aten/src/ATen/native/Linear.cpp

#include <ATen/ATen.h>
#include <ATen/native/Resize.h>
#include <ATen/NativeFunctions.h>
#include <ATen/native/xnnpack/Engine.h>
#include <ATen/WrapDimUtilsMulti.h>
#include <c10/macros/Macros.h>
#include <c10/util/irange.h>
#include <c10/util/MaybeOwned.h>

#include <array>
#include <cctype>
#include <cstddef>
#include <sstream>
#include <string>
#include <vector>

namespace at { namespace native {

Tensor linear(const Tensor& input, const Tensor& weight, const c10::optional<Tensor>& bias_opt) {
  // See [Note: hacky wrapper removal for optional tensor]
  auto bias = bias_opt.has_value()
    ? c10::MaybeOwned<Tensor>::borrowed(*bias_opt)
    : c10::MaybeOwned<Tensor>::owned(c10::in_place);

  if (input.is_mkldnn()) {
    return at::mkldnn_linear(input, weight, *bias);
  }
#if defined(C10_MOBILE)
  if (xnnpack::use_linear(input, weight, *bias)) {
    return xnnpack::linear(input, weight, *bias);
  }
#endif
  if (input.dim() == 2 && bias->defined()) {
    // Fused op is marginally faster.
    return at::addmm(*bias, input, weight.t());
  }
  auto output = at::matmul(input, weight.t());
  if (bias->defined()) {
    output.add_(*bias);
  }
  return output;
}

// sumproduct_pair computes `(left*right).sum(sumdims)` by means of permutation and
// batch matrix multiplication
// its main purpose is to provide a pairwise reduction for einsum
static Tensor sumproduct_pair(const Tensor& left_, const Tensor& right_, IntArrayRef sum_dims_, bool keepdim) {
  // assumes that tensors have been pre-unsqueezed (so that all dimensions match - after broadcasting)
  // but makes no other assumptions on the order of dimensions
  TORCH_CHECK(left_.dim()==right_.dim(), "number of dimensions must match");
  if (sum_dims_.size() == 0)
    return at::mul(left_, right_);
  int64_t dim = left_.dim();
  auto sum_dims = at::dim_list_to_bitset(sum_dims_, dim);
  // dimensions that will be part of the output (i.e. not summed over) in three vectors
  // dims in lro appear in left, right and output, similarly lo: left and output, ro: right and output
  // also the sizes are kept track of for reshaping
  std::vector<int64_t> lro, lo, ro;
  int64_t lro_size = 1, lo_size = 1, ro_size = 1, sum_size = 1;
  Tensor left = left_;
  Tensor right = right_;
  for (const auto i : c10::irange(dim)) {
    auto sl = left.size(i)>1;
    auto sr = right.size(i)>1;
    if (sum_dims[i]) { // first dimensions that will be summed over after multiplication
      if (sl && sr) {  // dimensions nontrivially in both left and right must be of the same size
        TORCH_CHECK(left.size(i)==right.size(i), "non-broadcast dimensions must match");
        sum_size *= left.size(i);
      } else if (sl) { // if it is only in one of left and right, we can sum right away
        left = left.sum(i, true);
      } else if (sr) {
        right = right.sum(i, true);
      }
    } else if (sl && sr) { // now deal with dimensions  dimensions that will be in the output
      // dimensions nontrivially in both left and right must be of the same size
      TORCH_CHECK(left.size(i)==right.size(i), "non-broadcast dimensions must match");
      lro.push_back(i);
      lro_size *= left.size(i);
    } else if (sl) { // keep track of dimensions appearing only once
      lo.push_back(i);
      lo_size *= left.size(i);
    } else {
      ro.push_back(i);
      ro_size *= right.size(i);
    }
  }
  // we now work with the following permutations / shapes.
  // the pipeline is permute inputs -> reshape inputs -> batch matrix mul -> reshape(view) output -> permute output
  // output: "lro, lo, 1-for-summed-dims, ro" with orgiginal shape dimensions
  // left:   "lro, lo, summed" permuted with lpermutation and the three flattened
  // right:  "lro, summed, ro" permuted with rpermutation and the three flattened
  // then the permuted output is a view of bmm(left, right)
  // finally, opermutation reverts the permutation to the original order of dimensions
  std::vector<int64_t> out_size;
  // NOLINTNEXTLINE(performance-inefficient-vector-operation)
  for (auto& d : lro) out_size.push_back(left.size(d));
  for (auto& d : lo) out_size.push_back(left.size(d));
  for (auto& d : sum_dims_) { out_size.push_back(1); (void)(d); }; // avoid warining about not using d
  for (auto& d : ro) out_size.push_back(right.size(d));

