A roadmap for the computation of persistent homology

By Nina Otter, M. A. Porter, Ulrike Tillmann, Peter Grindrod, Heather A. Harrington. EPJ Data Science, 2015

873 citations recorded by the source corpus.

Persistent homology (PH) is a method used in topological data analysis (TDA) to study qualitative features of data that persist across multiple scales. It is robust to perturbations of input data, independent of dimensions and coordinates, and provides a compact representation of the qualitative features of the input. The computation of PH is an open area with numerous important and fascinating challenges. The field of PH computation is evolving rapidly, and new algorithms and software implementations are being updated and released at a rapid pace. The purposes of our article are to (1) introduce theory and computational methods for PH to a broad range of computational scientists and (2) provide benchmarks of state-of-the-art implementations for the computation of PH. We give a friendly introduction to PH, navigate the pipeline for the computation of PH with an eye towards applications, and use a range of synthetic and real-world data sets to evaluate currently available open-source implementations for the computation of PH. Based on our benchmarking, we indicate which algorithms and implementations are best suited to different types of data sets. In an accompanying tutorial, we provide guidelines for the computation of PH. We make publicly available all scripts that we wrote for the tutorial, and we make available the processed version of the data sets used in the benchmarking.

Subfields: Computational Theory and Mathematics, Computer Vision and Pattern Recognition, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/edb8feef72708cf56d17c307b757292396dc6033

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Citations in the corpus

Higher order interactions in complex networks of phase oscillators promote abrupt synchronization switching (2019, background): “les), etc. comprise the simplicial complex. While simplicial complexes have been proven to be very useful for analysis and computation in high dimensional data sets, e.g., using persistent homologies [15], little is understood about their role in shaping dynamical processes, save for a handful of ex”
Networks beyond pairwise interactions: structure and dynamics (2020, background): “[127] Nina Otter, Mason A Porter, Ulrike Tillmann, Peter Grindrod, and Heather A Harrington.”
Random Walks on Simplicial Complexes and the normalized Hodge Laplacian (2018, background): “For the case where the data is a point cloud, finding fast algorithms for constructing SCs (or sequences of SCs) is an active area of research [31,34,43,67,98,127].”
What are higher-order networks? (2021, methodology): “Persistent homology is computable, with many software implementations available, and we refer to the following tutorials and “user guides” to more information [211,223,235].”
Two’s company, three (or more) is a simplex (2016, background): “A comprehensive survey on the state-of-the-art in homology software with benchmarks as of late 2015 appears in (Otter et al. 2015).”
A Topological Loss Function for Deep-Learning based Image Segmentation using Persistent Homology (2019, methodology): “Below we give an overview of PH as it applies to our method, but for more thorough reviews we direct the reader to [11], [25], [26].”