/equation": "\\begin{equation}\n\n\\label{E-2-2}\n\nE_\\alpha = \\frac{1}{V_0} \\left(\\frac{\\partial^2 U}{\\partial \\alpha^2}\\right)_{\\alpha=1}\n\n\\ ,\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n\\label{E-2-2b}\n\n\\vec{x}_i \\rightarrow \\vec{x}_i + (\\alpha - 1) \\\n\n\\vec{e}_{\\alpha}\\cdot\\vec{x}_i \\ \\vec{e}_{\\alpha}\n\n\\ .\n\n\\end{equation}"}
{"table/equation": "\\begin{table}[h!]\n\n\\centering\n\n\\caption{\\label{baro-table-1}\n\nComparison of Young's moduli calculated with three distinct methods\n\nfor single-layer graphene (SLG) as well as carbon nanomembranes 1 \\& 2, \n\ncompare Figs.~\\xref{graphene} and \\xref{CNM}, for temperatures in the range of 50~K to 350~K.\n\nAll values are in GPa.} \n\n\\begin{tabular}{|l|c|c|c|} %  \n\n\\hline\n\n\\textbf{Method} & \\textbf{SLG} & \\textbf{CNM1} & \\textbf{CNM2} \\\\\n\n\\hline\n\nScaling           &     $1088\\pm11$        &    $240\\pm5$         &     $253\\pm3$  \\\\\n\n\\hline\n\nStress-Strain     &   $854\\pm5.2$          &     $35.835\\pm0.835$        & $35.71\\pm5.13$     \\\\\n\n\\hline\n\nBarostated dyn., PC in z &     $854\\pm23$        &     $33\\pm5$        &  $31\\pm3$   \\\\\n\n\\hline\n\nBarostated dyn., without PC in z&   $854\\pm19$          &  $32\\pm7$ & $28\\pm3$            \\\\\n\n\\hline\n\n\\end{tabular}\n\n\\label{table}\n\n\\end{table}"}
{"table/equation": "\\begin{equation}\n\n\\label{E-2-2}\n\nE_\\alpha = \\frac{1}{V_0} \\left(\\frac{\\partial^2 U}{\\partial \\alpha^2}\\right)_{\\alpha=1}\n\n\\ ,\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n\\label{E-2-2b}\n\n\\vec{x}_i \\rightarrow \\vec{x}_i + (\\alpha - 1) \\\n\n\\vec{e}_{\\alpha}\\cdot\\vec{x}_i \\ \\vec{e}_{\\alpha}\n\n\\ .\n\n\\end{equation}"}

{"table/equation": "\\begin{table}[t] \\centering\n\n    \\begin{tabular}{|c|c|c|c|}\n\n\t\\hline\n\n\t\\hline\n\n\t$\\nu_{\\mu}$ &  $\\bar \\nu_{\\mu}$ & $\\nu_e$ &  $\\bar \\nu_e$\\\\\n\n\t\\hline\n\n\t669.0 &13.7 & 5.9& 0.3\\\\\n\n\t\\hline\\hline\n\n    \\end{tabular}\n\n    \\caption{\\label{tab:rates}\\sl%\n\n      Nominal performance of the CNGS reference beam~\\cite{fluxes}.\n\n      The total non-oscillated CC event rates are calculated assuming\n\n      $10^{19}$ pot and 1 Kton lead target mass.}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[t] \\centering\n\n    \\begin{tabular}{|c|c|c|c|c|c|}\n\n\t\\hline\\hline\n\n\t$(\\theta_{13}; \\theta_{14}; \\theta_{24}; \\theta_{34})$ & $N_\\tau$ & background & $(\\theta_{13}; \\theta_{14}; \\theta_{24}; \\theta_{34})$ & $N_\\tau$ & background \\\\\n\n\t\\hline\n\n\t$(5^\\circ;  5^\\circ;  5^\\circ; 20^\\circ)$    &     8.9    & 1.0 & $(10^\\circ; 5^\\circ;  5^\\circ; 20^\\circ)$    &     8.5    & 1.0 \\\\  \n\n\t$(5^\\circ;  5^\\circ;  5^\\circ; 30^\\circ)$    &     6.9    & 1.0 & $(10^\\circ; 5^\\circ;  5^\\circ; 30^\\circ)$    &     6.5    & 1.0 \\\\\n\n\t$(5^\\circ;  5^\\circ; 10^\\circ; 20^\\circ)$    &     8.3    & 1.0 & $(10^\\circ; 5^\\circ; 10^\\circ; 20^\\circ)$    &     7.9    & 1.0 \\\\\n\n\t$(5^\\circ;  5^\\circ; 10^\\circ; 30^\\circ)$    &    10.5    & 1.0 & $(10^\\circ; 5^\\circ; 10^\\circ; 30^\\circ)$    &    10.3    & 1.0 \\\\\n\n\t3 families                          &    15.1    & 1.0 &          3 families                          &    14.4    & 1.0 \\\\\n\n\t\\hline \\hline\n\n    \\end{tabular}\n\n    \\caption{\\label{tab:rates3}\\sl%\n\n      Event rates and expected background for the $\\nu_\\mu \\to\n\n      \\nu_\\tau$ channel in the OPERA detector, for different values of\n\n      $\\theta_{14}, \\theta_{24}$ and $\\theta_{34}$ in the (3+1)\n\n      scheme. The other unknown angle, $\\theta_{13}$ has been fixed\n\n      to: $\\theta_{13} = 5^\\circ, 10^\\circ$. The CP-violating phases\n\n      are: $\\delta_1 = \\delta_2 = 0$; $\\delta_3 = 90^\\circ$.  As a\n\n      reference, the expected value in the case of standard\n\n      three-family oscillation (\\textit{i.e.}, for $\\theta_{i4} = 0$) is shown\n\n      for maximal CP-violating phase $\\delta$. The rates are computed\n\n      according to Eq.~\\eqref{eq:signaltau}.}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[t] \\centering\n\n    \\begin{tabular}{|c|c|c|c|c|c|}\n\n\t\\hline\\hline\n\n\t$(\\theta_{13}; \\theta_{14}; \\theta_{24}; \\theta_{34})$ & $N_e$ & $\\nu_e^{CC}$ & $\\nu_\\mu^{NC}$ & $\\tau \\to e$ & $\\nu_\\mu^{CC}$  \\\\ \\hline\n\n\t$(5^\\circ; 5^\\circ; 5^\\circ; 20^\\circ)$   &  3.5 & 19.4 & 5.3 & 2.8 & 0.9 \\\\\n\n\t$(5^\\circ; 5^\\circ; 5^\\circ; 30^\\circ)$   &  3.5 & 19.4 & 5.3 & 2.1 & 0.9 \\\\\n\n\t$(5^\\circ; 5^\\circ; 10^\\circ; 20^\\circ)$  &  2.4 & 19.4 & 5.3 & 2.3 & 0.9 \\\\\n\n\t$(5^\\circ; 5^\\circ; 10^\\circ; 30^\\circ)$  &  2.4 & 19.4 & 5.3 & 2.4 & 0.9 \\\\\n\n\t3 families                        &  3.7 & 19.7 & 5.3 & 4.6 & 0.9 \\\\\n\n\t\\hline\n\n\t$(10^\\circ; 5^\\circ; 5^\\circ; 20^\\circ)$  & 10.6 & 19.4 & 5.3 & 2.7 & 0.9 \\\\\n\n\t$(10^\\circ; 5^\\circ; 5^\\circ; 30^\\circ)$  & 10.4 & 19.4 & 5.3 & 2.0 & 0.9 \\\\\n\n\t$(10^\\circ; 5^\\circ; 10^\\circ; 20^\\circ)$ &  8.8 & 19.4 & 5.3 & 2.2 & 0.9 \\\\\n\n\t$(10^\\circ; 5^\\circ; 10^\\circ; 30^\\circ)$ &  8.6 & 19.4 & 5.3 & 2.4 & 0.9 \\\\\n\n\t3 families                        & 15.1 & 19.7 & 5.3 & 4.8 & 0.9 \\\\\n\n\t\\hline \\hline\n\n    \\end{tabular}\n\n    \\caption{\\label{tab:rates2}\\sl%\n\n      Event rates and expected background for the $\\nu_\\mu \\to \\nu_e$\n\n      channel in the OPERA detector, for different values of \n\n      $\\theta_{14}, \\theta_{24}$ and $\\theta_{34}$ in the (3+1)\n\n      scheme. The other unknown angle, $\\theta_{13}$, has been fixed\n\n      to: $\\theta_{13} = 5^\\circ, 10^\\circ$. The CP-violating phases\n\n      are: $\\delta_1 = \\delta_2 = 0$; $\\delta_3 = 90^\\circ$.  As a\n\n      reference, the expected value in the case of standard\n\n      three-family oscillation(\\textit{i.e.}, for $\\theta_{i4} = 0$) is shown\n\n      for maximal CP-violating phase $\\delta$.  The rates are computed\n\n      according to Eq.~\\eqref{eq:signale}. Backgrounds have been\n\n      computed following Ref.~\\cite{Komatsu:2002sz}.}\n\n\\end{table}"}
{"table/equation": "\\begin{equation}\n\n    \\eta_s = \\sum_{i \\,\\in\\, \\text{sol}} |U_{s i}|^2\n\n    \\qquad\\text{and}\\qquad\n\n    c_s = \\sum_{j \\,\\in\\, \\text{atm}} |U_{s j}|^2\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:3+1param}\n\n    U_\\text{SBL} =\n\n    R_{14}(\\theta_{14}) \\; R_{24}(\\theta_{24}) \\; R_{34}(\\theta_{34}) \\; \n\n    R_{23}(\\theta_{23} ,\\, \\delta_3) \\; R_{13}(\\theta_{13} ,\\, \\delta_2) \\;\n\n    R_{12}(\\theta_{12} ,\\, \\delta_1) \\,.\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:3+1param2}\n\n    U_\\text{atm} =\n\n    R_{34}(\\theta_{34}) \\; R_{24}(\\theta_{24}) \\;\n\n    R_{23}(\\theta_{23} ,\\, \\delta_3) \\;\n\n    R_{14}(\\theta_{14}) \\; R_{13}(\\theta_{13} ,\\, \\delta_2) \\; \n\n    R_{12}(\\theta_{12} ,\\, \\delta_1) \\,.\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:pee}\n\n    P_{ee} = 1 - \\sin^2 2 \\theta_{14} \\sin^2 \\frac{\\Delta_\\Sbl L }{2} \n\n    - c^4_{14} \\sin^2 2 \\theta_{13} \\sin^2 \\frac{\\Delta_\\Atm L}{2} \\, ,\n\n\\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    P(\\nu_\\alpha \\to \\nu_\\beta) &= \\delta_{\\alpha \\beta}\n\n    - 4 \\Re \\left[ U_{\\alpha 3} U^\\star_{\\beta 3} \\left(\n\n    \\delta_{\\alpha\\beta} - U^\\star_{\\alpha 3} U_{\\beta 3}\n\n    - U^\\star_{\\alpha 4} U_{\\beta 4} \\right) \\right]\n\n    \\sin^2 \\frac{\\Delta_{23} L }{2}\n\n    \\\\\n\n    &- 2 \\Re \\left[ \\delta_{\\alpha\\beta} U_{\\alpha 4} U^\\star_{\\beta 4}\n\n    - |U_{\\alpha 4}|^2 |U_{\\beta 4}|^2 \\right]\n\n    \\pm 2 \\Im \\left[ U_{\\alpha 3} U^\\star_{\\beta 3}\n\n    U^\\star_{\\alpha 4} U_{\\beta 4} \\right] \\sin \\Delta_{23} L \\,,\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    P_{\\mu\\mu} &= 1 - 2 c^2_{14} s^2_{24} (1 - c^2_{14} s^2_{24})\n\n    - 4 \\big\\lbrace s^2_{23} c^2_{24} \\left[ c^2_{24} (c^2_{23}\n\n    - s^2_{13}) - s^2_{14} s^2_{24} \\right]\n\n    \\\\    \n\n    &- 2 c^3_{24} s_{23} (1 - 2 s^2_{23}) s_{13} s_{14} s_{24}\n\n    \\cos(\\delta_2 - \\delta_3) \\big\\rbrace\n\n    \\sin^2 \\frac{\\Delta_\\Atm L }{2} \\,.\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:pmumu}\n\n    P_{\\mu\\mu} = 1 - 2 s^2_{24}\n\n    - 4 s^2_{23} \\left[ c^2_{23} (1 - 2 s^2_{24}) - s^2_{13} \\right]\n\n    \\sin^2 \\frac{\\Delta_\\Atm L }{2} \\,.\n\n\\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    \\label{eq:pmus}\n\n    P_{\\mu s} &= 2 c^2_{34} s^2_{24} + \\big\\lbrace\n\n    \\sin^2 2 \\theta_{23} (c^4_{13} c^2_{24} s^2_{34}\n\n    - c^2_{34} s^2_{24})\n\n    \\\\\n\n    &+ 2  c_{34} \\sin 2 \\theta_{23} s_{34} \\left[\n\n    s_{24} (1 -2 s^2_{23}) \\cos \\delta_3\n\n    + 2 s_{23} s_{13} s_{14} \\cos \\delta_2 \\right] \\big\\rbrace\n\n    \\sin^2 \\frac{\\Delta_\\Atm L }{2}\n\n    \\\\\n\n    &\\pm c_{34} \\sin 2 \\theta_{23} s_{24} s_{34}\n\n    \\sin\\delta_3 \\sin\\Delta_\\Atm L \\,.\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    \\label{eq:pmue}\n\n    P_{\\mu e} &= 4 \\left\\lbrace s^2_{23} s^2_{13}\n\n    [1 - s^2_{13} - s^2_{14} - s^2_{24}] \n\n    + s_{23} s_{13} s_{14} s_{24} \\cos(\\delta_2 - \\delta_3) \\right\\rbrace\n\n    \\sin^2 \\frac{\\Delta_\\Atm L}{2}\n\n    \\\\\n\n    & \\pm 2 s_{23} s_{13} s_{14} s_{24} \\sin(\\delta_2 - \\delta_3)\n\n    \\sin\\Delta_\\Atm L + 2 s^2_{14} s^2_{24} \\,.\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    \\label{eq:pmutau}\n\n    P_{\\mu \\tau} &= 2 s^2_{24} s^2_{34} + \\big\\lbrace \\sin^2 2 \\theta_{23}\n\n    [c^4_{13} c^2_{24} c^2_{34} - s^2_{24} s^2_{34}]\n\n    \\\\\n\n    &- 4 \\sin 2 \\theta_{23} s_{13} s_{14} [s_{23} s_{34} \\cos\\delta_2\n\n    + c_{23} s_{24} \\cos(\\delta_2 - \\delta_3)]\n\n    \\\\\n\n    &+ 2 \\sin 2 \\theta_{23} s_{24} s_{34} c^2_{13} c^2_{24} c_{34}\n\n    [c^2_{14} - 2 c^2_{13} s^2_{23}] \\cos\\delta_3 \\big\\rbrace\n\n    \\sin^2 \\frac{\\Delta_\\Atm L}{2}\n\n    \\\\\n\n    &\\mp \\sin 2 \\theta_{23} s_{24} s_{34} c^2_{13} c^2_{14} c^2_{24} c_{34}\n\n    \\sin\\delta_3 \\sin\\Delta_\\Atm L \\,.\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    P^{3\\nu}_{\\mu\\tau} = P_{\\mu\\tau} (\\theta_{i4} = 0)\n\n    \\simeq c^4_{13} \\sin^2 2 \\theta_{23}\n\n    \\sin^2 \\frac{\\Delta_\\Atm L}{2} \\,.\n\n\\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    \\Delta P_{\\mu \\tau} &\\equiv P_{\\mu \\tau} - P^{3\\nu}_{\\mu\\tau}\n\n    \\\\\n\n    &= \\left\\lbrace - (s^2_{24} + s^2_{34}) \\sin^2 2 \\theta_{23} \n\n    + 2 \\sin 2 \\theta_{23} (1 - 2 s^2_{13}) s_{24} s_{34}\n\n    \\cos\\delta_3 \\right\\rbrace \\sin^2 \\frac{\\Delta_\\Atm L}{2}\n\n    \\\\\n\n    &\\mp \\sin 2 \\theta_{23} s_{24} s_{34} \\sin\\delta_3\n\n    \\sin \\Delta_\\Atm L + \\dots\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    \\label{eq:dpmue}\n\n    \\Delta P_{\\mu e} &\\equiv P_{\\mu e} - P_{\\mu e}(\\theta_{i4} = 0)\n\n    \\\\\n\n    &= s_{23} s_{13} s_{14} s_{24} \\cos(\\delta_2 - \\delta_3)\n\n    \\sin^2 \\frac{\\Delta_\\Atm L}{2}\n\n    \\\\\n\n    &\\pm 2 s_{23} s_{13} s_{14} s_{24}\n\n    \\sin(\\delta_2 - \\delta_3) \\sin \\Delta_\\Atm L + \\dots\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    N = \\int \\frac{d \\phi_{\\nu_\\alpha}(E)}{dE} \\,\n\n    \\sigma_{\\nu_\\alpha}(E) \\, dE \\,,\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:signaltau}\n\n    N_{\\mu \\tau} = A \\int \\frac{ d\\phi_{\\nu_{\\mu}}(E)}{dE} \\,\n\n    P_{\\mu \\tau}(E) \\sigma_{\\nu_\\tau}^{CC}(E) \\,\n\n    \\varepsilon_{\\mu \\tau} \\, dE \\,.\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:signale}\n\n    N_{\\mu e} = A \\int \\frac{ d\\phi_{\\nu_{\\mu}}(E)}{dE} \\,\n\n    P_{\\mu e}(E) \\sigma_{\\nu_{e}}^{CC}(E) \\,\n\n    \\varepsilon_{\\mu e}(E) \\, dE \\,,\n\n\\end{equation}"}