  std::vector<int64_t> lpermutation(lro);
  lpermutation.insert(lpermutation.end(), lo.begin(), lo.end());
  lpermutation.insert(lpermutation.end(), sum_dims_.begin(), sum_dims_.end());
  lpermutation.insert(lpermutation.end(), ro.begin(), ro.end());

  std::vector<int64_t> rpermutation(lro);
  rpermutation.insert(rpermutation.end(), sum_dims_.begin(), sum_dims_.end());
  rpermutation.insert(rpermutation.end(), ro.begin(), ro.end());
  rpermutation.insert(rpermutation.end(), lo.begin(), lo.end());

  std::vector<int64_t> opermutation(lro.size()+lo.size()+sum_dims_.size()+ro.size(), -1);
  {
    int64_t i = 0;

    for (auto it = lro.cbegin(); it != lro.cend(); i++, it++) {
      opermutation[*it] = i;
    }
    for (auto it = lo.cbegin(); it != lo.cend(); i++, it++) {
      opermutation[*it] = i;
    }
    for (auto it = sum_dims_.cbegin(); it != sum_dims_.cend(); i++, it++) {
      opermutation[*it] = i;
    }
    for (auto it = ro.cbegin(); it != ro.cend(); i++, it++) {
      opermutation[*it] = i;
    }
  }

  // now we can execute the operations above
  left = left.permute(lpermutation).reshape({lro_size, lo_size, sum_size});
  right = right.permute(rpermutation).reshape({lro_size, sum_size, ro_size});
  Tensor result = at::bmm(left, right);
  result = result.view(out_size).permute(opermutation);

  // finally squeeze summed dimensions if desired
  if (! keepdim) {
    auto sizes = result.sizes().vec();
    // NOLINTNEXTLINE(bugprone-narrowing-conversions,cppcoreguidelines-narrowing-conversions)
    for (int i = dim-1; i>=0; i--) {
      if (sum_dims[i]) {
        sizes.erase(sizes.begin() + i);
      }
    }
    result = result.view(sizes);
  }
  return result;
}

namespace {

bool einsum_check_label(unsigned char label) {
  return std::isalpha(label);
}

uint8_t einsum_label_to_index(unsigned char label) {
  constexpr uint8_t NUM_OF_LETTERS = 'z' - 'a' + 1;
  return std::isupper(label) ? label - 'A' : NUM_OF_LETTERS + (label - 'a');
}

unsigned char einsum_index_to_label(uint8_t index) {
  constexpr uint8_t NUM_OF_LETTERS = 'z' - 'a' + 1;
  return index < NUM_OF_LETTERS ? index + 'A' : index - NUM_OF_LETTERS + 'a';
}

} // namespace

// There are roughly three parts to compute einsum:
// 1. Parse equation to extract the labels for each input operand and output
// 2. Unsqueeze missing dimensions from input operands and permute to align them
// 3. Compute result by multiplying input operands and summing contraction
//    dimensions We do the last part by reducing to bmm.
Tensor einsum(c10::string_view equation, TensorList operands) {
  TORCH_CHECK(!operands.empty(), "einsum(): must provide at least one operand");

  // Code used to identify ELLIPSIS ("...")
  constexpr uint8_t ELLIPSIS = 52;

  // Find arrow (->) to split equation into lhs and rhs
  const auto arrow_pos = equation.find("->");
  const auto lhs = equation.substr(0, arrow_pos);

  const auto num_ops = operands.size();

