PersistenceImages swMATH ID: 41418 Software Authors: Henry Adams, Sofya Chepushtanova, Tegan Emerson, Eric Hanson, Michael Kirby, Francis Motta, Rachel Neville, Chris Peterson, Patrick Shipman, Lori Ziegelmeier Description: Persistence Images: A Stable Vector Representation of Persistent Homology. Many datasets can be viewed as a noisy sampling of an underlying space, and tools from topological data analysis can characterize this structure for the purpose of knowledge discovery. One such tool is persistent homology, which provides a multiscale description of the homological features within a dataset. A useful representation of this homological information is a persistence diagram (PD). Efforts have been made to map PDs into spaces with additional structure valuable to machine learning tasks. We convert a PD to a finite-dimensional vector representation which we call a persistence image (PI), and prove the stability of this transformation with respect to small perturbations in the inputs. The discriminatory power of PIs is compared against existing methods, showing significant performance gains. We explore the use of PIs with vector-based machine learning tools, such as linear sparse support vector machines, which identify features containing discriminating topological information. Finally, high accuracy inference of parameter values from the dynamic output of a discrete dynamical system (the linked twist map) and a partial differential equation (the anisotropic Kuramoto-Sivashinsky equation) provide a novel application of the discriminatory power of PIs. Homepage: https://arxiv.org/abs/1507.06217 Source Code: https://gitlab.com/csu-tda/PersistenceImages Related Software: GitHub; Ripser; PHAT; Scikit; Gudhi; Persistence Landscape; TDA; Dionysus; TopologyNet; javaPlex; Perseus; PersistenceCurves; Ripser.py; CliqueTop; LearningAlgebraicVarieties; jHoles; SW1PerS; SciPy; AFLOW-SYM; Spglib Cited in: 42 Publications Standard Articles 1 Publication describing the Software, including 1 Publication in zbMATH Year Persistence images: a stable vector representation of persistent homology. Zbl 1431.68105Adams, Henry; Emerson, Tegan; Kirby, Michael; Neville, Rachel; Peterson, Chris; Shipman, Patrick; Chepushtanova, Sofya; Hanson, Eric; Motta, Francis; Ziegelmeier, Lori 2017 all top 5 Cited by 102 Authors 4 Maroulas, Vasileios 3 Adams, Henry 3 Divol, Vincent 3 Oballe, Christopher 3 Wei, Guowei 2 Bubenik, Peter 2 Cang, Zixuan 2 Chazal, Frédéric 2 Hiraoka, Yasuaki 2 Kusano, Genki 2 Neville, Rachel 2 Peterson, Chris 2 Ziegelmeier, Lori 1 Aminian, Manuchehr 1 Bendich, Paul 1 Berry, Eric A. 1 Beshkov, Kosio 1 Boothe, Dave 1 Brutti, Pierpaolo 1 Calcina, Sabrina S. 1 Carlsson, Gunnar E. 1 Chen, Yen-Chi 1 Chepushtanova, Sofya V. 1 Cherne, Alan 1 Chung, Yu-Min 1 Ciocanel, Maria-Veronica 1 Cisewski-Kehe, Jessi 1 Crawford, Lorin 1 Dawes, Adriana T. 1 Dong, Zhetong 1 Emerson, Tegan H. 1 Farnell, Elin 1 Fasy, Brittany Terese 1 Feng, Michelle 1 Feragen, Aasa 1 Franaszczuk, Piotr J. 1 Fukumizu, Kenji 1 Gameiro, Marcio F. 1 Grbić, Jelena 1 Guillemard, Mijail 1 Heo, Giseon 1 Hess, Kathryn P. 1 Hofer, Christoph D. 1 Hotz, Thomas 1 Huckemann, Stephan F. 1 Iske, Armin 1 Jaquette, Jonathan 1 Johnson, Megan 1 Juda, Mateusz 1 Juenemann, Riley 1 Jung, Jae-Hun 1 Kališnik Verovšek, Sara 1 Kerick, Scott 1 Kimura, Masao 1 Kwitt, Roland 1 Lacombe, Théo 1 Lawson, Austin 1 Leitte, Heike 1 Levrard, Clément 1 Leygonie, Jacob 1 Lin, Hongwei 1 McCleary, Alexander 1 Mckinley, Scott A. 1 Micucci, Cassie Putman 1 Mike, Joshua L. 1 Miller, Ezra 1 Mirth, Joshua 1 Monod, Anthea 1 Motta, Francis C. 1 Munch, Elizabeth 1 Nasrin, Farzana 1 Niethammer, Marc 1 Obayashi, Ippei 1 Oudot, Steve Yann 1 Padellini, Tullia 1 Patiño-Galindo, Juan Ángel 1 Polonik, Wolfgang 1 Porter, Mason Alexander 1 Pu, Junyu 1 Ramamurthy, Karthikeyan Natesan 1 Rieck, Bastian 1 Royer, Martin 1 Sadlo, Filip 1 Schweinhart, Benjamin 1 Seidl, Markus 1 Shipman, Patrick D. 1 Shonkwiler, Clayton 1 Singhal, Kritika 1 Skryzalin, Jacek 1 Som, Anirudh 1 Spannaus, Adam 1 Tiesinga, Paul H. E. 1 Tillmann, Ulrike 1 Topaz, Chad M. 1 Turaga, Pavan K. 1 Tymochko, Sarah 1 Vipond, Oliver 1 Wu, Jie 1 Xia, Kelin 1 Xian, Lu ...and 2 more Authors all top 5 Cited in 20 Serials 7 Journal of Applied and Computational Topology 5 Journal of Machine Learning Research (JMLR) 2 SIAM Journal on Applied Algebra and Geometry 2 Foundations of Data Science 2 SIAM Journal on Mathematics of Data Science 1 Biological Cybernetics 1 Bulletin of Mathematical Biology 1 Mathematics and Computers in Simulation 1 Computer Aided Geometric Design 1 Statistics 1 Japan Journal of Industrial and Applied Mathematics 1 SIAM Review 1 Advances in Computational Mathematics 1 Communications in Nonlinear Science and Numerical Simulation 1 Foundations of Computational Mathematics 1 Oberwolfach Reports 1 Advances in Data Analysis and Classification. ADAC 1 Electronic Journal of Statistics 1 Discrete and Continuous Dynamical Systems. Series S 1 Journal of the Korean Society for Industrial and Applied Mathematics all top 5 Cited in 21 Fields 29 Algebraic topology (55-XX) 26 Statistics (62-XX) 15 Computer science (68-XX) 7 Biology and other natural sciences (92-XX) 5 Dynamical systems and ergodic theory (37-XX) 5 Probability theory and stochastic processes (60-XX) 3 Numerical analysis (65-XX) 2 Algebraic geometry (14-XX) 2 Information and communication theory, circuits (94-XX) 1 General and overarching topics; collections (00-XX) 1 Combinatorics (05-XX) 1 Commutative algebra (13-XX) 1 Category theory; homological algebra (18-XX) 1 Measure and integration (28-XX) 1 Approximations and expansions (41-XX) 1 Harmonic analysis on Euclidean spaces (42-XX) 1 Calculus of variations and optimal control; optimization (49-XX) 1 Differential geometry (53-XX) 1 Manifolds and cell complexes (57-XX) 1 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 1 Systems theory; control (93-XX) Citations by Year