{"table/equation": "\\begin{table}[t] \\centering\n\n    \\begin{tabular}{|c|c|c|c|}\n\n\t\\hline\n\n\t\\hline\n\n\t$\\nu_{\\mu}$ &  $\\bar \\nu_{\\mu}$ & $\\nu_e$ &  $\\bar \\nu_e$\\\\\n\n\t\\hline\n\n\t669.0 &13.7 & 5.9& 0.3\\\\\n\n\t\\hline\\hline\n\n    \\end{tabular}\n\n    \\caption{\\label{tab:rates}\\sl%\n\n      Nominal performance of the CNGS reference beam~\\cite{fluxes}.\n\n      The total non-oscillated CC event rates are calculated assuming\n\n      $10^{19}$ pot and 1 Kton lead target mass.}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[t] \\centering\n\n    \\begin{tabular}{|c|c|c|c|c|c|}\n\n\t\\hline\\hline\n\n\t$(\\theta_{13}; \\theta_{14}; \\theta_{24}; \\theta_{34})$ & $N_\\tau$ & background & $(\\theta_{13}; \\theta_{14}; \\theta_{24}; \\theta_{34})$ & $N_\\tau$ & background \\\\\n\n\t\\hline\n\n\t$(5^\\circ;  5^\\circ;  5^\\circ; 20^\\circ)$    &     8.9    & 1.0 & $(10^\\circ; 5^\\circ;  5^\\circ; 20^\\circ)$    &     8.5    & 1.0 \\\\  \n\n\t$(5^\\circ;  5^\\circ;  5^\\circ; 30^\\circ)$    &     6.9    & 1.0 & $(10^\\circ; 5^\\circ;  5^\\circ; 30^\\circ)$    &     6.5    & 1.0 \\\\\n\n\t$(5^\\circ;  5^\\circ; 10^\\circ; 20^\\circ)$    &     8.3    & 1.0 & $(10^\\circ; 5^\\circ; 10^\\circ; 20^\\circ)$    &     7.9    & 1.0 \\\\\n\n\t$(5^\\circ;  5^\\circ; 10^\\circ; 30^\\circ)$    &    10.5    & 1.0 & $(10^\\circ; 5^\\circ; 10^\\circ; 30^\\circ)$    &    10.3    & 1.0 \\\\\n\n\t3 families                          &    15.1    & 1.0 &          3 families                          &    14.4    & 1.0 \\\\\n\n\t\\hline \\hline\n\n    \\end{tabular}\n\n    \\caption{\\label{tab:rates3}\\sl%\n\n      Event rates and expected background for the $\\nu_\\mu \\to\n\n      \\nu_\\tau$ channel in the OPERA detector, for different values of\n\n      $\\theta_{14}, \\theta_{24}$ and $\\theta_{34}$ in the (3+1)\n\n      scheme. The other unknown angle, $\\theta_{13}$ has been fixed\n\n      to: $\\theta_{13} = 5^\\circ, 10^\\circ$. The CP-violating phases\n\n      are: $\\delta_1 = \\delta_2 = 0$; $\\delta_3 = 90^\\circ$.  As a\n\n      reference, the expected value in the case of standard\n\n      three-family oscillation (\\textit{i.e.}, for $\\theta_{i4} = 0$) is shown\n\n      for maximal CP-violating phase $\\delta$. The rates are computed\n\n      according to Eq.~\\eqref{eq:signaltau}.}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[t] \\centering\n\n    \\begin{tabular}{|c|c|c|c|c|c|}\n\n\t\\hline\\hline\n\n\t$(\\theta_{13}; \\theta_{14}; \\theta_{24}; \\theta_{34})$ & $N_e$ & $\\nu_e^{CC}$ & $\\nu_\\mu^{NC}$ & $\\tau \\to e$ & $\\nu_\\mu^{CC}$  \\\\ \\hline\n\n\t$(5^\\circ; 5^\\circ; 5^\\circ; 20^\\circ)$   &  3.5 & 19.4 & 5.3 & 2.8 & 0.9 \\\\\n\n\t$(5^\\circ; 5^\\circ; 5^\\circ; 30^\\circ)$   &  3.5 & 19.4 & 5.3 & 2.1 & 0.9 \\\\\n\n\t$(5^\\circ; 5^\\circ; 10^\\circ; 20^\\circ)$  &  2.4 & 19.4 & 5.3 & 2.3 & 0.9 \\\\\n\n\t$(5^\\circ; 5^\\circ; 10^\\circ; 30^\\circ)$  &  2.4 & 19.4 & 5.3 & 2.4 & 0.9 \\\\\n\n\t3 families                        &  3.7 & 19.7 & 5.3 & 4.6 & 0.9 \\\\\n\n\t\\hline\n\n\t$(10^\\circ; 5^\\circ; 5^\\circ; 20^\\circ)$  & 10.6 & 19.4 & 5.3 & 2.7 & 0.9 \\\\\n\n\t$(10^\\circ; 5^\\circ; 5^\\circ; 30^\\circ)$  & 10.4 & 19.4 & 5.3 & 2.0 & 0.9 \\\\\n\n\t$(10^\\circ; 5^\\circ; 10^\\circ; 20^\\circ)$ &  8.8 & 19.4 & 5.3 & 2.2 & 0.9 \\\\\n\n\t$(10^\\circ; 5^\\circ; 10^\\circ; 30^\\circ)$ &  8.6 & 19.4 & 5.3 & 2.4 & 0.9 \\\\\n\n\t3 families                        & 15.1 & 19.7 & 5.3 & 4.8 & 0.9 \\\\\n\n\t\\hline \\hline\n\n    \\end{tabular}\n\n    \\caption{\\label{tab:rates2}\\sl%\n\n      Event rates and expected background for the $\\nu_\\mu \\to \\nu_e$\n\n      channel in the OPERA detector, for different values of \n\n      $\\theta_{14}, \\theta_{24}$ and $\\theta_{34}$ in the (3+1)\n\n      scheme. The other unknown angle, $\\theta_{13}$, has been fixed\n\n      to: $\\theta_{13} = 5^\\circ, 10^\\circ$. The CP-violating phases\n\n      are: $\\delta_1 = \\delta_2 = 0$; $\\delta_3 = 90^\\circ$.  As a\n\n      reference, the expected value in the case of standard\n\n      three-family oscillation(\\textit{i.e.}, for $\\theta_{i4} = 0$) is shown\n\n      for maximal CP-violating phase $\\delta$.  The rates are computed\n\n      according to Eq.~\\eqref{eq:signale}. Backgrounds have been\n\n      computed following Ref.~\\cite{Komatsu:2002sz}.}\n\n\\end{table}"}
{"table/equation": "\\begin{equation}\n\n    \\eta_s = \\sum_{i \\,\\in\\, \\text{sol}} |U_{s i}|^2\n\n    \\qquad\\text{and}\\qquad\n\n    c_s = \\sum_{j \\,\\in\\, \\text{atm}} |U_{s j}|^2\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:3+1param}\n\n    U_\\text{SBL} =\n\n    R_{14}(\\theta_{14}) \\; R_{24}(\\theta_{24}) \\; R_{34}(\\theta_{34}) \\; \n\n    R_{23}(\\theta_{23} ,\\, \\delta_3) \\; R_{13}(\\theta_{13} ,\\, \\delta_2) \\;\n\n    R_{12}(\\theta_{12} ,\\, \\delta_1) \\,.\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:3+1param2}\n\n    U_\\text{atm} =\n\n    R_{34}(\\theta_{34}) \\; R_{24}(\\theta_{24}) \\;\n\n    R_{23}(\\theta_{23} ,\\, \\delta_3) \\;\n\n    R_{14}(\\theta_{14}) \\; R_{13}(\\theta_{13} ,\\, \\delta_2) \\; \n\n    R_{12}(\\theta_{12} ,\\, \\delta_1) \\,.\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:pee}\n\n    P_{ee} = 1 - \\sin^2 2 \\theta_{14} \\sin^2 \\frac{\\Delta_\\Sbl L }{2} \n\n    - c^4_{14} \\sin^2 2 \\theta_{13} \\sin^2 \\frac{\\Delta_\\Atm L}{2} \\, ,\n\n\\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    P(\\nu_\\alpha \\to \\nu_\\beta) &= \\delta_{\\alpha \\beta}\n\n    - 4 \\Re \\left[ U_{\\alpha 3} U^\\star_{\\beta 3} \\left(\n\n    \\delta_{\\alpha\\beta} - U^\\star_{\\alpha 3} U_{\\beta 3}\n\n    - U^\\star_{\\alpha 4} U_{\\beta 4} \\right) \\right]\n\n    \\sin^2 \\frac{\\Delta_{23} L }{2}\n\n    \\\\\n\n    &- 2 \\Re \\left[ \\delta_{\\alpha\\beta} U_{\\alpha 4} U^\\star_{\\beta 4}\n\n    - |U_{\\alpha 4}|^2 |U_{\\beta 4}|^2 \\right]\n\n    \\pm 2 \\Im \\left[ U_{\\alpha 3} U^\\star_{\\beta 3}\n\n    U^\\star_{\\alpha 4} U_{\\beta 4} \\right] \\sin \\Delta_{23} L \\,,\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    P_{\\mu\\mu} &= 1 - 2 c^2_{14} s^2_{24} (1 - c^2_{14} s^2_{24})\n\n    - 4 \\big\\lbrace s^2_{23} c^2_{24} \\left[ c^2_{24} (c^2_{23}\n\n    - s^2_{13}) - s^2_{14} s^2_{24} \\right]\n\n    \\\\    \n\n    &- 2 c^3_{24} s_{23} (1 - 2 s^2_{23}) s_{13} s_{14} s_{24}\n\n    \\cos(\\delta_2 - \\delta_3) \\big\\rbrace\n\n    \\sin^2 \\frac{\\Delta_\\Atm L }{2} \\,.\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:pmumu}\n\n    P_{\\mu\\mu} = 1 - 2 s^2_{24}\n\n    - 4 s^2_{23} \\left[ c^2_{23} (1 - 2 s^2_{24}) - s^2_{13} \\right]\n\n    \\sin^2 \\frac{\\Delta_\\Atm L }{2} \\,.\n\n\\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    \\label{eq:pmus}\n\n    P_{\\mu s} &= 2 c^2_{34} s^2_{24} + \\big\\lbrace\n\n    \\sin^2 2 \\theta_{23} (c^4_{13} c^2_{24} s^2_{34}\n\n    - c^2_{34} s^2_{24})\n\n    \\\\\n\n    &+ 2  c_{34} \\sin 2 \\theta_{23} s_{34} \\left[\n\n    s_{24} (1 -2 s^2_{23}) \\cos \\delta_3\n\n    + 2 s_{23} s_{13} s_{14} \\cos \\delta_2 \\right] \\big\\rbrace\n\n    \\sin^2 \\frac{\\Delta_\\Atm L }{2}\n\n    \\\\\n\n    &\\pm c_{34} \\sin 2 \\theta_{23} s_{24} s_{34}\n\n    \\sin\\delta_3 \\sin\\Delta_\\Atm L \\,.\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    \\label{eq:pmue}\n\n    P_{\\mu e} &= 4 \\left\\lbrace s^2_{23} s^2_{13}\n\n    [1 - s^2_{13} - s^2_{14} - s^2_{24}] \n\n    + s_{23} s_{13} s_{14} s_{24} \\cos(\\delta_2 - \\delta_3) \\right\\rbrace\n\n    \\sin^2 \\frac{\\Delta_\\Atm L}{2}\n\n    \\\\\n\n    & \\pm 2 s_{23} s_{13} s_{14} s_{24} \\sin(\\delta_2 - \\delta_3)\n\n    \\sin\\Delta_\\Atm L + 2 s^2_{14} s^2_{24} \\,.\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    \\label{eq:pmutau}\n\n    P_{\\mu \\tau} &= 2 s^2_{24} s^2_{34} + \\big\\lbrace \\sin^2 2 \\theta_{23}\n\n    [c^4_{13} c^2_{24} c^2_{34} - s^2_{24} s^2_{34}]\n\n    \\\\\n\n    &- 4 \\sin 2 \\theta_{23} s_{13} s_{14} [s_{23} s_{34} \\cos\\delta_2\n\n    + c_{23} s_{24} \\cos(\\delta_2 - \\delta_3)]\n\n    \\\\\n\n    &+ 2 \\sin 2 \\theta_{23} s_{24} s_{34} c^2_{13} c^2_{24} c_{34}\n\n    [c^2_{14} - 2 c^2_{13} s^2_{23}] \\cos\\delta_3 \\big\\rbrace\n\n    \\sin^2 \\frac{\\Delta_\\Atm L}{2}\n\n    \\\\\n\n    &\\mp \\sin 2 \\theta_{23} s_{24} s_{34} c^2_{13} c^2_{14} c^2_{24} c_{34}\n\n    \\sin\\delta_3 \\sin\\Delta_\\Atm L \\,.\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    P^{3\\nu}_{\\mu\\tau} = P_{\\mu\\tau} (\\theta_{i4} = 0)\n\n    \\simeq c^4_{13} \\sin^2 2 \\theta_{23}\n\n    \\sin^2 \\frac{\\Delta_\\Atm L}{2} \\,.\n\n\\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    \\Delta P_{\\mu \\tau} &\\equiv P_{\\mu \\tau} - P^{3\\nu}_{\\mu\\tau}\n\n    \\\\\n\n    &= \\left\\lbrace - (s^2_{24} + s^2_{34}) \\sin^2 2 \\theta_{23} \n\n    + 2 \\sin 2 \\theta_{23} (1 - 2 s^2_{13}) s_{24} s_{34}\n\n    \\cos\\delta_3 \\right\\rbrace \\sin^2 \\frac{\\Delta_\\Atm L}{2}\n\n    \\\\\n\n    &\\mp \\sin 2 \\theta_{23} s_{24} s_{34} \\sin\\delta_3\n\n    \\sin \\Delta_\\Atm L + \\dots\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    \\label{eq:dpmue}\n\n    \\Delta P_{\\mu e} &\\equiv P_{\\mu e} - P_{\\mu e}(\\theta_{i4} = 0)\n\n    \\\\\n\n    &= s_{23} s_{13} s_{14} s_{24} \\cos(\\delta_2 - \\delta_3)\n\n    \\sin^2 \\frac{\\Delta_\\Atm L}{2}\n\n    \\\\\n\n    &\\pm 2 s_{23} s_{13} s_{14} s_{24}\n\n    \\sin(\\delta_2 - \\delta_3) \\sin \\Delta_\\Atm L + \\dots\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    N = \\int \\frac{d \\phi_{\\nu_\\alpha}(E)}{dE} \\,\n\n    \\sigma_{\\nu_\\alpha}(E) \\, dE \\,,\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:signaltau}\n\n    N_{\\mu \\tau} = A \\int \\frac{ d\\phi_{\\nu_{\\mu}}(E)}{dE} \\,\n\n    P_{\\mu \\tau}(E) \\sigma_{\\nu_\\tau}^{CC}(E) \\,\n\n    \\varepsilon_{\\mu \\tau} \\, dE \\,.\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:signale}\n\n    N_{\\mu e} = A \\int \\frac{ d\\phi_{\\nu_{\\mu}}(E)}{dE} \\,\n\n    P_{\\mu e}(E) \\sigma_{\\nu_{e}}^{CC}(E) \\,\n\n    \\varepsilon_{\\mu e}(E) \\, dE \\,,\n\n\\end{equation}"}
{"table/equation": "\\begin{table}[t] \\centering\n\n    \\begin{tabular}{|c|c|c|c|}\n\n\t\\hline\n\n\t\\hline\n\n\t$\\nu_{\\mu}$ &  $\\bar \\nu_{\\mu}$ & $\\nu_e$ &  $\\bar \\nu_e$\\\\\n\n\t\\hline\n\n\t669.0 &13.7 & 5.9& 0.3\\\\\n\n\t\\hline\\hline\n\n    \\end{tabular}\n\n    \\caption{\\label{tab:rates}\\sl%\n\n      Nominal performance of the CNGS reference beam~\\cite{fluxes}.\n\n      The total non-oscillated CC event rates are calculated assuming\n\n      $10^{19}$ pot and 1 Kton lead target mass.}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[t] \\centering\n\n    \\begin{tabular}{|c|c|c|c|c|c|}\n\n\t\\hline\\hline\n\n\t$(\\theta_{13}; \\theta_{14}; \\theta_{24}; \\theta_{34})$ & $N_\\tau$ & background & $(\\theta_{13}; \\theta_{14}; \\theta_{24}; \\theta_{34})$ & $N_\\tau$ & background \\\\\n\n\t\\hline\n\n\t$(5^\\circ;  5^\\circ;  5^\\circ; 20^\\circ)$    &     8.9    & 1.0 & $(10^\\circ; 5^\\circ;  5^\\circ; 20^\\circ)$    &     8.5    & 1.0 \\\\  \n\n\t$(5^\\circ;  5^\\circ;  5^\\circ; 30^\\circ)$    &     6.9    & 1.0 & $(10^\\circ; 5^\\circ;  5^\\circ; 30^\\circ)$    &     6.5    & 1.0 \\\\\n\n\t$(5^\\circ;  5^\\circ; 10^\\circ; 20^\\circ)$    &     8.3    & 1.0 & $(10^\\circ; 5^\\circ; 10^\\circ; 20^\\circ)$    &     7.9    & 1.0 \\\\\n\n\t$(5^\\circ;  5^\\circ; 10^\\circ; 30^\\circ)$    &    10.5    & 1.0 & $(10^\\circ; 5^\\circ; 10^\\circ; 30^\\circ)$    &    10.3    & 1.0 \\\\\n\n\t3 families                          &    15.1    & 1.0 &          3 families                          &    14.4    & 1.0 \\\\\n\n\t\\hline \\hline\n\n    \\end{tabular}\n\n    \\caption{\\label{tab:rates3}\\sl%\n\n      Event rates and expected background for the $\\nu_\\mu \\to\n\n      \\nu_\\tau$ channel in the OPERA detector, for different values of\n\n      $\\theta_{14}, \\theta_{24}$ and $\\theta_{34}$ in the (3+1)\n\n      scheme. The other unknown angle, $\\theta_{13}$ has been fixed\n\n      to: $\\theta_{13} = 5^\\circ, 10^\\circ$. The CP-violating phases\n\n      are: $\\delta_1 = \\delta_2 = 0$; $\\delta_3 = 90^\\circ$.  As a\n\n      reference, the expected value in the case of standard\n\n      three-family oscillation (\\textit{i.e.}, for $\\theta_{i4} = 0$) is shown\n\n      for maximal CP-violating phase $\\delta$. The rates are computed\n\n      according to Eq.~\\eqref{eq:signaltau}.}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[t] \\centering\n\n    \\begin{tabular}{|c|c|c|c|c|c|}\n\n\t\\hline\\hline\n\n\t$(\\theta_{13}; \\theta_{14}; \\theta_{24}; \\theta_{34})$ & $N_e$ & $\\nu_e^{CC}$ & $\\nu_\\mu^{NC}$ & $\\tau \\to e$ & $\\nu_\\mu^{CC}$  \\\\ \\hline\n\n\t$(5^\\circ; 5^\\circ; 5^\\circ; 20^\\circ)$   &  3.5 & 19.4 & 5.3 & 2.8 & 0.9 \\\\\n\n\t$(5^\\circ; 5^\\circ; 5^\\circ; 30^\\circ)$   &  3.5 & 19.4 & 5.3 & 2.1 & 0.9 \\\\\n\n\t$(5^\\circ; 5^\\circ; 10^\\circ; 20^\\circ)$  &  2.4 & 19.4 & 5.3 & 2.3 & 0.9 \\\\\n\n\t$(5^\\circ; 5^\\circ; 10^\\circ; 30^\\circ)$  &  2.4 & 19.4 & 5.3 & 2.4 & 0.9 \\\\\n\n\t3 families                        &  3.7 & 19.7 & 5.3 & 4.6 & 0.9 \\\\\n\n\t\\hline\n\n\t$(10^\\circ; 5^\\circ; 5^\\circ; 20^\\circ)$  & 10.6 & 19.4 & 5.3 & 2.7 & 0.9 \\\\\n\n\t$(10^\\circ; 5^\\circ; 5^\\circ; 30^\\circ)$  & 10.4 & 19.4 & 5.3 & 2.0 & 0.9 \\\\\n\n\t$(10^\\circ; 5^\\circ; 10^\\circ; 20^\\circ)$ &  8.8 & 19.4 & 5.3 & 2.2 & 0.9 \\\\\n\n\t$(10^\\circ; 5^\\circ; 10^\\circ; 30^\\circ)$ &  8.6 & 19.4 & 5.3 & 2.4 & 0.9 \\\\\n\n\t3 families                        & 15.1 & 19.7 & 5.3 & 4.8 & 0.9 \\\\\n\n\t\\hline \\hline\n\n    \\end{tabular}\n\n    \\caption{\\label{tab:rates2}\\sl%\n\n      Event rates and expected background for the $\\nu_\\mu \\to \\nu_e$\n\n      channel in the OPERA detector, for different values of \n\n      $\\theta_{14}, \\theta_{24}$ and $\\theta_{34}$ in the (3+1)\n\n      scheme. The other unknown angle, $\\theta_{13}$, has been fixed\n\n      to: $\\theta_{13} = 5^\\circ, 10^\\circ$. The CP-violating phases\n\n      are: $\\delta_1 = \\delta_2 = 0$; $\\delta_3 = 90^\\circ$.  As a\n\n      reference, the expected value in the case of standard\n\n      three-family oscillation(\\textit{i.e.}, for $\\theta_{i4} = 0$) is shown\n\n      for maximal CP-violating phase $\\delta$.  The rates are computed\n\n      according to Eq.~\\eqref{eq:signale}. Backgrounds have been\n\n      computed following Ref.~\\cite{Komatsu:2002sz}.}\n\n\\end{table}"}