  // Convert labels for input operands into an index in [0, 52) and store
  // them in op_labels for each operand along with ELLIPSIS if present.
  std::vector<std::vector<uint8_t>> op_labels(num_ops);
  bool found_ell = false;
  std::size_t curr_op = 0;
  for (auto i = decltype(lhs.length()){0}; i < lhs.length(); ++i) {
    const unsigned char label = lhs[i];
    switch (label) {
      case ' ':
        // Ignore spaces
        break;

      case '.':
        TORCH_CHECK(
            // Only one ellipsis per operand can be given
            !found_ell,
            "einsum(): found \'.\' for operand ",
            curr_op,
            " for which an ellipsis was already found");
        TORCH_CHECK(
            // Ensure it's a valid ellipsis
            i + 2 < lhs.length() && lhs[++i] == '.' && lhs[++i] == '.',
            "einsum(): found \'.\' for operand ",
            curr_op,
            " that is not part of any ellipsis");
        op_labels[curr_op].push_back(ELLIPSIS);
        found_ell = true;
        break;

      case ',':
        // Move onto next operand
        ++curr_op;
        TORCH_CHECK(
            curr_op < num_ops,
            "einsum(): fewer operands were provided than specified in the equation");
        found_ell = false;
        break;

      default:
        // Parse label
        TORCH_CHECK(
            einsum_check_label(label),
            "einsum(): invalid subscript given at index ",
            i,
            " in the equation string, subscripts must be in [a-zA-Z]");
        op_labels[curr_op].push_back(einsum_label_to_index(label));
    }
  }

  TORCH_CHECK(
      curr_op == num_ops - 1,
      "einsum(): more operands were provided than specified in the equation");

  // Labels must be within [a-zA-Z].
  constexpr uint8_t TOTAL_LABELS = 52;
  std::vector<int64_t> label_count(TOTAL_LABELS, 0);

  // The maximum number of dimensions covered by any ellipsis, needed when
  // unsqueezing missing dimensions from operands to permute and broadcast
  int64_t ell_num_dim = 0;

  // Compute label frequency and number of dimensions covered by ellipsis
  // We do this after parsing labels to make it more readable and simpler
  // to compute the number of dimensions covered by ellipsis.
  for(const auto i : c10::irange(num_ops)) {
    const auto operand = operands[i];
    const auto labels = op_labels[i];
    const auto ndims = operand.dim();
    int64_t nlabels = static_cast<int64_t>(labels.size());
    bool has_ellipsis = false;

    for (const auto& label : labels) {
      if (label == ELLIPSIS) {
        --nlabels;
        has_ellipsis = true;
        ell_num_dim = std::max(ell_num_dim, ndims - nlabels);
      } else {
        ++label_count[label];
      }
    }

    TORCH_CHECK(
        has_ellipsis ? nlabels <= ndims : nlabels == ndims,
        "einsum(): the number of subscripts in the equation (",
        nlabels,
        has_ellipsis ? ") is more than the number of dimensions ("
                     : ") does not match the number of dimensions (",
        ndims,
        ") for operand ",
        i,
        has_ellipsis ? "" : " and no ellipsis was given");
  }

  // We want to align the dimensions of every input tensor to have
  // shape out_dims + sum_dims. For this, we create a mapping of label
  // to index into the permuted shape.
  std::vector<int64_t> label_perm_index(TOTAL_LABELS, -1);

  // Current index in the permuted shape
  int64_t perm_index = 0;

  // Start index of ellipsis dimensions in the permuted shape
  int64_t ell_index = 0;
  found_ell = false;

  if (arrow_pos == std::string::npos) {
    // Implicit output is ellipsis (...) + labels seen only once
    perm_index = ell_num_dim;
    found_ell = true;
    for (const auto label : c10::irange(TOTAL_LABELS)) {
      if (label_count[label] == 1) {
        label_perm_index[label] = perm_index++;
      }
    }
  } else {
    // Parse explicit output
    const auto rhs = equation.substr(arrow_pos + 2);
    for (auto i = decltype(rhs.length()){0}; i < rhs.length(); ++i) {
      const unsigned char label = rhs[i];
      switch (label) {
        case ' ':
          // Ignore spaces
          break;

