paper.bib

@article{bauer2017ripser,
  title={Ripser: a lean C++ code for the computation of Vietoris-Rips persistence barcodes},
  author={Bauer, Ulrich},
  journal={Software available at https://github.com/Ripser/ripser},
  year={2017}
}

@article{scikit-learn,
 title={Scikit-learn: Machine Learning in {P}ython},
 author={Pedregosa, F. and Varoquaux, G. and Gramfort, A. and Michel, V.
         and Thirion, B. and Grisel, O. and Blondel, M. and Prettenhofer, P.
         and Weiss, R. and Dubourg, V. and Vanderplas, J. and Passos, A. and
         Cournapeau, D. and Brucher, M. and Perrot, M. and Duchesnay, E.},
 journal={Journal of Machine Learning Research},
 volume={12},
 pages={2825--2830},
 year={2011}
}

@Article{obayashi2018persistence,
  author="Obayashi, Ippei
  and Hiraoka, Yasuaki
  and Kimura, Masao",
  title="Persistence diagrams with linear machine learning models",
  journal="Journal of Applied and Computational Topology",
  year="2018",
  month="Jun",
  day="01",
  volume="1",
  number="3",
  pages="421--449",
  abstract="Persistence diagrams have been widely recognized as a compact descriptor for characterizing multiscale topological features in data. When many datasets are available, statistical features embedded in those persistence diagrams can be extracted by applying machine learnings. In particular, the ability for explicitly analyzing the inverse in the original data space from those statistical features of persistence diagrams is significantly important for practical applications. In this paper, we propose a unified method for the inverse analysis by combining linear machine learning models with persistence images. The method is applied to point clouds and cubical sets, showing the ability of the statistical inverse analysis and its advantages.",
  issn="2367-1734",
  doi="10.1007/s41468-018-0013-5",
  url="https://doi.org/10.1007/s41468-018-0013-5"
}

@Article{de2011persistent,
  author="de Silva, Vin
  and Morozov, Dmitriy
  and Vejdemo-Johansson, Mikael",
  title="Persistent Cohomology and Circular Coordinates",
  journal="Discrete {\&} Computational Geometry",
  year="2011",
  month="Jun",
  day="01",
  volume="45",
  number="4",
  pages="737--759",
  abstract="Nonlinear dimensionality reduction (NLDR) algorithms such as Isomap, LLE, and Laplacian Eigenmaps address the problem of representing high-dimensional nonlinear data in terms of low-dimensional coordinates which represent the intrinsic structure of the data. This paradigm incorporates the assumption that real-valued coordinates provide a rich enough class of functions to represent the data faithfully and efficiently. On the other hand, there are simple structures which challenge this assumption: the circle, for example, is one-dimensional, but its faithful representation requires two real coordinates. In this work, we present a strategy for constructing circle-valued functions on a statistical data set. We develop a machinery of persistent cohomology to identify candidates for significant circle-structures in the data, and we use harmonic smoothing and integration to obtain the circle-valued coordinate functions themselves. We suggest that this enriched class of coordinate functions permits a precise NLDR analysis of a broader range of realistic data sets.",
  issn="1432-0444",
  doi="10.1007/s00454-011-9344-x",
  url="https://doi.org/10.1007/s00454-011-9344-x"
}

@Article{perea2018multiscale,
  author="Perea, Jose A.",
  title="Multiscale Projective Coordinates via Persistent Cohomology of Sparse Filtrations",
  journal="Discrete {\&} Computational Geometry",
  year="2018",
  month="Jan",
  day="01",
  volume="59",
  number="1",
  pages="175--225",
  abstract="We present a framework which leverages the underlying topology of a data set, in order to produce appropriate coordinate representations. In particular, we show how to construct maps to real and complex projective spaces, given appropriate persistent cohomology classes. An initial map is obtained in two steps: First, the persistent cohomology of a sparse filtration is used to compute systems of transition functions for (real and complex) line bundles over neighborhoods of the data. Next, the transition functions are used to produce explicit classifying maps for the induced bundles. A framework for dimensionality reduction in projective space (Principal Projective Components) is also developed, aimed at decreasing the target dimension of the original map. Several examples are provided as well as theorems addressing choices in the construction.",
  issn="1432-0444",
  doi="10.1007/s00454-017-9927-2",
  url="https://doi.org/10.1007/s00454-017-9927-2"
}