{"table/equation": "\\begin{equation}\n\n    \\eta_s = \\sum_{i \\,\\in\\, \\text{sol}} |U_{s i}|^2\n\n    \\qquad\\text{and}\\qquad\n\n    c_s = \\sum_{j \\,\\in\\, \\text{atm}} |U_{s j}|^2\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:3+1param}\n\n    U_\\text{SBL} =\n\n    R_{14}(\\theta_{14}) \\; R_{24}(\\theta_{24}) \\; R_{34}(\\theta_{34}) \\; \n\n    R_{23}(\\theta_{23} ,\\, \\delta_3) \\; R_{13}(\\theta_{13} ,\\, \\delta_2) \\;\n\n    R_{12}(\\theta_{12} ,\\, \\delta_1) \\,.\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:3+1param2}\n\n    U_\\text{atm} =\n\n    R_{34}(\\theta_{34}) \\; R_{24}(\\theta_{24}) \\;\n\n    R_{23}(\\theta_{23} ,\\, \\delta_3) \\;\n\n    R_{14}(\\theta_{14}) \\; R_{13}(\\theta_{13} ,\\, \\delta_2) \\; \n\n    R_{12}(\\theta_{12} ,\\, \\delta_1) \\,.\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:pee}\n\n    P_{ee} = 1 - \\sin^2 2 \\theta_{14} \\sin^2 \\frac{\\Delta_\\Sbl L }{2} \n\n    - c^4_{14} \\sin^2 2 \\theta_{13} \\sin^2 \\frac{\\Delta_\\Atm L}{2} \\, ,\n\n\\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    P(\\nu_\\alpha \\to \\nu_\\beta) &= \\delta_{\\alpha \\beta}\n\n    - 4 \\Re \\left[ U_{\\alpha 3} U^\\star_{\\beta 3} \\left(\n\n    \\delta_{\\alpha\\beta} - U^\\star_{\\alpha 3} U_{\\beta 3}\n\n    - U^\\star_{\\alpha 4} U_{\\beta 4} \\right) \\right]\n\n    \\sin^2 \\frac{\\Delta_{23} L }{2}\n\n    \\\\\n\n    &- 2 \\Re \\left[ \\delta_{\\alpha\\beta} U_{\\alpha 4} U^\\star_{\\beta 4}\n\n    - |U_{\\alpha 4}|^2 |U_{\\beta 4}|^2 \\right]\n\n    \\pm 2 \\Im \\left[ U_{\\alpha 3} U^\\star_{\\beta 3}\n\n    U^\\star_{\\alpha 4} U_{\\beta 4} \\right] \\sin \\Delta_{23} L \\,,\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    P_{\\mu\\mu} &= 1 - 2 c^2_{14} s^2_{24} (1 - c^2_{14} s^2_{24})\n\n    - 4 \\big\\lbrace s^2_{23} c^2_{24} \\left[ c^2_{24} (c^2_{23}\n\n    - s^2_{13}) - s^2_{14} s^2_{24} \\right]\n\n    \\\\    \n\n    &- 2 c^3_{24} s_{23} (1 - 2 s^2_{23}) s_{13} s_{14} s_{24}\n\n    \\cos(\\delta_2 - \\delta_3) \\big\\rbrace\n\n    \\sin^2 \\frac{\\Delta_\\Atm L }{2} \\,.\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:pmumu}\n\n    P_{\\mu\\mu} = 1 - 2 s^2_{24}\n\n    - 4 s^2_{23} \\left[ c^2_{23} (1 - 2 s^2_{24}) - s^2_{13} \\right]\n\n    \\sin^2 \\frac{\\Delta_\\Atm L }{2} \\,.\n\n\\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    \\label{eq:pmus}\n\n    P_{\\mu s} &= 2 c^2_{34} s^2_{24} + \\big\\lbrace\n\n    \\sin^2 2 \\theta_{23} (c^4_{13} c^2_{24} s^2_{34}\n\n    - c^2_{34} s^2_{24})\n\n    \\\\\n\n    &+ 2  c_{34} \\sin 2 \\theta_{23} s_{34} \\left[\n\n    s_{24} (1 -2 s^2_{23}) \\cos \\delta_3\n\n    + 2 s_{23} s_{13} s_{14} \\cos \\delta_2 \\right] \\big\\rbrace\n\n    \\sin^2 \\frac{\\Delta_\\Atm L }{2}\n\n    \\\\\n\n    &\\pm c_{34} \\sin 2 \\theta_{23} s_{24} s_{34}\n\n    \\sin\\delta_3 \\sin\\Delta_\\Atm L \\,.\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    \\label{eq:pmue}\n\n    P_{\\mu e} &= 4 \\left\\lbrace s^2_{23} s^2_{13}\n\n    [1 - s^2_{13} - s^2_{14} - s^2_{24}] \n\n    + s_{23} s_{13} s_{14} s_{24} \\cos(\\delta_2 - \\delta_3) \\right\\rbrace\n\n    \\sin^2 \\frac{\\Delta_\\Atm L}{2}\n\n    \\\\\n\n    & \\pm 2 s_{23} s_{13} s_{14} s_{24} \\sin(\\delta_2 - \\delta_3)\n\n    \\sin\\Delta_\\Atm L + 2 s^2_{14} s^2_{24} \\,.\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    \\label{eq:pmutau}\n\n    P_{\\mu \\tau} &= 2 s^2_{24} s^2_{34} + \\big\\lbrace \\sin^2 2 \\theta_{23}\n\n    [c^4_{13} c^2_{24} c^2_{34} - s^2_{24} s^2_{34}]\n\n    \\\\\n\n    &- 4 \\sin 2 \\theta_{23} s_{13} s_{14} [s_{23} s_{34} \\cos\\delta_2\n\n    + c_{23} s_{24} \\cos(\\delta_2 - \\delta_3)]\n\n    \\\\\n\n    &+ 2 \\sin 2 \\theta_{23} s_{24} s_{34} c^2_{13} c^2_{24} c_{34}\n\n    [c^2_{14} - 2 c^2_{13} s^2_{23}] \\cos\\delta_3 \\big\\rbrace\n\n    \\sin^2 \\frac{\\Delta_\\Atm L}{2}\n\n    \\\\\n\n    &\\mp \\sin 2 \\theta_{23} s_{24} s_{34} c^2_{13} c^2_{14} c^2_{24} c_{34}\n\n    \\sin\\delta_3 \\sin\\Delta_\\Atm L \\,.\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    P^{3\\nu}_{\\mu\\tau} = P_{\\mu\\tau} (\\theta_{i4} = 0)\n\n    \\simeq c^4_{13} \\sin^2 2 \\theta_{23}\n\n    \\sin^2 \\frac{\\Delta_\\Atm L}{2} \\,.\n\n\\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    \\Delta P_{\\mu \\tau} &\\equiv P_{\\mu \\tau} - P^{3\\nu}_{\\mu\\tau}\n\n    \\\\\n\n    &= \\left\\lbrace - (s^2_{24} + s^2_{34}) \\sin^2 2 \\theta_{23} \n\n    + 2 \\sin 2 \\theta_{23} (1 - 2 s^2_{13}) s_{24} s_{34}\n\n    \\cos\\delta_3 \\right\\rbrace \\sin^2 \\frac{\\Delta_\\Atm L}{2}\n\n    \\\\\n\n    &\\mp \\sin 2 \\theta_{23} s_{24} s_{34} \\sin\\delta_3\n\n    \\sin \\Delta_\\Atm L + \\dots\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    \\label{eq:dpmue}\n\n    \\Delta P_{\\mu e} &\\equiv P_{\\mu e} - P_{\\mu e}(\\theta_{i4} = 0)\n\n    \\\\\n\n    &= s_{23} s_{13} s_{14} s_{24} \\cos(\\delta_2 - \\delta_3)\n\n    \\sin^2 \\frac{\\Delta_\\Atm L}{2}\n\n    \\\\\n\n    &\\pm 2 s_{23} s_{13} s_{14} s_{24}\n\n    \\sin(\\delta_2 - \\delta_3) \\sin \\Delta_\\Atm L + \\dots\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    N = \\int \\frac{d \\phi_{\\nu_\\alpha}(E)}{dE} \\,\n\n    \\sigma_{\\nu_\\alpha}(E) \\, dE \\,,\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:signaltau}\n\n    N_{\\mu \\tau} = A \\int \\frac{ d\\phi_{\\nu_{\\mu}}(E)}{dE} \\,\n\n    P_{\\mu \\tau}(E) \\sigma_{\\nu_\\tau}^{CC}(E) \\,\n\n    \\varepsilon_{\\mu \\tau} \\, dE \\,.\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:signale}\n\n    N_{\\mu e} = A \\int \\frac{ d\\phi_{\\nu_{\\mu}}(E)}{dE} \\,\n\n    P_{\\mu e}(E) \\sigma_{\\nu_{e}}^{CC}(E) \\,\n\n    \\varepsilon_{\\mu e}(E) \\, dE \\,,\n\n\\end{equation}"}
{"table/equation": "\\begin{table}[t] \\centering\n\n    \\begin{tabular}{|c|c|c|c|}\n\n\t\\hline\n\n\t\\hline\n\n\t$\\nu_{\\mu}$ &  $\\bar \\nu_{\\mu}$ & $\\nu_e$ &  $\\bar \\nu_e$\\\\\n\n\t\\hline\n\n\t669.0 &13.7 & 5.9& 0.3\\\\\n\n\t\\hline\\hline\n\n    \\end{tabular}\n\n    \\caption{\\label{tab:rates}\\sl%\n\n      Nominal performance of the CNGS reference beam~\\cite{fluxes}.\n\n      The total non-oscillated CC event rates are calculated assuming\n\n      $10^{19}$ pot and 1 Kton lead target mass.}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[t] \\centering\n\n    \\begin{tabular}{|c|c|c|c|c|c|}\n\n\t\\hline\\hline\n\n\t$(\\theta_{13}; \\theta_{14}; \\theta_{24}; \\theta_{34})$ & $N_\\tau$ & background & $(\\theta_{13}; \\theta_{14}; \\theta_{24}; \\theta_{34})$ & $N_\\tau$ & background \\\\\n\n\t\\hline\n\n\t$(5^\\circ;  5^\\circ;  5^\\circ; 20^\\circ)$    &     8.9    & 1.0 & $(10^\\circ; 5^\\circ;  5^\\circ; 20^\\circ)$    &     8.5    & 1.0 \\\\  \n\n\t$(5^\\circ;  5^\\circ;  5^\\circ; 30^\\circ)$    &     6.9    & 1.0 & $(10^\\circ; 5^\\circ;  5^\\circ; 30^\\circ)$    &     6.5    & 1.0 \\\\\n\n\t$(5^\\circ;  5^\\circ; 10^\\circ; 20^\\circ)$    &     8.3    & 1.0 & $(10^\\circ; 5^\\circ; 10^\\circ; 20^\\circ)$    &     7.9    & 1.0 \\\\\n\n\t$(5^\\circ;  5^\\circ; 10^\\circ; 30^\\circ)$    &    10.5    & 1.0 & $(10^\\circ; 5^\\circ; 10^\\circ; 30^\\circ)$    &    10.3    & 1.0 \\\\\n\n\t3 families                          &    15.1    & 1.0 &          3 families                          &    14.4    & 1.0 \\\\\n\n\t\\hline \\hline\n\n    \\end{tabular}\n\n    \\caption{\\label{tab:rates3}\\sl%\n\n      Event rates and expected background for the $\\nu_\\mu \\to\n\n      \\nu_\\tau$ channel in the OPERA detector, for different values of\n\n      $\\theta_{14}, \\theta_{24}$ and $\\theta_{34}$ in the (3+1)\n\n      scheme. The other unknown angle, $\\theta_{13}$ has been fixed\n\n      to: $\\theta_{13} = 5^\\circ, 10^\\circ$. The CP-violating phases\n\n      are: $\\delta_1 = \\delta_2 = 0$; $\\delta_3 = 90^\\circ$.  As a\n\n      reference, the expected value in the case of standard\n\n      three-family oscillation (\\textit{i.e.}, for $\\theta_{i4} = 0$) is shown\n\n      for maximal CP-violating phase $\\delta$. The rates are computed\n\n      according to Eq.~\\eqref{eq:signaltau}.}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[t] \\centering\n\n    \\begin{tabular}{|c|c|c|c|c|c|}\n\n\t\\hline\\hline\n\n\t$(\\theta_{13}; \\theta_{14}; \\theta_{24}; \\theta_{34})$ & $N_e$ & $\\nu_e^{CC}$ & $\\nu_\\mu^{NC}$ & $\\tau \\to e$ & $\\nu_\\mu^{CC}$  \\\\ \\hline\n\n\t$(5^\\circ; 5^\\circ; 5^\\circ; 20^\\circ)$   &  3.5 & 19.4 & 5.3 & 2.8 & 0.9 \\\\\n\n\t$(5^\\circ; 5^\\circ; 5^\\circ; 30^\\circ)$   &  3.5 & 19.4 & 5.3 & 2.1 & 0.9 \\\\\n\n\t$(5^\\circ; 5^\\circ; 10^\\circ; 20^\\circ)$  &  2.4 & 19.4 & 5.3 & 2.3 & 0.9 \\\\\n\n\t$(5^\\circ; 5^\\circ; 10^\\circ; 30^\\circ)$  &  2.4 & 19.4 & 5.3 & 2.4 & 0.9 \\\\\n\n\t3 families                        &  3.7 & 19.7 & 5.3 & 4.6 & 0.9 \\\\\n\n\t\\hline\n\n\t$(10^\\circ; 5^\\circ; 5^\\circ; 20^\\circ)$  & 10.6 & 19.4 & 5.3 & 2.7 & 0.9 \\\\\n\n\t$(10^\\circ; 5^\\circ; 5^\\circ; 30^\\circ)$  & 10.4 & 19.4 & 5.3 & 2.0 & 0.9 \\\\\n\n\t$(10^\\circ; 5^\\circ; 10^\\circ; 20^\\circ)$ &  8.8 & 19.4 & 5.3 & 2.2 & 0.9 \\\\\n\n\t$(10^\\circ; 5^\\circ; 10^\\circ; 30^\\circ)$ &  8.6 & 19.4 & 5.3 & 2.4 & 0.9 \\\\\n\n\t3 families                        & 15.1 & 19.7 & 5.3 & 4.8 & 0.9 \\\\\n\n\t\\hline \\hline\n\n    \\end{tabular}\n\n    \\caption{\\label{tab:rates2}\\sl%\n\n      Event rates and expected background for the $\\nu_\\mu \\to \\nu_e$\n\n      channel in the OPERA detector, for different values of \n\n      $\\theta_{14}, \\theta_{24}$ and $\\theta_{34}$ in the (3+1)\n\n      scheme. The other unknown angle, $\\theta_{13}$, has been fixed\n\n      to: $\\theta_{13} = 5^\\circ, 10^\\circ$. The CP-violating phases\n\n      are: $\\delta_1 = \\delta_2 = 0$; $\\delta_3 = 90^\\circ$.  As a\n\n      reference, the expected value in the case of standard\n\n      three-family oscillation(\\textit{i.e.}, for $\\theta_{i4} = 0$) is shown\n\n      for maximal CP-violating phase $\\delta$.  The rates are computed\n\n      according to Eq.~\\eqref{eq:signale}. Backgrounds have been\n\n      computed following Ref.~\\cite{Komatsu:2002sz}.}\n\n\\end{table}"}
{"table/equation": "\\begin{equation}\n\n    \\eta_s = \\sum_{i \\,\\in\\, \\text{sol}} |U_{s i}|^2\n\n    \\qquad\\text{and}\\qquad\n\n    c_s = \\sum_{j \\,\\in\\, \\text{atm}} |U_{s j}|^2\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:3+1param}\n\n    U_\\text{SBL} =\n\n    R_{14}(\\theta_{14}) \\; R_{24}(\\theta_{24}) \\; R_{34}(\\theta_{34}) \\; \n\n    R_{23}(\\theta_{23} ,\\, \\delta_3) \\; R_{13}(\\theta_{13} ,\\, \\delta_2) \\;\n\n    R_{12}(\\theta_{12} ,\\, \\delta_1) \\,.\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:3+1param2}\n\n    U_\\text{atm} =\n\n    R_{34}(\\theta_{34}) \\; R_{24}(\\theta_{24}) \\;\n\n    R_{23}(\\theta_{23} ,\\, \\delta_3) \\;\n\n    R_{14}(\\theta_{14}) \\; R_{13}(\\theta_{13} ,\\, \\delta_2) \\; \n\n    R_{12}(\\theta_{12} ,\\, \\delta_1) \\,.\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:pee}\n\n    P_{ee} = 1 - \\sin^2 2 \\theta_{14} \\sin^2 \\frac{\\Delta_\\Sbl L }{2} \n\n    - c^4_{14} \\sin^2 2 \\theta_{13} \\sin^2 \\frac{\\Delta_\\Atm L}{2} \\, ,\n\n\\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    P(\\nu_\\alpha \\to \\nu_\\beta) &= \\delta_{\\alpha \\beta}\n\n    - 4 \\Re \\left[ U_{\\alpha 3} U^\\star_{\\beta 3} \\left(\n\n    \\delta_{\\alpha\\beta} - U^\\star_{\\alpha 3} U_{\\beta 3}\n\n    - U^\\star_{\\alpha 4} U_{\\beta 4} \\right) \\right]\n\n    \\sin^2 \\frac{\\Delta_{23} L }{2}\n\n    \\\\\n\n    &- 2 \\Re \\left[ \\delta_{\\alpha\\beta} U_{\\alpha 4} U^\\star_{\\beta 4}\n\n    - |U_{\\alpha 4}|^2 |U_{\\beta 4}|^2 \\right]\n\n    \\pm 2 \\Im \\left[ U_{\\alpha 3} U^\\star_{\\beta 3}\n\n    U^\\star_{\\alpha 4} U_{\\beta 4} \\right] \\sin \\Delta_{23} L \\,,\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    P_{\\mu\\mu} &= 1 - 2 c^2_{14} s^2_{24} (1 - c^2_{14} s^2_{24})\n\n    - 4 \\big\\lbrace s^2_{23} c^2_{24} \\left[ c^2_{24} (c^2_{23}\n\n    - s^2_{13}) - s^2_{14} s^2_{24} \\right]\n\n    \\\\    \n\n    &- 2 c^3_{24} s_{23} (1 - 2 s^2_{23}) s_{13} s_{14} s_{24}\n\n    \\cos(\\delta_2 - \\delta_3) \\big\\rbrace\n\n    \\sin^2 \\frac{\\Delta_\\Atm L }{2} \\,.\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:pmumu}\n\n    P_{\\mu\\mu} = 1 - 2 s^2_{24}\n\n    - 4 s^2_{23} \\left[ c^2_{23} (1 - 2 s^2_{24}) - s^2_{13} \\right]\n\n    \\sin^2 \\frac{\\Delta_\\Atm L }{2} \\,.\n\n\\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    \\label{eq:pmus}\n\n    P_{\\mu s} &= 2 c^2_{34} s^2_{24} + \\big\\lbrace\n\n    \\sin^2 2 \\theta_{23} (c^4_{13} c^2_{24} s^2_{34}\n\n    - c^2_{34} s^2_{24})\n\n    \\\\\n\n    &+ 2  c_{34} \\sin 2 \\theta_{23} s_{34} \\left[\n\n    s_{24} (1 -2 s^2_{23}) \\cos \\delta_3\n\n    + 2 s_{23} s_{13} s_{14} \\cos \\delta_2 \\right] \\big\\rbrace\n\n    \\sin^2 \\frac{\\Delta_\\Atm L }{2}\n\n    \\\\\n\n    &\\pm c_{34} \\sin 2 \\theta_{23} s_{24} s_{34}\n\n    \\sin\\delta_3 \\sin\\Delta_\\Atm L \\,.