        case '.':
          TORCH_CHECK(
              // There can only be one ellipsis in the output
              !found_ell,
              "einsum(): found \'.\' for output but an ellipsis (...) was already found");
          TORCH_CHECK(
              // Ensure ellipsis is correct
              i + 2 < rhs.length() && rhs[++i] == '.' && rhs[++i] == '.',
              "einsum(): found \'.\' for output that is not part of any ellipsis (...)");
          ell_index = perm_index;
          perm_index += ell_num_dim;
          found_ell = true;
          break;

        default:
          TORCH_CHECK(
              einsum_check_label(label),
              "einsum(): invalid subscript given at index ",
            lhs.size() + 2 + i,
              " in the equation string, subscripts must be in [a-zA-Z]");
          const auto index = einsum_label_to_index(label);
          TORCH_CHECK(
              // Ensure label appeared at least once for some input operand and at
              // most once for the output
              label_count[index] > 0 && label_perm_index[index] == -1,
              "einsum(): output subscript ",
              label,
              label_perm_index[index] > -1
                  ? " appears more than once in the output"
                  : " does not appear in the equation for any input operand");
          label_perm_index[index] = perm_index++;
      }
    }
  }

  // Save output size before adding contraction dims (dims to sum out)
  const int64_t out_size = perm_index;

  // If ellipsis is not part of the output, add to contraction dimensions
  if (!found_ell) {
    ell_index = perm_index;
    perm_index += ell_num_dim;
  }

  // Add contraction labels (labels not present in output)
  for (const auto label : c10::irange(TOTAL_LABELS)) {
    if (label_count[label] > 0 && label_perm_index[label] == -1) {
      label_perm_index[label] = perm_index++;
    }
  }

  // Here we unsqueeze missing dimensions to make all operands have the same
  // number of dimensions. We take diagonals for repeated labels within the
  // same operand. Finally we permute the operands to align dimensions as
  // per the perm_out_index we computed above.
  std::vector<Tensor> permuted_operands;
  for (const auto i: c10::irange(num_ops)) {
    std::vector<int64_t> perm_shape(perm_index, -1);
    std::vector<int64_t> label_dim(TOTAL_LABELS, -1);
    Tensor operand = operands[i];
    const auto labels = op_labels[i];
    const auto original_sizes = operand.sizes();

    int64_t j = 0;
    for (const auto& label : labels) {
      if (label == ELLIPSIS) {
        // Add missing dimensions covered by the ellipsis
        const auto num_missing_dim =
            ell_num_dim - (original_sizes.size() - labels.size() + 1);
        for (const auto k : c10::irange(num_missing_dim)) {
          (void)k; //Suppress unused warning
          operand = operand.unsqueeze(j);
        }
        for (const auto k : c10::irange(ell_num_dim)) {
          perm_shape[ell_index + k] = j++;
        }
      } else if (label_dim[label] != -1) {
        // Repeated label, take diagonal
        const auto dim = label_dim[label];
        TORCH_CHECK(
            operand.size(j) == operand.size(dim),
            "einsum(): subscript ",
            einsum_index_to_label(label),
            " is repeated for operand ",
            i,
            " but the sizes don't match, ",
            operand.size(j),
            " != ",
            operand.size(dim));
        operand = operand.diagonal(0, dim, j).movedim(-1, dim);
      } else {
        // Lookup output index for label
        label_dim[label] = j;
        perm_shape[label_perm_index[label]] = j++;
      }
    }

    // Add dimensions for missing labels
    for (int64_t& index : perm_shape) {
      if (index == -1) {
        operand = operand.unsqueeze(-1);
        index = j++;
      }
    }