@article{cavannageometric,
  title={A Geometric Perspective on Sparse Filtrations},
  author={Cavanna, Nicholas J and Jahanseir, Mahmoodreza and Sheehy, Donald R},
  journal={Proceedings of the Canadian Conference in Computational Geometry},
  year={2015}
}

% TDA / Applications

@book{edelsbrunner2010computational,
  title={Computational topology: an introduction},
  author={Edelsbrunner, Herbert and Harer, John},
  year={2010},
  publisher={American Mathematical Soc.}
}

@article{carlsson2009topology,
  title={Topology and data},
  author={Carlsson, Gunnar},
  journal={Bulletin of the American Mathematical Society},
  volume={46},
  number={2},
  pages={255--308},
  year={2009}
}

@Article{perea2015sliding,
  author="Perea, Jose A.
  and Harer, John",
  title="Sliding Windows and Persistence: An Application of Topological Methods to Signal Analysis",
  journal="Foundations of Computational Mathematics",
  year="2015",
  month="Jun",
  day="01",
  volume="15",
  number="3",
  pages="799--838",
  abstract="We develop in this paper a theoretical framework for the topological study of time series data. Broadly speaking, we describe geometrical and topological properties of sliding window embeddings, as seen through the lens of persistent homology. In particular, we show that maximum persistence at the point-cloud level can be used to quantify periodicity at the signal level, prove structural and convergence theorems for the resulting persistence diagrams, and derive estimates for their dependency on window size and embedding dimension. We apply this methodology to quantifying periodicity in synthetic data sets and compare the results with those obtained using state-of-the-art methods in gene expression analysis. We call this new method SW1PerS, which stands for Sliding Windows and 1-Dimensional Persistence Scoring.",
  issn="1615-3383",
  doi="10.1007/s10208-014-9206-z",
  url="https://doi.org/10.1007/s10208-014-9206-z"
}

@InProceedings{wu2017optimal,
  author="Wu, Pengxiang
  and Chen, Chao
  and Wang, Yusu
  and Zhang, Shaoting
  and Yuan, Changhe
  and Qian, Zhen
  and Metaxas, Dimitris
  and Axel, Leon",
  editor="Niethammer, Marc
  and Styner, Martin
  and Aylward, Stephen
  and Zhu, Hongtu
  and Oguz, Ipek
  and Yap, Pew-Thian
  and Shen, Dinggang",
  title="Optimal Topological Cycles and Their Application in Cardiac Trabeculae Restoration",
  booktitle="Information Processing in Medical Imaging",
  year="2017",
  publisher="Springer International Publishing",
  address="Cham",
  pages="80--92",
  abstract="In cardiac image analysis, it is important yet challenging to reconstruct the trabeculae, namely, fine muscle columns whose ends are attached to the ventricular walls. To extract these fine structures, traditional image segmentation methods are insufficient. In this paper, we propose a novel method to jointly detect salient topological handles and compute the optimal representations of them. The detected handles are considered hypothetical trabeculae structures. They are further screened using a classifier and are then included in the final segmentation. We show in experiments the significance of our contribution compared with previous standard segmentation methods without topological priors, as well as with previous topological method in which non-optimal representations of topological handles are used.",
  isbn="978-3-319-59050-9"
}