\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    \\label{eq:pmue}\n\n    P_{\\mu e} &= 4 \\left\\lbrace s^2_{23} s^2_{13}\n\n    [1 - s^2_{13} - s^2_{14} - s^2_{24}] \n\n    + s_{23} s_{13} s_{14} s_{24} \\cos(\\delta_2 - \\delta_3) \\right\\rbrace\n\n    \\sin^2 \\frac{\\Delta_\\Atm L}{2}\n\n    \\\\\n\n    & \\pm 2 s_{23} s_{13} s_{14} s_{24} \\sin(\\delta_2 - \\delta_3)\n\n    \\sin\\Delta_\\Atm L + 2 s^2_{14} s^2_{24} \\,.\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    \\label{eq:pmutau}\n\n    P_{\\mu \\tau} &= 2 s^2_{24} s^2_{34} + \\big\\lbrace \\sin^2 2 \\theta_{23}\n\n    [c^4_{13} c^2_{24} c^2_{34} - s^2_{24} s^2_{34}]\n\n    \\\\\n\n    &- 4 \\sin 2 \\theta_{23} s_{13} s_{14} [s_{23} s_{34} \\cos\\delta_2\n\n    + c_{23} s_{24} \\cos(\\delta_2 - \\delta_3)]\n\n    \\\\\n\n    &+ 2 \\sin 2 \\theta_{23} s_{24} s_{34} c^2_{13} c^2_{24} c_{34}\n\n    [c^2_{14} - 2 c^2_{13} s^2_{23}] \\cos\\delta_3 \\big\\rbrace\n\n    \\sin^2 \\frac{\\Delta_\\Atm L}{2}\n\n    \\\\\n\n    &\\mp \\sin 2 \\theta_{23} s_{24} s_{34} c^2_{13} c^2_{14} c^2_{24} c_{34}\n\n    \\sin\\delta_3 \\sin\\Delta_\\Atm L \\,.\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    P^{3\\nu}_{\\mu\\tau} = P_{\\mu\\tau} (\\theta_{i4} = 0)\n\n    \\simeq c^4_{13} \\sin^2 2 \\theta_{23}\n\n    \\sin^2 \\frac{\\Delta_\\Atm L}{2} \\,.\n\n\\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    \\Delta P_{\\mu \\tau} &\\equiv P_{\\mu \\tau} - P^{3\\nu}_{\\mu\\tau}\n\n    \\\\\n\n    &= \\left\\lbrace - (s^2_{24} + s^2_{34}) \\sin^2 2 \\theta_{23} \n\n    + 2 \\sin 2 \\theta_{23} (1 - 2 s^2_{13}) s_{24} s_{34}\n\n    \\cos\\delta_3 \\right\\rbrace \\sin^2 \\frac{\\Delta_\\Atm L}{2}\n\n    \\\\\n\n    &\\mp \\sin 2 \\theta_{23} s_{24} s_{34} \\sin\\delta_3\n\n    \\sin \\Delta_\\Atm L + \\dots\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation} \\begin{split}\n\n    \\label{eq:dpmue}\n\n    \\Delta P_{\\mu e} &\\equiv P_{\\mu e} - P_{\\mu e}(\\theta_{i4} = 0)\n\n    \\\\\n\n    &= s_{23} s_{13} s_{14} s_{24} \\cos(\\delta_2 - \\delta_3)\n\n    \\sin^2 \\frac{\\Delta_\\Atm L}{2}\n\n    \\\\\n\n    &\\pm 2 s_{23} s_{13} s_{14} s_{24}\n\n    \\sin(\\delta_2 - \\delta_3) \\sin \\Delta_\\Atm L + \\dots\n\n\\end{split} \\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    N = \\int \\frac{d \\phi_{\\nu_\\alpha}(E)}{dE} \\,\n\n    \\sigma_{\\nu_\\alpha}(E) \\, dE \\,,\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:signaltau}\n\n    N_{\\mu \\tau} = A \\int \\frac{ d\\phi_{\\nu_{\\mu}}(E)}{dE} \\,\n\n    P_{\\mu \\tau}(E) \\sigma_{\\nu_\\tau}^{CC}(E) \\,\n\n    \\varepsilon_{\\mu \\tau} \\, dE \\,.\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq:signale}\n\n    N_{\\mu e} = A \\int \\frac{ d\\phi_{\\nu_{\\mu}}(E)}{dE} \\,\n\n    P_{\\mu e}(E) \\sigma_{\\nu_{e}}^{CC}(E) \\,\n\n    \\varepsilon_{\\mu e}(E) \\, dE \\,,\n\n\\end{equation}"}
{"table/equation": "\\begin{table}[th]\n\n    \\centering\n\n    \\caption{Architecture for synthetic dataset models used in the experiments.}\n\n    \\resizebox{0.75\\linewidth}{!}{ \n\n        \\begin{tabular}{lllllll} \n\n            \\toprule \n\n            \\cellcolor{lightblue}\\textbf{Layer} & Input & FC1 & FC2 & FC3 & FC4 & FC5 \\\\ \\midrule \n\n            \\cellcolor{lightblue}\\textbf{Dimensions} & [5] & [5, 32] & [32, 64] & [64, 128] & [128, 32] & [32, 4] \\\\ \\bottomrule\n\n        \\end{tabular}\n\n    }\n\n    \\label{tab: synthetic-arch}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[H]\n\n    \\centering\n\n    \\caption{Architecture for CIFAR-MNIST dataset models. Every convolutional layer is followed by a max pooling layer with kernel size 2 and stride 2. }\n\n    \\resizebox{\\linewidth}{!}{ \n\n        \\begin{tabular}{lllllll}\n\n            \\toprule\n\n             \\cellcolor{lightblue}\\textbf{Layer} & Input & Conv1 & Conv2 & Conv3 & Flatten & FC \\\\ \\midrule \n\n             \\cellcolor{lightblue}\\textbf{Dimensions} & [3, 32, 32] & [3, 32, 3, 3] & [32, 64, 3, 3] & [64, 128, 3, 3] & [2048] & [2048, 20] \\\\ \\bottomrule \n\n        \\end{tabular}\n\n    }\n\n    \\label{tab: conv-arch}\n\n\\end{table}"}
{"table/equation": "\\begin{equation}\n\n    \\begin{split}\n\n        x^{\\star} \\leftarrow  \\argmin_{x \\in \\mathbb{R}^d} \\Bigg[ f(x) \\coloneqq \\frac{1}{N} \\sum_{i=1}^N f_i(x) \\Bigg], \n\n        \\label{eq: optimization-problem}\n\n    \\end{split}\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    x_{i}^{t,k} = x_{i}^{t,k-1} - \\eta_{\\ell} \\nabla f_i\\big(x_i^{t,k-1}, \\xi_i^{t,k-1}\\big), \\quad x_{i}^{t} = x_{i}^{t,K}, \\text{ and  } x_g^{t} = x_g^{t-1} - \\eta_g (x_g^{t-1} - \\frac{1}{N} \\sum_{i=1}^N x_{i}^{t}) , \n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq: mask learning}\n\n    \\min_{x, \\{m_i\\}_{i\\in[N]}} \\frac{1}{N} \\sum_{i=1}^N f_i(x \\odot m_i),\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    x_{i}^{t,k} = x_{i}^{t,k-1} - \\eta_{\\ell} \\nabla f_i\\big(x_i^{t,k-1} \\odot m_i^t, \\xi_i^{t,k-1}\\big) \\; \\text{and} \\; x_g^{t} = x_g^{t-1} - \\eta_g (x_g^{t-1} - \\frac{\\sum_{i \\in [N]}x_{i}^{t} \\odot m_i^t}{\\sum_{i \\in [N]}m_i^t}).\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq: mask score learning}\n\n    \\min_{x, \\{s_i\\}_{i\\in[N]}} \\frac{1}{N} \\sum_{i=1}^N f_i(x \\odot \\mathcal{G}(s_i)).\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\mathcal{L}_s = f_i\\Big(x_i^{t,k} \\odot \\mathcal{G}\\left(s_i^{t,k}\\right), \\xi_i^{t,k-1}\\Big) - \\lambda \\|\\sigma\\left(s_{i}^{t,k}\\right) - \\theta_{g \\backslash \\{i\\}}^{t}\\|_2^2; \\qquad s_{i}^{t,k+1} \\gets s_{i}^{t,k} - \\eta_s \\nabla_{s} \\mathcal{L}_s, \n\n    \\label{eq: mask-objective-func}\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\|\\sigma\\left(s_{i}^{t,k}\\right) - \\theta_{g \\backslash \\{i\\}}^{t}\\|_2^2 \\leq h. \n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\mathcal{L}_x = f_i\\Big(x_i^{t,k} \\odot \\mathcal{G}\\left(s_i^{t,k}\\right), \\xi_i^{t,k-1}\\Big); \\qquad x_{i}^{t,k+1} \\gets x_{i}^{t,k} - \\eta_{\\ell} \\nabla_{x} \\mathcal{L}_x,\n\n\\end{equation}"}
{"table/equation": "\\begin{table}[t]\n\n    \\centering\n\n    \\caption{Ablation study of the parameters $\\lambda$ and $\\tau$ on the CIFAR-MNIST dataset with $N=2$ participants over three independent runs. The first column ($\\tau=0.0$) corresponds to the FedAvg algorithm. The last row $(-)$ presents results for the \\texttt{FedPeWS-Fixed} algorithm.} \n\n    \\resizebox{\\linewidth}{!}{ \n\n    \\begin{tabular}{r|cccccc}\n\n        \\toprule\n\n        & \\multicolumn{6}{c}{\\cellcolor{lightblue}\\textbf{Proportion of warmup rounds $\\tau = W/T$}} \\\\ \\cmidrule{2-7} \n\n        $\\mathbf{\\lambda} (\\downarrow) , \\tau (\\rightarrow)$ & $\\tau=0.0$ (FedAvg) & $\\tau=0.2$ & $\\tau=0.4$ & $\\tau=0.5$ & $\\tau=0.6$ & $\\tau=0.8$ \\\\ \\midrule \n\n        0.0 & \\multirow{12}{*}{$71.23 \\pm 0.71$} & $68.01 \\pm 0.88$ & $66.01\\pm0.48$ & $65.96\\pm1.03$ & $65.40\\pm1.95$ & $65.77\\pm0.41$ \\\\ \\cmidrule{3-7}\n\n        0.1 & & $68.77\\pm1.09$ & $70.39\\pm1.00$ & $70.61\\pm1.17$ & $69.48\\pm0.37$ & $70.36\\pm2.06$ \\\\\n\n        0.3 & & $70.86\\pm0.23$ & $73.00\\pm0.65$ & $73.91\\pm0.71$ & $73.26\\pm0.46$ & $73.02\\pm1.05$ \\\\\n\n        0.5 & & $71.43\\pm0.56$ & $74.17\\pm0.91$ & $73.84\\pm0.12$ & $73.66\\pm0.88$ & $75.05\\pm0.45$ \\\\\n\n        1.0 & & $72.26\\pm0.54$ & $74.46\\pm0.44$ & $74.54\\pm0.76$ & $74.91\\pm0.42$ & $73.81\\pm0.87$ \\\\\n\n        2.0 & & $72.61\\pm0.79$ & $73.68\\pm0.17$ & \\cellcolor{lightgreen} $\\mathbf{75.35\\pm0.50}$ & $74.76\\pm0.54$ & $74.46\\pm0.56$ \\\\\n\n        5.0 & & $72.60\\pm0.45$ & \\cellcolor{lightgreen} $\\mathbf{75.22\\pm0.33}$ & \\cellcolor{lightgreen} $\\mathbf{75.00\\pm0.74}$ & $75.01\\pm0.71$ & $73.96\\pm1.60$ \\\\\n\n        10.0 & & $72.29\\pm0.48$ & \\cellcolor{lightgreen} $\\mathbf{74.97\\pm0.65}$ & $74.31\\pm0.95$ & $74.03\\pm0.30$ & $71.91\\pm2.69$ \\\\\n\n        100.0 & & $71.64\\pm0.47$ & $72.92\\pm0.39$ & $73.96\\pm0.65$ & $73.13\\pm0.73$ & $72.43\\pm3.55$ \\\\\n\n        1000.0 & & $71.58\\pm0.53$ & $73.18\\pm0.73$ & $73.32\\pm1.70$ & $73.87\\pm1.16$ & $72.52\\pm1.92$ \\\\ \\cmidrule{3-7} \n\n        $-$ & & $72.72\\pm0.44$ & \\cellcolor{lightgreen} $\\mathbf{75.22\\pm0.19}$ & \\cellcolor{lightgreen} $\\mathbf{75.05\\pm0.42}$ & $72.51\\pm3.89$ & $73.77\\pm0.20$ \\\\ \n\n        \\bottomrule\n\n    \\end{tabular}\n\n    }\n\n    \\label{tab: cifar-mnist-ablation}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[H]\n\n    \\centering\n\n    \\caption{Comparison of approaches for handling data heterogeneity in federated learning.} \n\n    \\resizebox{0.8\\linewidth}{!}{ \n\n        \\begin{tabular}{lcccc}\n\n            \\toprule\n\n             & \\multirow{2}{*}{\\parbox{2.1cm}{\\centering \\textbf{Shared \\\\ Global Model}}} & \\multirow{2}{*}{\\parbox{2.5cm}{\\centering \\textbf{Level of Mask \\\\ Personalization}}} & \\multirow{2}{*}{\\parbox{2cm}{\\centering {\\textbf{Learnable \\\\ Mask}}}} & \\multirow{2}{*}{\\parbox{2cm}{\\centering {\\textbf{Learnable \\\\ Parameters}}}} \\\\ \n\n             & & & & \\\\ \\midrule \n\n             FedPM \\citep{isik2022sparse} & \\checkmark & parameter-level & \\checkmark & $\\bm{\\times}$ \\\\ \n\n             IST \\citep{yuan2022distributed} & \\checkmark & neuron-level & $\\bm{\\times}$ (random) & \\checkmark \\\\ \n\n             LTH \\citep{frankle2018lottery} & $\\bm{\\times}$ & parameter-level & $\\bm{\\times}$ (pruned) & \\checkmark \\\\ \n\n             FedWeIT \\citep{yoon2021fedweit} & $\\bm{\\times}$ & parameter-level & \\checkmark & \\checkmark \\\\ \n\n             FjORD \\citep{horvath2021fjord} & \\checkmark & parameter-level & $\\bm{\\times}$ (slimmed) & \\checkmark \\\\ \\midrule \n\n             \\rowcolor{lightgreen} \\texttt{FedPeWS} (ours) & \\checkmark & neuron-level & \\checkmark & \\checkmark \\\\ \\bottomrule \n\n        \\end{tabular}\n\n    }\n\n    \\label{tab: methods-comparison}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[b]\n\n    \\centering\n\n    \\caption{Ablation study of the parameters $\\lambda$ and $\\tau$ on the combination of \\{Path-OCT-Tissue\\}MNIST datasets with $N=3$ participants over three independent runs. The first column ($\\tau=0.0$) corresponds to the FedAvg algorithm. The last row $(-)$ presents results for the \\texttt{FedPeWS-Fixed} algorithm.} \n\n    \\resizebox{\\linewidth}{!}{ \n\n    \\begin{tabular}{r|cccccc}\n\n        \\toprule\n\n        & \\multicolumn{6}{c}{\\cellcolor{lightblue}\\textbf{Proportion of warmup rounds $\\tau = W/T$}} \\\\ \\cmidrule{2-7} \n\n        ${\\lambda} (\\downarrow) , \\tau (\\rightarrow)$ & $\\tau=0.0$ (FedAvg) & $\\tau=0.2$ & $\\tau=0.4$ & $\\tau=0.5$ & $\\tau=0.6$ & $\\tau=0.8$ \\\\ \\midrule \n\n        0.0 & \\multirow{12}{*}{$52.25\\pm0.57$} & $53.04\\pm2.08$ & $50.35\\pm0.61$ & $47.83\\pm1.04$ & $47.89\\pm1.17$ & $46.80\\pm1.76$ \\\\ \\cmidrule{3-7}\n\n        0.1 & & $52.52\\pm1.79$ & $51.72\\pm0.63$ & $49.83\\pm1.83$ & $48.83\\pm2.51$ & $47.50\\pm1.07$ \\\\\n\n        0.3 & & $52.74\\pm1.75$ & $54.36\\pm0.80$ & $51.07\\pm0.25$ & $50.94\\pm2.22$ & $50.62\\pm1.91$ \\\\\n\n        0.5 & & $54.30\\pm2.08$ & $54.42\\pm1.43$ & $52.02\\pm2.53$ & $51.43\\pm1.60$ & $48.97\\pm0.80$ \\\\\n\n        1.0 & & \\cellcolor{lightgreen} $\\mathbf{54.89\\pm0.72}$ & $53.59\\pm1.40$ & $51.49\\pm1.19$ & $50.29\\pm2.70$ & $52.21\\pm1.02$ \\\\\n\n        2.0 & & \\cellcolor{lightgreen} $\\mathbf{54.75\\pm1.12}$ & \\cellcolor{lightgreen} $\\mathbf{54.45\\pm1.78}$ & $53.02\\pm0.45$ & $52.26\\pm1.81$ & $52.49\\pm0.49$ \\\\ \n\n        5.0 & & \\cellcolor{lightgreen} $\\mathbf{54.91\\pm0.91}$ & \\cellcolor{lightgreen} $\\mathbf{54.99\\pm0.90}$ & $52.41\\pm0.45$ & $52.95\\pm1.49$ & $52.53\\pm1.35$ \\\\ \n\n        10.0 & & \\cellcolor{lightgreen} $\\mathbf{55.12\\pm1.16}$ & \\cellcolor{lightgreen} $\\mathbf{55.03\\pm1.39}$ & $52.85\\pm0.95$ & $50.82\\pm3.03$ & $52.27\\pm1.11$ \\\\ \n\n        100.0 & & $54.22\\pm1.74$ & $52.31\\pm2.55$ & $52.10\\pm0.87$ & $49.52\\pm4.34$ & $51.46\\pm2.92$ \\\\ \n\n        1000.0 & & $53.45\\pm1.65$ & $53.82\\pm2.16$ & $51.16\\pm1.92$ & $51.30\\pm2.02$ & $51.19\\pm1.05$ \\\\ \\cmidrule{3-7} \n\n        $-$ & & $53.69\\pm0.77$ & $51.78\\pm0.44$ & $50.24\\pm1.88$ & $51.12\\pm0.72$ & $49.87\\pm1.23$ \\\\ \n\n        \\bottomrule\n\n    \\end{tabular}\n\n    }\n\n    \\label{tab: pot-ablation}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[t]\n\n    \\centering\n\n    \\caption{The required number of collaboration rounds to reach target accuracy $\\upsilon$ $\\%$ and the final accuracy after $T$ rounds. The results are averaged over 3 seeds. \n\n    {\\color{red} $\\times$} indicates that the algorithm cannot reach target accuracy $\\upsilon$ within $T$ rounds and {\\color{red} NA} means that it reaches $\\upsilon$ only in one random seed. } \n\n    \\resizebox{\\linewidth}{!