    permuted_operands.push_back(operand.permute(perm_shape));
  }

  // Check if operands broadcast and keep track of last operand with
  // dimension size != 1 for optimizing reductions
  std::vector<std::size_t> dim_last_op(perm_index, 0);
  bool has_zero_size_dim = false;
  for (const auto dim : c10::irange(perm_index)) {
    auto broadcast_size = permuted_operands[0].size(dim);
    for (const auto i: c10::irange(1, num_ops)) {
      const auto dim_size = permuted_operands[i].size(dim);
      if (broadcast_size != dim_size && broadcast_size != 1 && dim_size != 1) {
        std::ostringstream msg;
        msg << "einsum(): operands do not broadcast with remapped shapes [original->remapped]:";
        for (const auto j: c10::irange(num_ops)) {
          msg << " " << operands[j].sizes() << "->"
              << permuted_operands[j].sizes();
        }
        TORCH_CHECK(false, msg.str());
      }
      if (dim_size != 1) {
        broadcast_size = dim_size;
        dim_last_op[dim] = i;
      }
    }
    has_zero_size_dim |= broadcast_size == 0;
  }

  // Compute result
  Tensor result = permuted_operands[0];

  // Fast path for when an operand has zero sized dim
  if (has_zero_size_dim) {
    std::vector<int64_t> out_shape(out_size);
    for (const auto i : c10::irange(out_size)) {
      out_shape[i] = permuted_operands[dim_last_op[i]].size(i);
    }
    return at::zeros(out_shape, result.options());
  }

  // Sum out or squeeze dimensions that are size 1 for all later operands
  int64_t dim = out_size;
  for (int64_t i = dim; i < perm_index; ++i, ++dim) {
    if (dim_last_op[i] == 0) {
      if (result.size(dim) == 1) {
        result = result.squeeze(dim--);
      } else {
        result = result.sum(dim--);
      }
    }
  }

  for (const auto i: c10::irange(1, num_ops)) {
    Tensor operand = permuted_operands[i];
    std::vector<int64_t> sum_dims;

    // Sum out or squeeze dimensions that are size 1 for all later operands
    dim = out_size;
    for (int64_t j = dim; j < perm_index; ++j, ++dim) {
      if (dim_last_op[j] < i) {
        operand = operand.squeeze(dim);
        --dim;
      } else if (dim_last_op[j] == i) {
        if (result.size(dim) == 1) {
          operand = operand.sum(dim);
          result = result.squeeze(dim);
          --dim;
        } else {
          sum_dims.push_back(dim);
        }
      }
    }

    // Multiply tensors and sum out dimensions in sum_dims
    if (sum_dims.empty()) {
      result = result.mul(operand);
    } else if (sum_dims.size() == result.sizes().size()) {
      result = result.flatten().dot(operand.flatten());
    } else {
      result = sumproduct_pair(result, operand, sum_dims, false);
    }
  }

  return result;
}

// _trilinear computes a trilinear einstein sum with an unrolled dimension
// the result is `(i1.unsqueeze(expand1)*i2.unsqueeze(expand2)*i2.unsqueeze(expand3)).sum(sumdim)`
// the computation is unrolled in the unroll_dim dimension
// its main purpose is to unify the computations in bilinear and bilinear_backward
Tensor _trilinear(const Tensor& i1_, const Tensor& i2_, const Tensor& i3_,
                  IntArrayRef expand1_, IntArrayRef expand2_, IntArrayRef expand3_,
                  IntArrayRef sumdim_, int64_t unroll_dim) {
  int64_t total_dim = i1_.dim()+expand1_.size();
  TORCH_CHECK((unroll_dim >= 0) && (unroll_dim < total_dim), "unroll_dim must be in [0,", total_dim-1, "]");
  auto expand1 = at::dim_list_to_bitset(expand1_, total_dim);
  auto expand2 = at::dim_list_to_bitset(expand2_, total_dim);
  auto expand3 = at::dim_list_to_bitset(expand3_, total_dim);
  auto sumdim  = at::dim_list_to_bitset(sumdim_,  total_dim);
  Tensor i1 = i1_;
  Tensor i2 = i2_;
  Tensor i3 = i3_;
  std::vector<int64_t> output_size;
  std::vector<int64_t> sum_dims_12, sum_dims_23;
  int64_t unroll_size = -1;
  // asserts...
  for (const auto i : c10::irange(total_dim)) {
    int64_t s = 0;
    if (expand1[i]) {
      i1 = i1.unsqueeze(i);
    } else  {
      s = i1.size(i);
    }
    if (expand2[i]) {
      i2 = i2.unsqueeze(i);
    } else  {
      s = i2.size(i);
    }
    if (expand3[i]) {
      i3 = i3.unsqueeze(i);
      if (sumdim[i] && (i != unroll_dim))
        sum_dims_12.push_back(i);
    } else  {
      s = i3.size(i);
      if (sumdim[i] && (i != unroll_dim))
        sum_dims_23.push_back(i);
    }
    output_size.push_back(sumdim[i] ? 1 : s);
    if (i == unroll_dim)
      unroll_size = s;
  }
  int64_t slicemul1 = (expand1[unroll_dim] ? 0 : 1);
  int64_t slicemul2 = (expand2[unroll_dim] ? 0 : 1);
  int64_t slicemul3 = (expand3[unroll_dim] ? 0 : 1);