@article{stolz2017persistent,
  title={Persistent homology of time-dependent functional networks constructed from coupled time series},
  author={Stolz, Bernadette J and Harrington, Heather A and Porter, Mason A},
  journal={Chaos: An Interdisciplinary Journal of Nonlinear Science},
  volume={27},
  number={4},
  pages={047410},
  year={2017},
  publisher={AIP Publishing}
}

@article {iyer2010imaging,
	author = {Iyer-Pascuzzi, Anjali S. and Symonova, Olga and Mileyko, Yuriy and Hao, Yueling and Belcher, Heather and Harer, John and Weitz, Joshua S. and Benfey, Philip N.},
	title = {Imaging and Analysis Platform for Automatic Phenotyping and Trait Ranking of Plant Root Systems},
	volume = {152},
	number = {3},
	pages = {1148--1157},
	year = {2010},
	doi = {10.1104/pp.109.150748},
	publisher = {American Society of Plant Biologists},
	abstract = {The ability to nondestructively image and automatically phenotype complex root systems, like those of rice (Oryza sativa), is fundamental to identifying genes underlying root system architecture (RSA). Although root systems are central to plant fitness, identifying genes responsible for RSA remains an underexplored opportunity for crop improvement. Here we describe a nondestructive imaging and analysis system for automated phenotyping and trait ranking of RSA. Using this system, we image rice roots from 12 genotypes. We automatically estimate RSA traits previously identified as important to plant function. In addition, we expand the suite of features examined for RSA to include traits that more comprehensively describe monocot RSA but that are difficult to measure with traditional methods. Using 16 automatically acquired phenotypic traits for 2,297 images from 118 individuals, we observe (1) wide variation in phenotypes among the genotypes surveyed; and (2) greater intergenotype variance of RSA features than variance within a genotype. RSA trait values are integrated into a computational pipeline that utilizes supervised learning methods to determine which traits best separate two genotypes, and then ranks the traits according to their contribution to each pairwise comparison. This trait-ranking step identifies candidate traits for subsequent quantitative trait loci analysis and demonstrates that depth and average radius are key contributors to differences in rice RSA within our set of genotypes. Our results suggest a strong genetic component underlying rice RSA. This work enables the automatic phenotyping of RSA of individuals within mapping populations, providing an integrative framework for quantitative trait loci analysis of RSA.},
	issn = {0032-0889},
	URL = {http://www.plantphysiol.org/content/152/3/1148},
	eprint = {http://www.plantphysiol.org/content/152/3/1148.full.pdf},
	journal = {Plant Physiology}
}
@article {giusti2015clique,
	author = {Giusti, Chad and Pastalkova, Eva and Curto, Carina and Itskov, Vladimir},
	title = {Clique topology reveals intrinsic geometric structure in neural correlations},
	volume = {112},
	number = {44},
	pages = {13455--13460},
	year = {2015},
	doi = {10.1073/pnas.1506407112},
	publisher = {National Academy of Sciences},
	abstract = {Detecting structure in neural activity is critical for understanding the function of neural circuits. The coding properties of neurons are typically investigated by correlating their responses to external stimuli. It is not clear, however, if the structure of neural activity can be inferred intrinsically, without a priori knowledge of the relevant stimuli. We introduce a novel method, called clique topology, that detects intrinsic structure in neural activity that is invariant under nonlinear monotone transformations. Using pairwise correlations of neurons in the hippocampus, we demonstrate that our method is capable of detecting geometric structure from neural activity alone, without appealing to external stimuli or receptive fields.Detecting meaningful structure in neural activity and connectivity data is challenging in the presence of hidden nonlinearities, where traditional eigenvalue-based methods may be misleading. We introduce a novel approach to matrix analysis, called clique topology, that extracts features of the data invariant under nonlinear monotone transformations. These features can be used to detect both random and geometric structure, and depend only on the relative ordering of matrix entries. We then analyzed the activity of pyramidal neurons in rat hippocampus, recorded while the animal was exploring a 2D environment, and confirmed that our method is able to detect geometric organization using only the intrinsic pattern of neural correlations. Remarkably, we found similar results during nonspatial behaviors such as wheel running and rapid eye movement (REM) sleep. This suggests that the geometric structure of correlations is shaped by the underlying hippocampal circuits and is not merely a consequence of position coding. We propose that clique topology is a powerful new tool for matrix analysis in biological settings, where the relationship of observed quantities to more meaningful variables is often nonlinear and unknown.},
	issn = {0027-8424},
	URL = {http://www.pnas.org/content/112/44/13455},
	eprint = {http://www.pnas.org/content/112/44/13455.full.pdf},
	journal = {Proceedings of the National Academy of Sciences}
}