}{ \n\n        \\begin{tabular}{cr|ccccc} \n\n        \\toprule\n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Dataset $/$ Batch size}} & \\multicolumn{3}{c}{Synthetic-32K, $32$} & \\multicolumn{1}{c}{Synthetic-3.2K, $8$} \n\n        \\\\ \\midrule \n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Parameters} $\\{\\eta_g / \\lambda / \\tau\\}$} & $\\{1.0 / 5.0 / 0.125\\}$ & $\\{0.5/2.0/0.2\\}$ & $\\{0.25/1.0/0.1875\\}$ & \\multicolumn{1}{c}{$\\{0.1/2.0/0.1\\}$} \\\\ \\midrule\n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Target accuracy $\\upsilon (\\%)$}} & $99$ & $90$ & $75$ & $99$ \n\n        \\\\ \\midrule \n\n        \\multirow{2}{*}{\\parbox{3.5cm}{\\centering \\textbf{No. of rounds to \\\\ reach target accuracy}}} & FedAvg & $148\\pm3.79$ & \\color{red} \\cellcolor{lightred} $199\\pm \\text{NA}$ & \\color{red} \\cellcolor{lightred} $\\times$ & \\color{red} \\cellcolor{lightred} $371\\pm\\text{NA}$ \n\n        \\\\ \\cmidrule{2-6}\n\n         & \\texttt{FedPeWS} & \\cellcolor{lightgreen} $\\mathbf{115\\pm7.21}$ & \\cellcolor{lightgreen} $\\mathbf{182\\pm6.81}$ & \\cellcolor{lightgreen} $\\mathbf{286\\pm7.93}$ & \\cellcolor{lightgreen} $\\mathbf{301\\pm10.59}$ \n\n        \\\\ \\midrule \n\n        \\multirow{2}{*}{\\parbox{3.75cm}{\\centering \\textbf{Final accuracy after \\\\ $T$ collaboration rounds}}} & FedAvg & \\multicolumn{1}{c}{$99.94\\pm0.05$} & $91.40\\pm7.25$ & $67.64\\pm0.90$ & $97.33\\pm3.89$ \n\n        \\\\ \\cmidrule{2-6}\n\n        & \\texttt{FedPeWS} & \\multicolumn{1}{c}{\\cellcolor{lightgreen} $\\mathbf{99.96\\pm0.01}$} & \\cellcolor{lightgreen} $\\mathbf{99.49\\pm0.60}$ & \\cellcolor{lightgreen} $\\mathbf{83.50\\pm3.52}$ & \\cellcolor{lightgreen} $\\mathbf{99.66\\pm0.19}$ \n\n        \\\\ \\bottomrule \n\n        \\end{tabular}\n\n    }\n\n    \\label{table: convergence-table}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[t]\n\n    \\centering\n\n    \\caption{The required number of collaboration rounds to reach target accuracy $\\upsilon$ $\\%$ using FedProx algorithm and the final accuracy after $T$ rounds. The results are averaged over 3 seeds. \n\n    {\\color{red} $\\times$} indicates that the algorithm cannot reach target accuracy $\\upsilon$ within $T$ rounds.} \n\n    \\resizebox{\\linewidth}{!}{ \n\n        \\begin{tabular}{cr|ccccc} \n\n        \\toprule\n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Dataset $/$ Batch size}} & \\multicolumn{3}{c}{Synthetic-32K, $32$} & \\multicolumn{1}{c}{Synthetic-3.2K, $8$} \n\n        \\\\ \\midrule \n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Parameters} $\\{\\eta_g / \\lambda / \\tau\\}$} & $\\{1.0 / 0.1 / 0.125\\}$ & $\\{0.5/0.1/0.2\\}$ & $\\{0.25/1.0/0.1875\\}$ & \\multicolumn{1}{c}{$\\{0.1/1.0/0.1\\}$} \\\\ \\midrule\n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Target accuracy $\\upsilon (\\%)$}} & $99$ & $90$ & $75$ & $99$ \n\n        \\\\ \\midrule \n\n        \\multirow{2}{*}{\\parbox{3.5cm}{\\centering \\textbf{No. of rounds to \\\\ reach target accuracy}}} & FedProx & $138\\pm13.22$ & \\cellcolor{lightred} \\color{red} $\\times$ & \\cellcolor{lightred} \\color{red} $\\times$ & $362\\pm20.00$ \n\n        \\\\ \\cmidrule{2-6} \n\n         & \\texttt{FedPeWS} & \\cellcolor{lightgreen} $\\mathbf{115\\pm5.29}$ & \\cellcolor{lightgreen} $\\mathbf{211\\pm16.52}$ & \\cellcolor{lightgreen} $\\mathbf{314\\pm27.83}$ & \\cellcolor{lightgreen} $\\mathbf{344\\pm27.30}$ \n\n        \\\\ \\midrule \n\n        \\multirow{2}{*}{\\parbox{3.75cm}{\\centering \\textbf{Final accuracy after \\\\ $T$ collaboration rounds}}} & FedProx & \\multicolumn{1}{c}{$99.95\\pm0.02$} & $82.43\\pm7.98$ & $69.26\\pm6.03$ &\\cellcolor{lightgreen}  $\\mathbf{99.92\\pm0.06}$ \n\n        \\\\ \\cmidrule{2-6}\n\n        & \\texttt{FedPeWS} & \\multicolumn{1}{c}{\\cellcolor{lightgreen} $\\mathbf{99.96\\pm0.01}$} & \\cellcolor{lightgreen} $\\mathbf{98.40\\pm1.84}$ & \\cellcolor{lightgreen} $\\mathbf{90.40\\pm3.91}$ & \\cellcolor{lightgreen} $\\mathbf{99.92\\pm0.07}$ \n\n        \\\\ \\bottomrule \n\n        \\end{tabular}\n\n    }\n\n    \\label{table: convergence-table-fedprox}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[th]\n\n    \\centering\n\n    \\caption{Architecture for synthetic dataset models used in the experiments.}\n\n    \\resizebox{0.75\\linewidth}{!}{ \n\n        \\begin{tabular}{lllllll} \n\n            \\toprule \n\n            \\cellcolor{lightblue}\\textbf{Layer} & Input & FC1 & FC2 & FC3 & FC4 & FC5 \\\\ \\midrule \n\n            \\cellcolor{lightblue}\\textbf{Dimensions} & [5] & [5, 32] & [32, 64] & [64, 128] & [128, 32] & [32, 4] \\\\ \\bottomrule\n\n        \\end{tabular}\n\n    }\n\n    \\label{tab: synthetic-arch}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[H]\n\n    \\centering\n\n    \\caption{Architecture for CIFAR-MNIST dataset models. Every convolutional layer is followed by a max pooling layer with kernel size 2 and stride 2. }\n\n    \\resizebox{\\linewidth}{!}{ \n\n        \\begin{tabular}{lllllll}\n\n            \\toprule\n\n             \\cellcolor{lightblue}\\textbf{Layer} & Input & Conv1 & Conv2 & Conv3 & Flatten & FC \\\\ \\midrule \n\n             \\cellcolor{lightblue}\\textbf{Dimensions} & [3, 32, 32] & [3, 32, 3, 3] & [32, 64, 3, 3] & [64, 128, 3, 3] & [2048] & [2048, 20] \\\\ \\bottomrule \n\n        \\end{tabular}\n\n    }\n\n    \\label{tab: conv-arch}\n\n\\end{table}"}
{"table/equation": "\\begin{equation}\n\n    \\begin{split}\n\n        x^{\\star} \\leftarrow  \\argmin_{x \\in \\mathbb{R}^d} \\Bigg[ f(x) \\coloneqq \\frac{1}{N} \\sum_{i=1}^N f_i(x) \\Bigg], \n\n        \\label{eq: optimization-problem}\n\n    \\end{split}\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    x_{i}^{t,k} = x_{i}^{t,k-1} - \\eta_{\\ell} \\nabla f_i\\big(x_i^{t,k-1}, \\xi_i^{t,k-1}\\big), \\quad x_{i}^{t} = x_{i}^{t,K}, \\text{ and  } x_g^{t} = x_g^{t-1} - \\eta_g (x_g^{t-1} - \\frac{1}{N} \\sum_{i=1}^N x_{i}^{t}) , \n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq: mask learning}\n\n    \\min_{x, \\{m_i\\}_{i\\in[N]}} \\frac{1}{N} \\sum_{i=1}^N f_i(x \\odot m_i),\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    x_{i}^{t,k} = x_{i}^{t,k-1} - \\eta_{\\ell} \\nabla f_i\\big(x_i^{t,k-1} \\odot m_i^t, \\xi_i^{t,k-1}\\big) \\; \\text{and} \\; x_g^{t} = x_g^{t-1} - \\eta_g (x_g^{t-1} - \\frac{\\sum_{i \\in [N]}x_{i}^{t} \\odot m_i^t}{\\sum_{i \\in [N]}m_i^t}).\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq: mask score learning}\n\n    \\min_{x, \\{s_i\\}_{i\\in[N]}} \\frac{1}{N} \\sum_{i=1}^N f_i(x \\odot \\mathcal{G}(s_i)).\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\mathcal{L}_s = f_i\\Big(x_i^{t,k} \\odot \\mathcal{G}\\left(s_i^{t,k}\\right), \\xi_i^{t,k-1}\\Big) - \\lambda \\|\\sigma\\left(s_{i}^{t,k}\\right) - \\theta_{g \\backslash \\{i\\}}^{t}\\|_2^2; \\qquad s_{i}^{t,k+1} \\gets s_{i}^{t,k} - \\eta_s \\nabla_{s} \\mathcal{L}_s, \n\n    \\label{eq: mask-objective-func}\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\|\\sigma\\left(s_{i}^{t,k}\\right) - \\theta_{g \\backslash \\{i\\}}^{t}\\|_2^2 \\leq h. \n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\mathcal{L}_x = f_i\\Big(x_i^{t,k} \\odot \\mathcal{G}\\left(s_i^{t,k}\\right), \\xi_i^{t,k-1}\\Big); \\qquad x_{i}^{t,k+1} \\gets x_{i}^{t,k} - \\eta_{\\ell} \\nabla_{x} \\mathcal{L}_x,\n\n\\end{equation}"}
{"table/equation": "\\begin{table}[t]\n\n    \\centering\n\n    \\caption{Ablation study of the parameters $\\lambda$ and $\\tau$ on the CIFAR-MNIST dataset with $N=2$ participants over three independent runs. The first column ($\\tau=0.0$) corresponds to the FedAvg algorithm. The last row $(-)$ presents results for the \\texttt{FedPeWS-Fixed} algorithm.} \n\n    \\resizebox{\\linewidth}{!}{ \n\n    \\begin{tabular}{r|cccccc}\n\n        \\toprule\n\n        & \\multicolumn{6}{c}{\\cellcolor{lightblue}\\textbf{Proportion of warmup rounds $\\tau = W/T$}} \\\\ \\cmidrule{2-7} \n\n        $\\mathbf{\\lambda} (\\downarrow) , \\tau (\\rightarrow)$ & $\\tau=0.0$ (FedAvg) & $\\tau=0.2$ & $\\tau=0.4$ & $\\tau=0.5$ & $\\tau=0.6$ & $\\tau=0.8$ \\\\ \\midrule \n\n        0.0 & \\multirow{12}{*}{$71.23 \\pm 0.71$} & $68.01 \\pm 0.88$ & $66.01\\pm0.48$ & $65.96\\pm1.03$ & $65.40\\pm1.95$ & $65.77\\pm0.41$ \\\\ \\cmidrule{3-7}\n\n        0.1 & & $68.77\\pm1.09$ & $70.39\\pm1.00$ & $70.61\\pm1.17$ & $69.48\\pm0.37$ & $70.36\\pm2.06$ \\\\\n\n        0.3 & & $70.86\\pm0.23$ & $73.00\\pm0.65$ & $73.91\\pm0.71$ & $73.26\\pm0.46$ & $73.02\\pm1.05$ \\\\\n\n        0.5 & & $71.43\\pm0.56$ & $74.17\\pm0.91$ & $73.84\\pm0.12$ & $73.66\\pm0.88$ & $75.05\\pm0.45$ \\\\\n\n        1.0 & & $72.26\\pm0.54$ & $74.46\\pm0.44$ & $74.54\\pm0.76$ & $74.91\\pm0.42$ & $73.81\\pm0.87$ \\\\\n\n        2.0 & & $72.61\\pm0.79$ & $73.68\\pm0.17$ & \\cellcolor{lightgreen} $\\mathbf{75.35\\pm0.50}$ & $74.76\\pm0.54$ & $74.46\\pm0.56$ \\\\\n\n        5.0 & & $72.60\\pm0.45$ & \\cellcolor{lightgreen} $\\mathbf{75.22\\pm0.33}$ & \\cellcolor{lightgreen} $\\mathbf{75.00\\pm0.74}$ & $75.01\\pm0.71$ & $73.96\\pm1.60$ \\\\\n\n        10.0 & & $72.29\\pm0.48$ & \\cellcolor{lightgreen} $\\mathbf{74.97\\pm0.65}$ & $74.31\\pm0.95$ & $74.03\\pm0.30$ & $71.91\\pm2.69$ \\\\\n\n        100.0 & & $71.64\\pm0.47$ & $72.92\\pm0.39$ & $73.96\\pm0.65$ & $73.13\\pm0.73$ & $72.43\\pm3.55$ \\\\\n\n        1000.0 & & $71.58\\pm0.53$ & $73.18\\pm0.73$ & $73.32\\pm1.70$ & $73.87\\pm1.16$ & $72.52\\pm1.92$ \\\\ \\cmidrule{3-7} \n\n        $-$ & & $72.72\\pm0.44$ & \\cellcolor{lightgreen} $\\mathbf{75.22\\pm0.19}$ & \\cellcolor{lightgreen} $\\mathbf{75.05\\pm0.42}$ & $72.51\\pm3.89$ & $73.77\\pm0.20$ \\\\ \n\n        \\bottomrule\n\n    \\end{tabular}\n\n    }\n\n    \\label{tab: cifar-mnist-ablation}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[H]\n\n    \\centering\n\n    \\caption{Comparison of approaches for handling data heterogeneity in federated learning.} \n\n    \\resizebox{0.8\\linewidth}{!}{ \n\n        \\begin{tabular}{lcccc}\n\n            \\toprule\n\n             & \\multirow{2}{*}{\\parbox{2.1cm}{\\centering \\textbf{Shared \\\\ Global Model}}} & \\multirow{2}{*}{\\parbox{2.5cm}{\\centering \\textbf{Level of Mask \\\\ Personalization}}} & \\multirow{2}{*}{\\parbox{2cm}{\\centering {\\textbf{Learnable \\\\ Mask}}}} & \\multirow{2}{*}{\\parbox{2cm}{\\centering {\\textbf{Learnable \\\\ Parameters}}}} \\\\ \n\n             & & & & \\\\ \\midrule \n\n             FedPM \\citep{isik2022sparse} & \\checkmark & parameter-level & \\checkmark & $\\bm{\\times}$ \\\\ \n\n             IST \\citep{yuan2022distributed} & \\checkmark & neuron-level & $\\bm{\\times}$ (random) & \\checkmark \\\\ \n\n             LTH \\citep{frankle2018lottery} & $\\bm{\\times}$ & parameter-level & $\\bm{\\times}$ (pruned) & \\checkmark \\\\ \n\n             FedWeIT \\citep{yoon2021fedweit} & $\\bm{\\times}$ & parameter-level & \\checkmark & \\checkmark \\\\ \n\n             FjORD \\citep{horvath2021fjord} & \\checkmark & parameter-level & $\\bm{\\times}$ (slimmed) & \\checkmark \\\\ \\midrule \n\n             \\rowcolor{lightgreen} \\texttt{FedPeWS} (ours) & \\checkmark & neuron-level & \\checkmark & \\checkmark \\\\ \\bottomrule \n\n        \\end{tabular}\n\n    }\n\n    \\label{tab: methods-comparison}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[b]\n\n    \\centering\n\n    \\caption{Ablation study of the parameters $\\lambda$ and $\\tau$ on the combination of \\{Path-OCT-Tissue\\}MNIST datasets with $N=3$ participants over three independent runs. The first column ($\\tau=0.0$) corresponds to the FedAvg algorithm. The last row $(-)$ presents results for the \\texttt{FedPeWS-Fixed} algorithm.} \n\n    \\resizebox{\\linewidth}{!}{ \n\n    \\begin{tabular}{r|cccccc}\n\n        \\toprule\n\n        & \\multicolumn{6}{c}{\\cellcolor{lightblue}\\textbf{Proportion of warmup rounds $\\tau = W/T$}} \\\\ \\cmidrule{2-7} \n\n        ${\\lambda} (\\downarrow) , \\tau (\\rightarrow)$ & $\\tau=0.0$ (FedAvg) & $\\tau=0.2$ & $\\tau=0.4$ & $\\tau=0.5$ & $\\tau=0.6$ & $\\tau=0.8$ \\\\ \\midrule \n\n        0.0 & \\multirow{12}{*}{$52.25\\pm0.57$} & $53.04\\pm2.08$ & $50.35\\pm0.61$ & $47.83\\pm1.04$ & $47.89\\pm1.17$ & $46.80\\pm1.76$ \\\\ \\cmidrule{3-7}\n\n        0.1 & & $52.52\\pm1.79$ & $51.72\\pm0.63$ & $49.83\\pm1.83$ & $48.83\\pm2.51$ & $47.50\\pm1.07$ \\\\\n\n        0.3 & & $52.74\\pm1.75$ & $54.36\\pm0.80$ & $51.07\\pm0.25$ & $50.94\\pm2.22$ & $50.62\\pm1.91$ \\\\\n\n        0.5 & & $54.30\\pm2.08$ & $54.42\\pm1.43$ & $52.02\\pm2.53$ & $51.43\\pm1.60$ & $48.97\\pm0.80$ \\\\\n\n        1.0 & & \\cellcolor{lightgreen} $\\mathbf{54.89\\pm0.72}$ & $53.59\\pm1.40$ & $51.49\\pm1.19$ & $50.29\\pm2.70$ & $52.21\\pm1.02$ \\\\\n\n        2.0 & & \\cellcolor{lightgreen} $\\mathbf{54.75\\pm1.12}$ & \\cellcolor{lightgreen} $\\mathbf{54.45\\pm1.78}$ & $53.02\\pm0.45$ & $52.26\\pm1.81$ & $52.49\\pm0.49$ \\\\ \n\n        5.0 & & \\cellcolor{lightgreen} $\\mathbf{54.91\\pm0.91}$ & \\cellcolor{lightgreen} $\\mathbf{54.99\\pm0.90}$ & $52.41\\pm0.45$ & $52.95\\pm1.49$ & $52.53\\pm1.35$ \\\\ \n\n        10.0 & & \\cellcolor{lightgreen} $\\mathbf{55.12\\pm1.16}$ & \\cellcolor{lightgreen} $\\mathbf{55.03\\pm1.39}$ & $52.85\\pm0.95$ & $50.82\\pm3.03$ & $52.27\\pm1.11$ \\\\ \n\n        100.0 & & $54.22\\pm1.74$ & $52.31\\pm2.55$ & $52.10\\pm0.87$ & $49.52\\pm4.34$ & $51.46\\pm2.92$ \\\\ \n\n        1000.0 & & $53.45\\pm1.65$ & $53.82\\pm2.16$ & $51.16\\pm1.92$ & $51.30\\pm2.02$ & $51.19\\pm1.05$ \\\\ \\cmidrule{3-7} \n\n        $-$ & & $53.69\\pm0.77$ & $51.78\\pm0.44$ & $50.24\\pm1.88$ & $51.12\\pm0.72$ & $49.87\\pm1.23$ \\\\ \n\n        \\bottomrule\n\n    \\end{tabular}\n\n    }\n\n    \\label{tab: pot-ablation}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[t]\n\n    \\centering\n\n    \\caption{The required number of collaboration rounds to reach target accuracy $\\upsilon$ $\\%$ and the final accuracy after $T$ rounds. The results are averaged over 3 seeds. \n\n    {\\color{red} $\\times$} indicates that the algorithm cannot reach target accuracy $\\upsilon$ within $T$ rounds and {\\color{red} NA} means that it reaches $\\upsilon$ only in one random seed. } \n\n    \\resizebox{\\linewidth}{!