  auto output = at::zeros(output_size, i1.options());

  // Three conditionals are necessary since this function is meant to work for both
  // forward and backward, which changes the dimensions of the inputs.
  // Note that if output has zero elems is because (at least) one of i1, i2, i3 has zero elems.
  if (i1.numel() != 0 && i2.numel() != 0 && i3.numel() != 0) {
    if (! sumdim[unroll_dim]) {
      for (const auto k : c10::irange(unroll_size)) {
        Tensor buf = at::native::sumproduct_pair(i1.narrow(unroll_dim, k * slicemul1, 1),
                                                 i2.narrow(unroll_dim, k * slicemul2, 1),
                                                 sum_dims_12, true);
        buf = at::native::sumproduct_pair(buf, i3.narrow(unroll_dim, k * slicemul3, 1), sum_dims_23, true);
        output.narrow(unroll_dim, k, 1).add_(buf);
      }
    }
    else {
      for (const auto k : c10::irange(unroll_size)) {
        Tensor buf = at::native::sumproduct_pair(i1.narrow(unroll_dim, k*slicemul1, 1),
                                                 i2.narrow(unroll_dim, k*slicemul2, 1), sum_dims_12, true);
        buf = at::native::sumproduct_pair(buf, i3.narrow(unroll_dim, k*slicemul3, 1), sum_dims_23, true);
        output.add_(buf);
      }
    }
  }
  for (int64_t i = output.dim()-1; i >= 0; i--)
    if (sumdim[i])
      output.squeeze_(i);
  return output;
}

Tensor bilinear(const Tensor& input1, const Tensor& input2, const Tensor& weight, const c10::optional<Tensor>& bias_opt) {
  // See [Note: hacky wrapper removal for optional tensor]
  c10::MaybeOwned<Tensor> bias_maybe_owned = at::borrow_from_optional_tensor(bias_opt);
  const Tensor& bias = *bias_maybe_owned;

  TORCH_CHECK(input1.dim() == input2.dim(), "bilinear(): input dimensions do not match: got ", input1.dim(), " and ", input2.dim());
  for (const auto i : c10::irange(input1.dim() - 1)) {
    TORCH_CHECK(input1.size(i) == input2.size(i),
              "bilinear(): input batch dimensions do not match at dim ", i, ": got ", input1.size(i), " and ", input2.size(i));
  }
  TORCH_CHECK(input1.size(input1.dim() - 1) == weight.size(1),
            "bilinear(): input1 size does not match weight size: got ",
            input1.size(input1.dim() - 1), " but expected ", weight.size(1));
  TORCH_CHECK(input2.size(input2.dim() - 1) == weight.size(2),
            "bilinear(): input2 size does not match weight size: got ",
            input2.size(input2.dim() - 1), " but expected ", weight.size(2));
  TORCH_CHECK(!bias.defined() || bias.size(0) == weight.size(0),
            "bilinear(): bias size does not match weight size: got ",
            bias.size(0), " but expected ", weight.size(0));

  std::vector<int64_t> output_size;
  auto size1 = input1.sizes();
  output_size.insert(output_size.end(), size1.begin(), size1.end() - 1);
  output_size.push_back(weight.size(0));
  auto input1_flattened = input1.reshape({-1, input1.size(-1)});
  auto input2_flattened = input2.reshape({-1, input2.size(-1)});
  Tensor output = at::_trilinear(input1_flattened, weight, input2_flattened, {1,3}, {0}, {1,2}, {2,3}).reshape(output_size);
  if (bias.defined()) {
    output = output + bias;
  }
  return output;
}