@article{kramar2013persistence,
  title = {Persistence of force networks in compressed granular media},
  author = {Kramar, M. and Goullet, A. and Kondic, L. and Mischaikow, K.},
  journal = {Phys. Rev. E},
  volume = {87},
  issue = {4},
  pages = {042207},
  numpages = {8},
  year = {2013},
  month = {Apr},
  publisher = {American Physical Society},
  doi = {10.1103/PhysRevE.87.042207},
  url = {https://link.aps.org/doi/10.1103/PhysRevE.87.042207}
}

@Article{perea2014klein,
  author="Perea, Jose A.
  and Carlsson, Gunnar",
  title="A Klein-Bottle-Based Dictionary for Texture Representation",
  journal="International Journal of Computer Vision",
  year="2014",
  month="Mar",
  day="01",
  volume="107",
  number="1",
  pages="75--97",
  issn="1573-1405",
  doi="10.1007/s11263-013-0676-2",
  url="https://doi.org/10.1007/s11263-013-0676-2"
}

@article{tralie2017quasi,
  author = {Christopher J. Tralie and Jose A. Perea},
  title = {(Quasi)Periodicity Quantification in Video Data, Using Topology},
  journal = {SIAM Journal on Imaging Sciences},
  volume = {11},
  number = {2},
  pages = {1049-1077},
  year = {2018},
  doi = {10.1137/17M1150736},
  URL = {https://doi.org/10.1137/17M1150736},
  eprint = { https://doi.org/10.1137/17M1150736}
}
@inproceedings{tralie2018slomoloops, 
  title={Topological Eulerian Synthesis of Slow Motion Periodic Videos},
  author={Tralie, Christopher J and Berger, Matthew},
  booktitle={IEEE International Conference on Image Processing},
  year={2018}
}

@article{tralie2018autism,
  author = {Tralie, Christopher J. and Matthew, Goodwin S. and Sapiro, Guillermo},
  title = {Automated Detection of Stereotypical Motor Movements in Children with Autism Spectrum Disorder Using Geometric Feature Fusion},
  journal = {International Society for Autism Research (INSAR)},
  year = {2018}
}

@article{bendich2016persistent,
  title = "Persistent homology analysis of brain artery trees",
  abstract = "New representations of tree-structured data objects, using ideas from topological data analysis, enable improved statistical analyses of a population of brain artery trees. A number of representations of each data tree arise from persistence diagrams that quantify branching and looping of vessels at multiple scales. Novel approaches to the statistical analysis, through various summaries of the persistence diagrams, lead to heightened correlations with covariates such as age and sex, relative to earlier analyses of this data set. The correlation with age continues to be significant even after controlling for correlations from earlier significant summaries.",
  keywords = "Angiography, Persistent homology, Statistics, Topological data analysis, Tree-structured data",
  author = "Paul Bendich and Marron, {J. S.} and Ezra Miller and Alex Pieloch and Sean Skwerer",
  year = "2016",
  month = "3",
  day = "1",
  doi = "10.1214/15-AOAS886",
  language = "English (US)",
  volume = "10",
  pages = "198--218",
  journal = "Annals of Applied Statistics",
  issn = "1932-6157",
  publisher = "Institute of Mathematical Statistics",
  number = "1",
}