}{ \n\n        \\begin{tabular}{cr|ccccc} \n\n        \\toprule\n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Dataset $/$ Batch size}} & \\multicolumn{3}{c}{Synthetic-32K, $32$} & \\multicolumn{1}{c}{Synthetic-3.2K, $8$} \n\n        \\\\ \\midrule \n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Parameters} $\\{\\eta_g / \\lambda / \\tau\\}$} & $\\{1.0 / 5.0 / 0.125\\}$ & $\\{0.5/2.0/0.2\\}$ & $\\{0.25/1.0/0.1875\\}$ & \\multicolumn{1}{c}{$\\{0.1/2.0/0.1\\}$} \\\\ \\midrule\n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Target accuracy $\\upsilon (\\%)$}} & $99$ & $90$ & $75$ & $99$ \n\n        \\\\ \\midrule \n\n        \\multirow{2}{*}{\\parbox{3.5cm}{\\centering \\textbf{No. of rounds to \\\\ reach target accuracy}}} & FedAvg & $148\\pm3.79$ & \\color{red} \\cellcolor{lightred} $199\\pm \\text{NA}$ & \\color{red} \\cellcolor{lightred} $\\times$ & \\color{red} \\cellcolor{lightred} $371\\pm\\text{NA}$ \n\n        \\\\ \\cmidrule{2-6}\n\n         & \\texttt{FedPeWS} & \\cellcolor{lightgreen} $\\mathbf{115\\pm7.21}$ & \\cellcolor{lightgreen} $\\mathbf{182\\pm6.81}$ & \\cellcolor{lightgreen} $\\mathbf{286\\pm7.93}$ & \\cellcolor{lightgreen} $\\mathbf{301\\pm10.59}$ \n\n        \\\\ \\midrule \n\n        \\multirow{2}{*}{\\parbox{3.75cm}{\\centering \\textbf{Final accuracy after \\\\ $T$ collaboration rounds}}} & FedAvg & \\multicolumn{1}{c}{$99.94\\pm0.05$} & $91.40\\pm7.25$ & $67.64\\pm0.90$ & $97.33\\pm3.89$ \n\n        \\\\ \\cmidrule{2-6}\n\n        & \\texttt{FedPeWS} & \\multicolumn{1}{c}{\\cellcolor{lightgreen} $\\mathbf{99.96\\pm0.01}$} & \\cellcolor{lightgreen} $\\mathbf{99.49\\pm0.60}$ & \\cellcolor{lightgreen} $\\mathbf{83.50\\pm3.52}$ & \\cellcolor{lightgreen} $\\mathbf{99.66\\pm0.19}$ \n\n        \\\\ \\bottomrule \n\n        \\end{tabular}\n\n    }\n\n    \\label{table: convergence-table}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[t]\n\n    \\centering\n\n    \\caption{The required number of collaboration rounds to reach target accuracy $\\upsilon$ $\\%$ using FedProx algorithm and the final accuracy after $T$ rounds. The results are averaged over 3 seeds. \n\n    {\\color{red} $\\times$} indicates that the algorithm cannot reach target accuracy $\\upsilon$ within $T$ rounds.} \n\n    \\resizebox{\\linewidth}{!}{ \n\n        \\begin{tabular}{cr|ccccc} \n\n        \\toprule\n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Dataset $/$ Batch size}} & \\multicolumn{3}{c}{Synthetic-32K, $32$} & \\multicolumn{1}{c}{Synthetic-3.2K, $8$} \n\n        \\\\ \\midrule \n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Parameters} $\\{\\eta_g / \\lambda / \\tau\\}$} & $\\{1.0 / 0.1 / 0.125\\}$ & $\\{0.5/0.1/0.2\\}$ & $\\{0.25/1.0/0.1875\\}$ & \\multicolumn{1}{c}{$\\{0.1/1.0/0.1\\}$} \\\\ \\midrule\n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Target accuracy $\\upsilon (\\%)$}} & $99$ & $90$ & $75$ & $99$ \n\n        \\\\ \\midrule \n\n        \\multirow{2}{*}{\\parbox{3.5cm}{\\centering \\textbf{No. of rounds to \\\\ reach target accuracy}}} & FedProx & $138\\pm13.22$ & \\cellcolor{lightred} \\color{red} $\\times$ & \\cellcolor{lightred} \\color{red} $\\times$ & $362\\pm20.00$ \n\n        \\\\ \\cmidrule{2-6} \n\n         & \\texttt{FedPeWS} & \\cellcolor{lightgreen} $\\mathbf{115\\pm5.29}$ & \\cellcolor{lightgreen} $\\mathbf{211\\pm16.52}$ & \\cellcolor{lightgreen} $\\mathbf{314\\pm27.83}$ & \\cellcolor{lightgreen} $\\mathbf{344\\pm27.30}$ \n\n        \\\\ \\midrule \n\n        \\multirow{2}{*}{\\parbox{3.75cm}{\\centering \\textbf{Final accuracy after \\\\ $T$ collaboration rounds}}} & FedProx & \\multicolumn{1}{c}{$99.95\\pm0.02$} & $82.43\\pm7.98$ & $69.26\\pm6.03$ &\\cellcolor{lightgreen}  $\\mathbf{99.92\\pm0.06}$ \n\n        \\\\ \\cmidrule{2-6}\n\n        & \\texttt{FedPeWS} & \\multicolumn{1}{c}{\\cellcolor{lightgreen} $\\mathbf{99.96\\pm0.01}$} & \\cellcolor{lightgreen} $\\mathbf{98.40\\pm1.84}$ & \\cellcolor{lightgreen} $\\mathbf{90.40\\pm3.91}$ & \\cellcolor{lightgreen} $\\mathbf{99.92\\pm0.07}$ \n\n        \\\\ \\bottomrule \n\n        \\end{tabular}\n\n    }\n\n    \\label{table: convergence-table-fedprox}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[th]\n\n    \\centering\n\n    \\caption{Architecture for synthetic dataset models used in the experiments.}\n\n    \\resizebox{0.75\\linewidth}{!}{ \n\n        \\begin{tabular}{lllllll} \n\n            \\toprule \n\n            \\cellcolor{lightblue}\\textbf{Layer} & Input & FC1 & FC2 & FC3 & FC4 & FC5 \\\\ \\midrule \n\n            \\cellcolor{lightblue}\\textbf{Dimensions} & [5] & [5, 32] & [32, 64] & [64, 128] & [128, 32] & [32, 4] \\\\ \\bottomrule\n\n        \\end{tabular}\n\n    }\n\n    \\label{tab: synthetic-arch}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[H]\n\n    \\centering\n\n    \\caption{Architecture for CIFAR-MNIST dataset models. Every convolutional layer is followed by a max pooling layer with kernel size 2 and stride 2. }\n\n    \\resizebox{\\linewidth}{!}{ \n\n        \\begin{tabular}{lllllll}\n\n            \\toprule\n\n             \\cellcolor{lightblue}\\textbf{Layer} & Input & Conv1 & Conv2 & Conv3 & Flatten & FC \\\\ \\midrule \n\n             \\cellcolor{lightblue}\\textbf{Dimensions} & [3, 32, 32] & [3, 32, 3, 3] & [32, 64, 3, 3] & [64, 128, 3, 3] & [2048] & [2048, 20] \\\\ \\bottomrule \n\n        \\end{tabular}\n\n    }\n\n    \\label{tab: conv-arch}\n\n\\end{table}"}
{"table/equation": "\\begin{equation}\n\n    \\begin{split}\n\n        x^{\\star} \\leftarrow  \\argmin_{x \\in \\mathbb{R}^d} \\Bigg[ f(x) \\coloneqq \\frac{1}{N} \\sum_{i=1}^N f_i(x) \\Bigg], \n\n        \\label{eq: optimization-problem}\n\n    \\end{split}\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    x_{i}^{t,k} = x_{i}^{t,k-1} - \\eta_{\\ell} \\nabla f_i\\big(x_i^{t,k-1}, \\xi_i^{t,k-1}\\big), \\quad x_{i}^{t} = x_{i}^{t,K}, \\text{ and  } x_g^{t} = x_g^{t-1} - \\eta_g (x_g^{t-1} - \\frac{1}{N} \\sum_{i=1}^N x_{i}^{t}) , \n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq: mask learning}\n\n    \\min_{x, \\{m_i\\}_{i\\in[N]}} \\frac{1}{N} \\sum_{i=1}^N f_i(x \\odot m_i),\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    x_{i}^{t,k} = x_{i}^{t,k-1} - \\eta_{\\ell} \\nabla f_i\\big(x_i^{t,k-1} \\odot m_i^t, \\xi_i^{t,k-1}\\big) \\; \\text{and} \\; x_g^{t} = x_g^{t-1} - \\eta_g (x_g^{t-1} - \\frac{\\sum_{i \\in [N]}x_{i}^{t} \\odot m_i^t}{\\sum_{i \\in [N]}m_i^t}).\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq: mask score learning}\n\n    \\min_{x, \\{s_i\\}_{i\\in[N]}} \\frac{1}{N} \\sum_{i=1}^N f_i(x \\odot \\mathcal{G}(s_i)).\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\mathcal{L}_s = f_i\\Big(x_i^{t,k} \\odot \\mathcal{G}\\left(s_i^{t,k}\\right), \\xi_i^{t,k-1}\\Big) - \\lambda \\|\\sigma\\left(s_{i}^{t,k}\\right) - \\theta_{g \\backslash \\{i\\}}^{t}\\|_2^2; \\qquad s_{i}^{t,k+1} \\gets s_{i}^{t,k} - \\eta_s \\nabla_{s} \\mathcal{L}_s, \n\n    \\label{eq: mask-objective-func}\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\|\\sigma\\left(s_{i}^{t,k}\\right) - \\theta_{g \\backslash \\{i\\}}^{t}\\|_2^2 \\leq h. \n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\mathcal{L}_x = f_i\\Big(x_i^{t,k} \\odot \\mathcal{G}\\left(s_i^{t,k}\\right), \\xi_i^{t,k-1}\\Big); \\qquad x_{i}^{t,k+1} \\gets x_{i}^{t,k} - \\eta_{\\ell} \\nabla_{x} \\mathcal{L}_x,\n\n\\end{equation}"}
{"table/equation": "\\begin{table}[t]\n\n    \\centering\n\n    \\caption{Ablation study of the parameters $\\lambda$ and $\\tau$ on the CIFAR-MNIST dataset with $N=2$ participants over three independent runs. The first column ($\\tau=0.0$) corresponds to the FedAvg algorithm. The last row $(-)$ presents results for the \\texttt{FedPeWS-Fixed} algorithm.} \n\n    \\resizebox{\\linewidth}{!}{ \n\n    \\begin{tabular}{r|cccccc}\n\n        \\toprule\n\n        & \\multicolumn{6}{c}{\\cellcolor{lightblue}\\textbf{Proportion of warmup rounds $\\tau = W/T$}} \\\\ \\cmidrule{2-7} \n\n        $\\mathbf{\\lambda} (\\downarrow) , \\tau (\\rightarrow)$ & $\\tau=0.0$ (FedAvg) & $\\tau=0.2$ & $\\tau=0.4$ & $\\tau=0.5$ & $\\tau=0.6$ & $\\tau=0.8$ \\\\ \\midrule \n\n        0.0 & \\multirow{12}{*}{$71.23 \\pm 0.71$} & $68.01 \\pm 0.88$ & $66.01\\pm0.48$ & $65.96\\pm1.03$ & $65.40\\pm1.95$ & $65.77\\pm0.41$ \\\\ \\cmidrule{3-7}\n\n        0.1 & & $68.77\\pm1.09$ & $70.39\\pm1.00$ & $70.61\\pm1.17$ & $69.48\\pm0.37$ & $70.36\\pm2.06$ \\\\\n\n        0.3 & & $70.86\\pm0.23$ & $73.00\\pm0.65$ & $73.91\\pm0.71$ & $73.26\\pm0.46$ & $73.02\\pm1.05$ \\\\\n\n        0.5 & & $71.43\\pm0.56$ & $74.17\\pm0.91$ & $73.84\\pm0.12$ & $73.66\\pm0.88$ & $75.05\\pm0.45$ \\\\\n\n        1.0 & & $72.26\\pm0.54$ & $74.46\\pm0.44$ & $74.54\\pm0.76$ & $74.91\\pm0.42$ & $73.81\\pm0.87$ \\\\\n\n        2.0 & & $72.61\\pm0.79$ & $73.68\\pm0.17$ & \\cellcolor{lightgreen} $\\mathbf{75.35\\pm0.50}$ & $74.76\\pm0.54$ & $74.46\\pm0.56$ \\\\\n\n        5.0 & & $72.60\\pm0.45$ & \\cellcolor{lightgreen} $\\mathbf{75.22\\pm0.33}$ & \\cellcolor{lightgreen} $\\mathbf{75.00\\pm0.74}$ & $75.01\\pm0.71$ & $73.96\\pm1.60$ \\\\\n\n        10.0 & & $72.29\\pm0.48$ & \\cellcolor{lightgreen} $\\mathbf{74.97\\pm0.65}$ & $74.31\\pm0.95$ & $74.03\\pm0.30$ & $71.91\\pm2.69$ \\\\\n\n        100.0 & & $71.64\\pm0.47$ & $72.92\\pm0.39$ & $73.96\\pm0.65$ & $73.13\\pm0.73$ & $72.43\\pm3.55$ \\\\\n\n        1000.0 & & $71.58\\pm0.53$ & $73.18\\pm0.73$ & $73.32\\pm1.70$ & $73.87\\pm1.16$ & $72.52\\pm1.92$ \\\\ \\cmidrule{3-7} \n\n        $-$ & & $72.72\\pm0.44$ & \\cellcolor{lightgreen} $\\mathbf{75.22\\pm0.19}$ & \\cellcolor{lightgreen} $\\mathbf{75.05\\pm0.42}$ & $72.51\\pm3.89$ & $73.77\\pm0.20$ \\\\ \n\n        \\bottomrule\n\n    \\end{tabular}\n\n    }\n\n    \\label{tab: cifar-mnist-ablation}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[H]\n\n    \\centering\n\n    \\caption{Comparison of approaches for handling data heterogeneity in federated learning.} \n\n    \\resizebox{0.8\\linewidth}{!}{ \n\n        \\begin{tabular}{lcccc}\n\n            \\toprule\n\n             & \\multirow{2}{*}{\\parbox{2.1cm}{\\centering \\textbf{Shared \\\\ Global Model}}} & \\multirow{2}{*}{\\parbox{2.5cm}{\\centering \\textbf{Level of Mask \\\\ Personalization}}} & \\multirow{2}{*}{\\parbox{2cm}{\\centering {\\textbf{Learnable \\\\ Mask}}}} & \\multirow{2}{*}{\\parbox{2cm}{\\centering {\\textbf{Learnable \\\\ Parameters}}}} \\\\ \n\n             & & & & \\\\ \\midrule \n\n             FedPM \\citep{isik2022sparse} & \\checkmark & parameter-level & \\checkmark & $\\bm{\\times}$ \\\\ \n\n             IST \\citep{yuan2022distributed} & \\checkmark & neuron-level & $\\bm{\\times}$ (random) & \\checkmark \\\\ \n\n             LTH \\citep{frankle2018lottery} & $\\bm{\\times}$ & parameter-level & $\\bm{\\times}$ (pruned) & \\checkmark \\\\ \n\n             FedWeIT \\citep{yoon2021fedweit} & $\\bm{\\times}$ & parameter-level & \\checkmark & \\checkmark \\\\ \n\n             FjORD \\citep{horvath2021fjord} & \\checkmark & parameter-level & $\\bm{\\times}$ (slimmed) & \\checkmark \\\\ \\midrule \n\n             \\rowcolor{lightgreen} \\texttt{FedPeWS} (ours) & \\checkmark & neuron-level & \\checkmark & \\checkmark \\\\ \\bottomrule \n\n        \\end{tabular}\n\n    }\n\n    \\label{tab: methods-comparison}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[b]\n\n    \\centering\n\n    \\caption{Ablation study of the parameters $\\lambda$ and $\\tau$ on the combination of \\{Path-OCT-Tissue\\}MNIST datasets with $N=3$ participants over three independent runs. The first column ($\\tau=0.0$) corresponds to the FedAvg algorithm. The last row $(-)$ presents results for the \\texttt{FedPeWS-Fixed} algorithm.} \n\n    \\resizebox{\\linewidth}{!