// implements tensordot, a matrix-multiplication-like contraction, but the dimensions given
// in the two dimension lists
Tensor tensordot(const Tensor& input1, const Tensor& input2, IntArrayRef dims1, IntArrayRef dims2) {
  TORCH_CHECK(dims1.size() == dims2.size(), "both dimension lists should have same length");
  int64_t csize = 1;  // total size of the contracted dimensions
  Tensor t1 = input1;
  Tensor t2 = input2;
  for (const auto i : c10::irange(dims1.size())) {
    int s1 = input1.size(dims1[i]);
    int s2 = input2.size(dims2[i]);
    if (s2 == 1) { // broadcasted dimensions can be summed right away
      t1 = t1.sum(dims1[i], true);
    } else if (s1 == 1) {
      t2 = t2.sum(dims2[i], true);
    } else {
      TORCH_CHECK(s1 == s2, "contracted dimensions need to match, but first has size ", s1, " in dim ", dims1[i],
               " and second has size ", s2, " in dim ", dims2[i]);
      csize *= s1;
    }
  }

  auto cdims1 = at::dim_list_to_bitset(dims1, input1.dim());
  auto cdims2 = at::dim_list_to_bitset(dims2, input2.dim());
  std::vector<int64_t> p1, p2, rsizes;  // p1, p2: input permutations, rsizes: sizes of the result
  p1.reserve(input1.dim());
  p2.reserve(input2.dim());
  rsizes.reserve(input1.dim() + input2.dim() - (int64_t) dims1.size());
  int64_t size1 = 1; // number of non-contracted elements in input1
  int64_t size2 = 1; // number of non-contracted elements in input2

  // fill the permutations and compute sizes
  for (const auto i : c10::irange(input1.dim())) {
    if (! cdims1[i]) {
      p1.emplace_back(i);
      size1 *= t1.size(i);
      rsizes.emplace_back(t1.size(i));
    }
  }
  for (const auto x : dims1) {
    p1.emplace_back(x);
  }
  for (const auto x : dims2) {
    p2.emplace_back(x);
  }
  for (const auto i : c10::irange(input2.dim())) {
    if (! cdims2[i]) {
      p2.emplace_back(i);
      size2 *= t2.size(i);
      rsizes.emplace_back(t2.size(i));
    }
  }
  // permut and reshape for matrix multiplication
  t1 = t1.permute(p1).reshape({size1, csize});
  t2 = t2.permute(p2).reshape({csize, size2});
  // multiply and reshape to target size
  return at::mm(t1, t2).reshape(rsizes);
}

Tensor &tensordot_out(const Tensor& input1, const Tensor& input2, IntArrayRef dims1, IntArrayRef dims2, Tensor& result) {
  Tensor result_tmp = at::native::tensordot(input1, input2, dims1, dims2);
  auto result_dtype = result_tmp.scalar_type();
  auto output_tensor_dtype = result.scalar_type();
  auto output_device = result.device();
  auto input1_device = input1.device();
  auto input2_device = input2.device();
  // check if the input & output tensors are on the same device.
  TORCH_CHECK(
    (output_device == input1_device) && (input1_device == input2_device),
    "tensordot: Expected the output and input tensors to be on the "
    "same device, but got the output tensor on ", output_device,
    ", input tensor a on ", input1_device, ", and input tensor b on ", input2_device);
  // check if the computed result has the same dtype as the out tensor
  // (because tensordot does not support type promotion)
  TORCH_CHECK(
    result_dtype == output_tensor_dtype, "tensordot",
    ": Expected the output tensor to have dtype ", result_dtype,
    ", but got an output tensor with dtype ", output_tensor_dtype);
  at::native::resize_output(result, result_tmp.sizes());
  result.copy_(result_tmp);
  return result;
}

}}  // namespace at::native