}{ \n\n    \\begin{tabular}{r|cccccc}\n\n        \\toprule\n\n        & \\multicolumn{6}{c}{\\cellcolor{lightblue}\\textbf{Proportion of warmup rounds $\\tau = W/T$}} \\\\ \\cmidrule{2-7} \n\n        ${\\lambda} (\\downarrow) , \\tau (\\rightarrow)$ & $\\tau=0.0$ (FedAvg) & $\\tau=0.2$ & $\\tau=0.4$ & $\\tau=0.5$ & $\\tau=0.6$ & $\\tau=0.8$ \\\\ \\midrule \n\n        0.0 & \\multirow{12}{*}{$52.25\\pm0.57$} & $53.04\\pm2.08$ & $50.35\\pm0.61$ & $47.83\\pm1.04$ & $47.89\\pm1.17$ & $46.80\\pm1.76$ \\\\ \\cmidrule{3-7}\n\n        0.1 & & $52.52\\pm1.79$ & $51.72\\pm0.63$ & $49.83\\pm1.83$ & $48.83\\pm2.51$ & $47.50\\pm1.07$ \\\\\n\n        0.3 & & $52.74\\pm1.75$ & $54.36\\pm0.80$ & $51.07\\pm0.25$ & $50.94\\pm2.22$ & $50.62\\pm1.91$ \\\\\n\n        0.5 & & $54.30\\pm2.08$ & $54.42\\pm1.43$ & $52.02\\pm2.53$ & $51.43\\pm1.60$ & $48.97\\pm0.80$ \\\\\n\n        1.0 & & \\cellcolor{lightgreen} $\\mathbf{54.89\\pm0.72}$ & $53.59\\pm1.40$ & $51.49\\pm1.19$ & $50.29\\pm2.70$ & $52.21\\pm1.02$ \\\\\n\n        2.0 & & \\cellcolor{lightgreen} $\\mathbf{54.75\\pm1.12}$ & \\cellcolor{lightgreen} $\\mathbf{54.45\\pm1.78}$ & $53.02\\pm0.45$ & $52.26\\pm1.81$ & $52.49\\pm0.49$ \\\\ \n\n        5.0 & & \\cellcolor{lightgreen} $\\mathbf{54.91\\pm0.91}$ & \\cellcolor{lightgreen} $\\mathbf{54.99\\pm0.90}$ & $52.41\\pm0.45$ & $52.95\\pm1.49$ & $52.53\\pm1.35$ \\\\ \n\n        10.0 & & \\cellcolor{lightgreen} $\\mathbf{55.12\\pm1.16}$ & \\cellcolor{lightgreen} $\\mathbf{55.03\\pm1.39}$ & $52.85\\pm0.95$ & $50.82\\pm3.03$ & $52.27\\pm1.11$ \\\\ \n\n        100.0 & & $54.22\\pm1.74$ & $52.31\\pm2.55$ & $52.10\\pm0.87$ & $49.52\\pm4.34$ & $51.46\\pm2.92$ \\\\ \n\n        1000.0 & & $53.45\\pm1.65$ & $53.82\\pm2.16$ & $51.16\\pm1.92$ & $51.30\\pm2.02$ & $51.19\\pm1.05$ \\\\ \\cmidrule{3-7} \n\n        $-$ & & $53.69\\pm0.77$ & $51.78\\pm0.44$ & $50.24\\pm1.88$ & $51.12\\pm0.72$ & $49.87\\pm1.23$ \\\\ \n\n        \\bottomrule\n\n    \\end{tabular}\n\n    }\n\n    \\label{tab: pot-ablation}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[t]\n\n    \\centering\n\n    \\caption{The required number of collaboration rounds to reach target accuracy $\\upsilon$ $\\%$ and the final accuracy after $T$ rounds. The results are averaged over 3 seeds. \n\n    {\\color{red} $\\times$} indicates that the algorithm cannot reach target accuracy $\\upsilon$ within $T$ rounds and {\\color{red} NA} means that it reaches $\\upsilon$ only in one random seed. } \n\n    \\resizebox{\\linewidth}{!}{ \n\n        \\begin{tabular}{cr|ccccc} \n\n        \\toprule\n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Dataset $/$ Batch size}} & \\multicolumn{3}{c}{Synthetic-32K, $32$} & \\multicolumn{1}{c}{Synthetic-3.2K, $8$} \n\n        \\\\ \\midrule \n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Parameters} $\\{\\eta_g / \\lambda / \\tau\\}$} & $\\{1.0 / 5.0 / 0.125\\}$ & $\\{0.5/2.0/0.2\\}$ & $\\{0.25/1.0/0.1875\\}$ & \\multicolumn{1}{c}{$\\{0.1/2.0/0.1\\}$} \\\\ \\midrule\n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Target accuracy $\\upsilon (\\%)$}} & $99$ & $90$ & $75$ & $99$ \n\n        \\\\ \\midrule \n\n        \\multirow{2}{*}{\\parbox{3.5cm}{\\centering \\textbf{No. of rounds to \\\\ reach target accuracy}}} & FedAvg & $148\\pm3.79$ & \\color{red} \\cellcolor{lightred} $199\\pm \\text{NA}$ & \\color{red} \\cellcolor{lightred} $\\times$ & \\color{red} \\cellcolor{lightred} $371\\pm\\text{NA}$ \n\n        \\\\ \\cmidrule{2-6}\n\n         & \\texttt{FedPeWS} & \\cellcolor{lightgreen} $\\mathbf{115\\pm7.21}$ & \\cellcolor{lightgreen} $\\mathbf{182\\pm6.81}$ & \\cellcolor{lightgreen} $\\mathbf{286\\pm7.93}$ & \\cellcolor{lightgreen} $\\mathbf{301\\pm10.59}$ \n\n        \\\\ \\midrule \n\n        \\multirow{2}{*}{\\parbox{3.75cm}{\\centering \\textbf{Final accuracy after \\\\ $T$ collaboration rounds}}} & FedAvg & \\multicolumn{1}{c}{$99.94\\pm0.05$} & $91.40\\pm7.25$ & $67.64\\pm0.90$ & $97.33\\pm3.89$ \n\n        \\\\ \\cmidrule{2-6}\n\n        & \\texttt{FedPeWS} & \\multicolumn{1}{c}{\\cellcolor{lightgreen} $\\mathbf{99.96\\pm0.01}$} & \\cellcolor{lightgreen} $\\mathbf{99.49\\pm0.60}$ & \\cellcolor{lightgreen} $\\mathbf{83.50\\pm3.52}$ & \\cellcolor{lightgreen} $\\mathbf{99.66\\pm0.19}$ \n\n        \\\\ \\bottomrule \n\n        \\end{tabular}\n\n    }\n\n    \\label{table: convergence-table}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[t]\n\n    \\centering\n\n    \\caption{The required number of collaboration rounds to reach target accuracy $\\upsilon$ $\\%$ using FedProx algorithm and the final accuracy after $T$ rounds. The results are averaged over 3 seeds. \n\n    {\\color{red} $\\times$} indicates that the algorithm cannot reach target accuracy $\\upsilon$ within $T$ rounds.} \n\n    \\resizebox{\\linewidth}{!}{ \n\n        \\begin{tabular}{cr|ccccc} \n\n        \\toprule\n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Dataset $/$ Batch size}} & \\multicolumn{3}{c}{Synthetic-32K, $32$} & \\multicolumn{1}{c}{Synthetic-3.2K, $8$} \n\n        \\\\ \\midrule \n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Parameters} $\\{\\eta_g / \\lambda / \\tau\\}$} & $\\{1.0 / 0.1 / 0.125\\}$ & $\\{0.5/0.1/0.2\\}$ & $\\{0.25/1.0/0.1875\\}$ & \\multicolumn{1}{c}{$\\{0.1/1.0/0.1\\}$} \\\\ \\midrule\n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Target accuracy $\\upsilon (\\%)$}} & $99$ & $90$ & $75$ & $99$ \n\n        \\\\ \\midrule \n\n        \\multirow{2}{*}{\\parbox{3.5cm}{\\centering \\textbf{No. of rounds to \\\\ reach target accuracy}}} & FedProx & $138\\pm13.22$ & \\cellcolor{lightred} \\color{red} $\\times$ & \\cellcolor{lightred} \\color{red} $\\times$ & $362\\pm20.00$ \n\n        \\\\ \\cmidrule{2-6} \n\n         & \\texttt{FedPeWS} & \\cellcolor{lightgreen} $\\mathbf{115\\pm5.29}$ & \\cellcolor{lightgreen} $\\mathbf{211\\pm16.52}$ & \\cellcolor{lightgreen} $\\mathbf{314\\pm27.83}$ & \\cellcolor{lightgreen} $\\mathbf{344\\pm27.30}$ \n\n        \\\\ \\midrule \n\n        \\multirow{2}{*}{\\parbox{3.75cm}{\\centering \\textbf{Final accuracy after \\\\ $T$ collaboration rounds}}} & FedProx & \\multicolumn{1}{c}{$99.95\\pm0.02$} & $82.43\\pm7.98$ & $69.26\\pm6.03$ &\\cellcolor{lightgreen}  $\\mathbf{99.92\\pm0.06}$ \n\n        \\\\ \\cmidrule{2-6}\n\n        & \\texttt{FedPeWS} & \\multicolumn{1}{c}{\\cellcolor{lightgreen} $\\mathbf{99.96\\pm0.01}$} & \\cellcolor{lightgreen} $\\mathbf{98.40\\pm1.84}$ & \\cellcolor{lightgreen} $\\mathbf{90.40\\pm3.91}$ & \\cellcolor{lightgreen} $\\mathbf{99.92\\pm0.07}$ \n\n        \\\\ \\bottomrule \n\n        \\end{tabular}\n\n    }\n\n    \\label{table: convergence-table-fedprox}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[th]\n\n    \\centering\n\n    \\caption{Architecture for synthetic dataset models used in the experiments.}\n\n    \\resizebox{0.75\\linewidth}{!}{ \n\n        \\begin{tabular}{lllllll} \n\n            \\toprule \n\n            \\cellcolor{lightblue}\\textbf{Layer} & Input & FC1 & FC2 & FC3 & FC4 & FC5 \\\\ \\midrule \n\n            \\cellcolor{lightblue}\\textbf{Dimensions} & [5] & [5, 32] & [32, 64] & [64, 128] & [128, 32] & [32, 4] \\\\ \\bottomrule\n\n        \\end{tabular}\n\n    }\n\n    \\label{tab: synthetic-arch}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[H]\n\n    \\centering\n\n    \\caption{Architecture for CIFAR-MNIST dataset models. Every convolutional layer is followed by a max pooling layer with kernel size 2 and stride 2. }\n\n    \\resizebox{\\linewidth}{!}{ \n\n        \\begin{tabular}{lllllll}\n\n            \\toprule\n\n             \\cellcolor{lightblue}\\textbf{Layer} & Input & Conv1 & Conv2 & Conv3 & Flatten & FC \\\\ \\midrule \n\n             \\cellcolor{lightblue}\\textbf{Dimensions} & [3, 32, 32] & [3, 32, 3, 3] & [32, 64, 3, 3] & [64, 128, 3, 3] & [2048] & [2048, 20] \\\\ \\bottomrule \n\n        \\end{tabular}\n\n    }\n\n    \\label{tab: conv-arch}\n\n\\end{table}"}
{"table/equation": "\\begin{equation}\n\n    \\begin{split}\n\n        x^{\\star} \\leftarrow  \\argmin_{x \\in \\mathbb{R}^d} \\Bigg[ f(x) \\coloneqq \\frac{1}{N} \\sum_{i=1}^N f_i(x) \\Bigg], \n\n        \\label{eq: optimization-problem}\n\n    \\end{split}\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    x_{i}^{t,k} = x_{i}^{t,k-1} - \\eta_{\\ell} \\nabla f_i\\big(x_i^{t,k-1}, \\xi_i^{t,k-1}\\big), \\quad x_{i}^{t} = x_{i}^{t,K}, \\text{ and  } x_g^{t} = x_g^{t-1} - \\eta_g (x_g^{t-1} - \\frac{1}{N} \\sum_{i=1}^N x_{i}^{t}) , \n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq: mask learning}\n\n    \\min_{x, \\{m_i\\}_{i\\in[N]}} \\frac{1}{N} \\sum_{i=1}^N f_i(x \\odot m_i),\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    x_{i}^{t,k} = x_{i}^{t,k-1} - \\eta_{\\ell} \\nabla f_i\\big(x_i^{t,k-1} \\odot m_i^t, \\xi_i^{t,k-1}\\big) \\; \\text{and} \\; x_g^{t} = x_g^{t-1} - \\eta_g (x_g^{t-1} - \\frac{\\sum_{i \\in [N]}x_{i}^{t} \\odot m_i^t}{\\sum_{i \\in [N]}m_i^t}).\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\label{eq: mask score learning}\n\n    \\min_{x, \\{s_i\\}_{i\\in[N]}} \\frac{1}{N} \\sum_{i=1}^N f_i(x \\odot \\mathcal{G}(s_i)).\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\mathcal{L}_s = f_i\\Big(x_i^{t,k} \\odot \\mathcal{G}\\left(s_i^{t,k}\\right), \\xi_i^{t,k-1}\\Big) - \\lambda \\|\\sigma\\left(s_{i}^{t,k}\\right) - \\theta_{g \\backslash \\{i\\}}^{t}\\|_2^2; \\qquad s_{i}^{t,k+1} \\gets s_{i}^{t,k} - \\eta_s \\nabla_{s} \\mathcal{L}_s, \n\n    \\label{eq: mask-objective-func}\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\|\\sigma\\left(s_{i}^{t,k}\\right) - \\theta_{g \\backslash \\{i\\}}^{t}\\|_2^2 \\leq h. \n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n    \\mathcal{L}_x = f_i\\Big(x_i^{t,k} \\odot \\mathcal{G}\\left(s_i^{t,k}\\right), \\xi_i^{t,k-1}\\Big); \\qquad x_{i}^{t,k+1} \\gets x_{i}^{t,k} - \\eta_{\\ell} \\nabla_{x} \\mathcal{L}_x,\n\n\\end{equation}"}
{"table/equation": "\\begin{table}[t]\n\n    \\centering\n\n    \\caption{Ablation study of the parameters $\\lambda$ and $\\tau$ on the CIFAR-MNIST dataset with $N=2$ participants over three independent runs. The first column ($\\tau=0.0$) corresponds to the FedAvg algorithm. The last row $(-)$ presents results for the \\texttt{FedPeWS-Fixed} algorithm.} \n\n    \\resizebox{\\linewidth}{!}{ \n\n    \\begin{tabular}{r|cccccc}\n\n        \\toprule\n\n        & \\multicolumn{6}{c}{\\cellcolor{lightblue}\\textbf{Proportion of warmup rounds $\\tau = W/T$}} \\\\ \\cmidrule{2-7} \n\n        $\\mathbf{\\lambda} (\\downarrow) , \\tau (\\rightarrow)$ & $\\tau=0.0$ (FedAvg) & $\\tau=0.2$ & $\\tau=0.4$ & $\\tau=0.5$ & $\\tau=0.6$ & $\\tau=0.8$ \\\\ \\midrule \n\n        0.0 & \\multirow{12}{*}{$71.23 \\pm 0.71$} & $68.01 \\pm 0.88$ & $66.01\\pm0.48$ & $65.96\\pm1.03$ & $65.40\\pm1.95$ & $65.77\\pm0.41$ \\\\ \\cmidrule{3-7}\n\n        0.1 & & $68.77\\pm1.09$ & $70.39\\pm1.00$ & $70.61\\pm1.17$ & $69.48\\pm0.37$ & $70.36\\pm2.06$ \\\\\n\n        0.3 & & $70.86\\pm0.23$ & $73.00\\pm0.65$ & $73.91\\pm0.71$ & $73.26\\pm0.46$ & $73.02\\pm1.05$ \\\\\n\n        0.5 & & $71.43\\pm0.56$ & $74.17\\pm0.91$ & $73.84\\pm0.12$ & $73.66\\pm0.88$ & $75.05\\pm0.45$ \\\\\n\n        1.0 & & $72.26\\pm0.54$ & $74.46\\pm0.44$ & $74.54\\pm0.76$ & $74.91\\pm0.42$ & $73.81\\pm0.87$ \\\\\n\n        2.0 & & $72.61\\pm0.79$ & $73.68\\pm0.17$ & \\cellcolor{lightgreen} $\\mathbf{75.35\\pm0.50}$ & $74.76\\pm0.54$ & $74.46\\pm0.56$ \\\\\n\n        5.0 & & $72.60\\pm0.45$ & \\cellcolor{lightgreen} $\\mathbf{75.22\\pm0.33}$ & \\cellcolor{lightgreen} $\\mathbf{75.00\\pm0.74}$ & $75.01\\pm0.71$ & $73.96\\pm1.60$ \\\\\n\n        10.0 & & $72.29\\pm0.48$ & \\cellcolor{lightgreen} $\\mathbf{74.97\\pm0.65}$ & $74.31\\pm0.95$ & $74.03\\pm0.30$ & $71.91\\pm2.69$ \\\\\n\n        100.0 & & $71.64\\pm0.47$ & $72.92\\pm0.39$ & $73.96\\pm0.65$ & $73.13\\pm0.73$ & $72.43\\pm3.55$ \\\\\n\n        1000.0 & & $71.58\\pm0.53$ & $73.18\\pm0.73$ & $73.32\\pm1.70$ & $73.87\\pm1.16$ & $72.52\\pm1.92$ \\\\ \\cmidrule{3-7} \n\n        $-$ & & $72.72\\pm0.44$ & \\cellcolor{lightgreen} $\\mathbf{75.22\\pm0.19}$ & \\cellcolor{lightgreen} $\\mathbf{75.05\\pm0.42}$ & $72.51\\pm3.89$ & $73.77\\pm0.20$ \\\\ \n\n        \\bottomrule\n\n    \\end{tabular}\n\n    }\n\n    \\label{tab: cifar-mnist-ablation}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[H]\n\n    \\centering\n\n    \\caption{Comparison of approaches for handling data heterogeneity in federated learning.} \n\n    \\resizebox{0.8\\linewidth}{!}{ \n\n        \\begin{tabular}{lcccc}\n\n            \\toprule\n\n             & \\multirow{2}{*}{\\parbox{2.1cm}{\\centering \\textbf{Shared \\\\ Global Model}}} & \\multirow{2}{*}{\\parbox{2.5cm}{\\centering \\textbf{Level of Mask \\\\ Personalization}}} & \\multirow{2}{*}{\\parbox{2cm}{\\centering {\\textbf{Learnable \\\\ Mask}}}} & \\multirow{2}{*}{\\parbox{2cm}{\\centering {\\textbf{Learnable \\\\ Parameters}}}} \\\\ \n\n             & & & & \\\\ \\midrule \n\n             FedPM \\citep{isik2022sparse} & \\checkmark & parameter-level & \\checkmark & $\\bm{\\times}$ \\\\ \n\n             IST \\citep{yuan2022distributed} & \\checkmark & neuron-level & $\\bm{\\times}$ (random) & \\checkmark \\\\ \n\n             LTH \\citep{frankle2018lottery} & $\\bm{\\times}$ & parameter-level & $\\bm{\\times}$ (pruned) & \\checkmark \\\\ \n\n             FedWeIT \\citep{yoon2021fedweit} & $\\bm{\\times}$ & parameter-level & \\checkmark & \\checkmark \\\\ \n\n             FjORD \\citep{horvath2021fjord} & \\checkmark & parameter-level & $\\bm{\\times}$ (slimmed) & \\checkmark \\\\ \\midrule \n\n             \\rowcolor{lightgreen} \\texttt{FedPeWS} (ours) & \\checkmark & neuron-level & \\checkmark & \\checkmark \\\\ \\bottomrule \n\n        \\end{tabular}\n\n    }\n\n    \\label{tab: methods-comparison}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[b]\n\n    \\centering\n\n    \\caption{Ablation study of the parameters $\\lambda$ and $\\tau$ on the combination of \\{Path-OCT-Tissue\\}MNIST datasets with $N=3$ participants over three independent runs. The first column ($\\tau=0.0$) corresponds to the FedAvg algorithm. The last row $(-)$ presents results for the \\texttt{FedPeWS-Fixed} algorithm.} \n\n    \\resizebox{\\linewidth}{!}{ \n\n    \\begin{tabular}{r|cccccc}\n\n        \\toprule\n\n        & \\multicolumn{6}{c}{\\cellcolor{lightblue}\\textbf{Proportion of warmup rounds $\\tau = W/T$}} \\\\ \\cmidrule{2-7} \n\n        ${\\lambda} (\\downarrow) , \\tau (\\rightarrow)$ & $\\tau=0.0$ (FedAvg) & $\\tau=0.2$ & $\\tau=0.4$ & $\\tau=0.5$ & $\\tau=0.6$ & $\\tau=0.8$ \\\\ \\midrule \n\n        0.0 & \\multirow{12}{*}{$52.25\\pm0.57$} & $53.04\\pm2.08$ & $50.35\\pm0.61$ & $47.83\\pm1.04$ & $47.89\\pm1.17$ & $46.80\\pm1.76$ \\\\ \\cmidrule{3-7}\n\n        0.1 & & $52.52\\pm1.79$ & $51.72\\pm0.63$ & $49.83\\pm1.83$ & $48.83\\pm2.51$ & $47.50\\pm1.07$ \\\\\n\n        0.3 & & $52.74\\pm1.75$ & $54.36\\pm0.80$ & $51.07\\pm0.25$ & $50.94\\pm2.22$ & $50.62\\pm1.91$ \\\\\n\n        0.5 & & $54.30\\pm2.08$ & $54.42\\pm1.43$ & $52.02\\pm2.53$ & $51.43\\pm1.60$ & $48.97\\pm0.80$ \\\\\n\n        1.0 & & \\cellcolor{lightgreen} $\\mathbf{54.89\\pm0.72}$ & $53.59\\pm1.40$ & $51.49\\pm1.19$ & $50.29\\pm2.70$ & $52.21\\pm1.02$ \\\\\n\n        2.0 & & \\cellcolor{lightgreen} $\\mathbf{54.75\\pm1.12}$ & \\cellcolor{lightgreen} $\\mathbf{54.45\\pm1.78}$ & $53.02\\pm0.45$ & $52.26\\pm1.81$ & $52.49\\pm0.49$ \\\\ \n\n        5.0 & & \\cellcolor{lightgreen} $\\mathbf{54.91\\pm0.91}$ & \\cellcolor{lightgreen} $\\mathbf{54.99\\pm0.90}$ & $52.41\\pm0.45$ & $52.95\\pm1.49$ & $52.53\\pm1.35$ \\\\ \n\n        10.0 & & \\cellcolor{lightgreen} $\\mathbf{55.12\\pm1.16}$ & \\cellcolor{lightgreen} $\\mathbf{55.03\\pm1.39}$ & $52.85\\pm0.95$ & $50.82\\pm3.03$ & $52.27\\pm1.11$ \\\\ \n\n        100.0 & & $54.22\\pm1.74$ & $52.31\\pm2.55$ & $52.10\\pm0.87$ & $49.52\\pm4.34$ & $51.46\\pm2.92$ \\\\ \n\n        1000.0 & & $53.45\\pm1.65$ & $53.82\\pm2.16$ & $51.16\\pm1.92$ & $51.30\\pm2.02$ & $51.19\\pm1.05$ \\\\ \\cmidrule{3-7} \n\n        $-$ & & $53.69\\pm0.77$ & $51.78\\pm0.44$ & $50.24\\pm1.88$ & $51.12\\pm0.72$ & $49.87\\pm1.23$ \\\\ \n\n        \\bottomrule\n\n    \\end{tabular}\n\n    }\n\n    \\label{tab: pot-ablation}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[t]\n\n    \\centering\n\n    \\caption{The required number of collaboration rounds to reach target accuracy $\\upsilon$ $\\%$ and the final accuracy after $T$ rounds. The results are averaged over 3 seeds. \n\n    {\\color{red} $\\times$} indicates that the algorithm cannot reach target accuracy $\\upsilon$ within $T$ rounds and {\\color{red} NA} means that it reaches $\\upsilon$ only in one random seed. } \n\n    \\resizebox{\\linewidth}{!}{ \n\n        \\begin{tabular}{cr|ccccc} \n\n        \\toprule\n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Dataset $/$ Batch size}} & \\multicolumn{3}{c}{Synthetic-32K, $32$} & \\multicolumn{1}{c}{Synthetic-3.2K, $8$} \n\n        \\\\ \\midrule \n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Parameters} $\\{\\eta_g / \\lambda / \\tau\\}$} & $\\{1.0 / 5.0 / 0.125\\}$ & $\\{0.5/2.0/0.2\\}$ & $\\{0.25/1.0/0.1875\\}$ & \\multicolumn{1}{c}{$\\{0.1/2.0/0.1\\}$} \\\\ \\midrule\n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Target accuracy $\\upsilon (\\%)$}} & $99$ & $90$ & $75$ & $99$ \n\n        \\\\ \\midrule \n\n        \\multirow{2}{*}{\\parbox{3.5cm}{\\centering \\textbf{No. of rounds to \\\\ reach target accuracy}}} & FedAvg & $148\\pm3.79$ & \\color{red} \\cellcolor{lightred} $199\\pm \\text{NA}$ & \\color{red} \\cellcolor{lightred} $\\times$ & \\color{red} \\cellcolor{lightred} $371\\pm\\text{NA}$ \n\n        \\\\ \\cmidrule{2-6}\n\n         & \\texttt{FedPeWS} & \\cellcolor{lightgreen} $\\mathbf{115\\pm7.21}$ & \\cellcolor{lightgreen} $\\mathbf{182\\pm6.81}$ & \\cellcolor{lightgreen} $\\mathbf{286\\pm7.93}$ & \\cellcolor{lightgreen} $\\mathbf{301\\pm10.59}$ \n\n        \\\\ \\midrule \n\n        \\multirow{2}{*}{\\parbox{3.75cm}{\\centering \\textbf{Final accuracy after \\\\ $T$ collaboration rounds}}} & FedAvg & \\multicolumn{1}{c}{$99.94\\pm0.05$} & $91.40\\pm7.25$ & $67.64\\pm0.90$ & $97.33\\pm3.89$ \n\n        \\\\ \\cmidrule{2-6}\n\n        & \\texttt{FedPeWS} & \\multicolumn{1}{c}{\\cellcolor{lightgreen} $\\mathbf{99.96\\pm0.01}$} & \\cellcolor{lightgreen} $\\mathbf{99.49\\pm0.60}$ & \\cellcolor{lightgreen} $\\mathbf{83.50\\pm3.52}$ & \\cellcolor{lightgreen} $\\mathbf{99.66\\pm0.19}$ \n\n        \\\\ \\bottomrule \n\n        \\end{tabular}\n\n    }\n\n    \\label{table: convergence-table}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[t]\n\n    \\centering\n\n    \\caption{The required number of collaboration rounds to reach target accuracy $\\upsilon$ $\\%$ using FedProx algorithm and the final accuracy after $T$ rounds. The results are averaged over 3 seeds. \n\n    {\\color{red} $\\times$} indicates that the algorithm cannot reach target accuracy $\\upsilon$ within $T$ rounds.} \n\n    \\resizebox{\\linewidth}{!}{ \n\n        \\begin{tabular}{cr|ccccc} \n\n        \\toprule\n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Dataset $/$ Batch size}} & \\multicolumn{3}{c}{Synthetic-32K, $32$} & \\multicolumn{1}{c}{Synthetic-3.2K, $8$} \n\n        \\\\ \\midrule \n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Parameters} $\\{\\eta_g / \\lambda / \\tau\\}$} & $\\{1.0 / 0.1 / 0.125\\}$ & $\\{0.5/0.1/0.2\\}$ & $\\{0.25/1.0/0.1875\\}$ & \\multicolumn{1}{c}{$\\{0.1/1.0/0.1\\}$} \\\\ \\midrule\n\n        \\rowcolor{lightblue} \\multicolumn{2}{r}{\\textbf{Target accuracy $\\upsilon (\\%)$}} & $99$ & $90$ & $75$ & $99$ \n\n        \\\\ \\midrule \n\n        \\multirow{2}{*}{\\parbox{3.5cm}{\\centering \\textbf{No. of rounds to \\\\ reach target accuracy}}} & FedProx & $138\\pm13.22$ & \\cellcolor{lightred} \\color{red} $\\times$ & \\cellcolor{lightred} \\color{red} $\\times$ & $362\\pm20.00$ \n\n        \\\\ \\cmidrule{2-6} \n\n         & \\texttt{FedPeWS} & \\cellcolor{lightgreen} $\\mathbf{115\\pm5.29}$ & \\cellcolor{lightgreen} $\\mathbf{211\\pm16.52}$ & \\cellcolor{lightgreen} $\\mathbf{314\\pm27.83}$ & \\cellcolor{lightgreen} $\\mathbf{344\\pm27.30}$ \n\n        \\\\ \\midrule \n\n        \\multirow{2}{*}{\\parbox{3.75cm}{\\centering \\textbf{Final accuracy after \\\\ $T$ collaboration rounds}}} & FedProx & \\multicolumn{1}{c}{$99.95\\pm0.02$} & $82.43\\pm7.98$ & $69.26\\pm6.03$ &\\cellcolor{lightgreen}  $\\mathbf{99.92\\pm0.06}$ \n\n        \\\\ \\cmidrule{2-6}\n\n        & \\texttt{FedPeWS} & \\multicolumn{1}{c}{\\cellcolor{lightgreen} $\\mathbf{99.96\\pm0.01}$} & \\cellcolor{lightgreen} $\\mathbf{98.40\\pm1.84}$ & \\cellcolor{lightgreen} $\\mathbf{90.40\\pm3.91}$ & \\cellcolor{lightgreen} $\\mathbf{99.92\\pm0.07}$ \n\n        \\\\ \\bottomrule \n\n        \\end{tabular}\n\n    }\n\n    \\label{table: convergence-table-fedprox}\n\n\\end{table}"}
{"table/equation": "\\begin{table}[h!]\n\n\\centering\n\n\\caption{\\label{baro-table-1}\n\nComparison of Young's moduli calculated with three distinct methods\n\nfor single-layer graphene (SLG) as well as carbon nanomembranes 1 \\& 2, \n\ncompare Figs.~\\xref{graphene} and \\xref{CNM}, for temperatures in the range of 50~K to 350~K.\n\nAll values are in GPa.} \n\n\\begin{tabular}{|l|c|c|c|} %  \n\n\\hline\n\n\\textbf{Method} & \\textbf{SLG} & \\textbf{CNM1} & \\textbf{CNM2} \\\\\n\n\\hline\n\nScaling           &     $1088\\pm11$        &    $240\\pm5$         &     $253\\pm3$  \\\\\n\n\\hline\n\nStress-Strain     &   $854\\pm5.2$          &     $35.835\\pm0.835$        & $35.71\\pm5.13$     \\\\\n\n\\hline\n\nBarostated dyn., PC in z &     $854\\pm23$        &     $33\\pm5$        &  $31\\pm3$   \\\\\n\n\\hline\n\nBarostated dyn., without PC in z&   $854\\pm19$          &  $32\\pm7$ & $28\\pm3$            \\\\\n\n\\hline\n\n\\end{tabular}\n\n\\label{table}\n\n\\end{table}"}
{"table/equation": "\\begin{equation}\n\n\\label{E-2-2}\n\nE_\\alpha = \\frac{1}{V_0} \\left(\\frac{\\partial^2 U}{\\partial \\alpha^2}\\right)_{\\alpha=1}\n\n\\ ,\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n\\label{E-2-2b}\n\n\\vec{x}_i \\rightarrow \\vec{x}_i + (\\alpha - 1) \\\n\n\\vec{e}_{\\alpha}\\cdot\\vec{x}_i \\ \\vec{e}_{\\alpha}\n\n\\ .\n\n\\end{equation}"}
{"table/equation": "\\begin{table}[h!]\n\n\\centering\n\n\\caption{\\label{baro-table-1}\n\nComparison of Young's moduli calculated with three distinct methods\n\nfor single-layer graphene (SLG) as well as carbon nanomembranes 1 \\& 2, \n\ncompare Figs.~\\xref{graphene} and \\xref{CNM}, for temperatures in the range of 50~K to 350~K.\n\nAll values are in GPa.} \n\n\\begin{tabular}{|l|c|c|c|} %  \n\n\\hline\n\n\\textbf{Method} & \\textbf{SLG} & \\textbf{CNM1} & \\textbf{CNM2} \\\\\n\n\\hline\n\nScaling           &     $1088\\pm11$        &    $240\\pm5$         &     $253\\pm3$  \\\\\n\n\\hline\n\nStress-Strain     &   $854\\pm5.2$          &     $35.835\\pm0.835$        & $35.71\\pm5.13$     \\\\\n\n\\hline\n\nBarostated dyn., PC in z &     $854\\pm23$        &     $33\\pm5$        &  $31\\pm3$   \\\\\n\n\\hline\n\nBarostated dyn., without PC in z&   $854\\pm19$          &  $32\\pm7$ & $28\\pm3$            \\\\\n\n\\hline\n\n\\end{tabular}\n\n\\label{table}\n\n\\end{table}"}
{"table/equation": "\\begin{equation}\n\n\\label{E-2-2}\n\nE_\\alpha = \\frac{1}{V_0} \\left(\\frac{\\partial^2 U}{\\partial \\alpha^2}\\right)_{\\alpha=1}\n\n\\ ,\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n\\label{E-2-2b}\n\n\\vec{x}_i \\rightarrow \\vec{x}_i + (\\alpha - 1) \\\n\n\\vec{e}_{\\alpha}\\cdot\\vec{x}_i \\ \\vec{e}_{\\alpha}\n\n\\ .\n\n\\end{equation}"}
{"table/equation": "\\begin{table}[h!]\n\n\\centering\n\n\\caption{\\label{baro-table-1}\n\nComparison of Young's moduli calculated with three distinct methods\n\nfor single-layer graphene (SLG) as well as carbon nanomembranes 1 \\& 2, \n\ncompare Figs.~\\xref{graphene} and \\xref{CNM}, for temperatures in the range of 50~K to 350~K.\n\nAll values are in GPa.} \n\n\\begin{tabular}{|l|c|c|c|} %  \n\n\\hline\n\n\\textbf{Method} & \\textbf{SLG} & \\textbf{CNM1} & \\textbf{CNM2} \\\\\n\n\\hline\n\nScaling           &     $1088\\pm11$        &    $240\\pm5$         &     $253\\pm3$  \\\\\n\n\\hline\n\nStress-Strain     &   $854\\pm5.2$          &     $35.835\\pm0.835$        & $35.71\\pm5.13$     \\\\\n\n\\hline\n\nBarostated dyn., PC in z &     $854\\pm23$        &     $33\\pm5$        &  $31\\pm3$   \\\\\n\n\\hline\n\nBarostated dyn., without PC in z&   $854\\pm19$          &  $32\\pm7$ & $28\\pm3$            \\\\\n\n\\hline\n\n\\end{tabular}\n\n\\label{table}\n\n\\end{table}"}
{"table/equation": "\\begin{equation}\n\n\\label{E-2-2}\n\nE_\\alpha = \\frac{1}{V_0} \\left(\\frac{\\partial^2 U}{\\partial \\alpha^2}\\right)_{\\alpha=1}\n\n\\ ,\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n\\label{E-2-2b}\n\n\\vec{x}_i \\rightarrow \\vec{x}_i + (\\alpha - 1) \\\n\n\\vec{e}_{\\alpha}\\cdot\\vec{x}_i \\ \\vec{e}_{\\alpha}\n\n\\ .\n\n\\end{equation}"}
{"table/equation": "\\begin{table}[h!]\n\n\\centering\n\n\\caption{\\label{baro-table-1}\n\nComparison of Young's moduli calculated with three distinct methods\n\nfor single-layer graphene (SLG) as well as carbon nanomembranes 1 \\& 2, \n\ncompare Figs.~\\xref{graphene} and \\xref{CNM}, for temperatures in the range of 50~K to 350~K.\n\nAll values are in GPa.} \n\n\\begin{tabular}{|l|c|c|c|} %  \n\n\\hline\n\n\\textbf{Method} & \\textbf{SLG} & \\textbf{CNM1} & \\textbf{CNM2} \\\\\n\n\\hline\n\nScaling           &     $1088\\pm11$        &    $240\\pm5$         &     $253\\pm3$  \\\\\n\n\\hline\n\nStress-Strain     &   $854\\pm5.2$          &     $35.835\\pm0.835$        & $35.71\\pm5.13$     \\\\\n\n\\hline\n\nBarostated dyn., PC in z &     $854\\pm23$        &     $33\\pm5$        &  $31\\pm3$   \\\\\n\n\\hline\n\nBarostated dyn., without PC in z&   $854\\pm19$          &  $32\\pm7$ & $28\\pm3$            \\\\\n\n\\hline\n\n\\end{tabular}\n\n\\label{table}\n\n\\end{table}"}
{"table/equation": "\\begin{equation}\n\n\\label{E-2-2}\n\nE_\\alpha = \\frac{1}{V_0} \\left(\\frac{\\partial^2 U}{\\partial \\alpha^2}\\right)_{\\alpha=1}\n\n\\ ,\n\n\\end{equation}"}
{"table/equation": "\\begin{equation}\n\n\\label{E-2-2b}\n\n\\vec{x}_i \\rightarrow \\vec{x}_i + (\\alpha - 1) \\\n\n\\vec{e}_{\\alpha}\\cdot\\vec{x}_i \\ \\vec{e}_{\\alpha}\n\n\\ .\n\n\\end{equation}"}