Optimal rates of convergence for persistence diagrams in Topological Data Analysis

Computational topology has recently known an important development toward data analysis, giving birth to the field of topological data analysis. Topological persistence, or persistent homology, appears as a fundamental tool in this field. In this paper, we study topological persistence in general metric spaces, with a statistical approach. We show that the use of persistent homology can be naturally considered in general statistical frameworks and persistence diagrams can be used as statistics with interesting convergence properties. Some numerical experiments are performed in various contexts to illustrate our results.

[1]  Sivaraman Balakrishnan,et al.  Confidence sets for persistence diagrams , 2013, The Annals of Statistics.

[2]  Vin de Silva,et al.  HOMOLOGICAL SENSOR NETWORKS , 2005 .

[3]  D. Burago,et al.  A Course in Metric Geometry , 2001 .

[4]  A. González,et al.  Set estimation: another bridge between statistics and geometry , 2009 .

[5]  A. Tsybakov,et al.  Minimax theory of image reconstruction , 1993 .

[6]  S. Mukherjee,et al.  Probability measures on the space of persistence diagrams , 2011 .

[7]  Sivaraman Balakrishnan,et al.  Statistical Inference For Persistent Homology , 2013, arXiv.org.

[8]  A. Cuevas,et al.  On Statistical Properties of Sets Fulfilling Rolling-Type Conditions , 2011, Advances in Applied Probability.

[9]  A. Cuevas,et al.  A plug-in approach to support estimation , 1997 .

[10]  Leonidas J. Guibas,et al.  Proximity of persistence modules and their diagrams , 2009, SCG '09.

[11]  Bruno Pelletier,et al.  Asymptotic Normality in Density Support Estimation , 2009 .

[12]  Frédéric Chazal,et al.  Convergence rates for persistence diagram estimation in topological data analysis , 2014, J. Mach. Learn. Res..

[13]  L. Devroye,et al.  Detection of Abnormal Behavior Via Nonparametric Estimation of the Support , 1980 .

[14]  Leonidas J. Guibas,et al.  Gromov‐Hausdorff Stable Signatures for Shapes using Persistence , 2009, Comput. Graph. Forum.

[15]  Leonidas J. Guibas,et al.  Persistence-based clustering in riemannian manifolds , 2011, SoCG '11.

[16]  Herbert Edelsbrunner,et al.  Topological persistence and simplification , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[17]  Peter Bubenik,et al.  A statistical approach to persistent homology , 2006, math/0607634.

[18]  Gunnar E. Carlsson,et al.  Topology and data , 2009 .

[19]  L. Dümbgen,et al.  RATES OF CONVERGENCE FOR RANDOM APPROXIMATIONS OF CONVEX SETS , 1996 .

[20]  A. Tsybakov.  On nonparametric estimation of density level sets , 1997 .

[21]  Robert D. Nowak,et al.  Adaptive Hausdorff Estimation of Density Level Sets , 2009, COLT.

[22]  Stephen Smale,et al.  A Topological View of Unsupervised Learning from Noisy Data , 2011, SIAM J. Comput..

[23]  Jianzhong Wang,et al.  Geometric Structure of High-Dimensional Data and Dimensionality Reduction , 2012 .

[24]  Afra Zomorodian,et al.  Computing Persistent Homology , 2004, SCG '04.

[25]  Sivaraman Balakrishnan,et al.  Minimax rates for homology inference , 2011, AISTATS.

[26]  Frédéric Chazal,et al.  Geometric Inference for Probability Measures , 2011, Found. Comput. Math..

[27]  Frédéric Chazal,et al.  Deconvolution for the Wasserstein Metric and Geometric Inference , 2011, GSI.

[28]  P. Massart,et al.  Concentration inequalities and model selection , 2007 .

[29]  D. Ringach,et al.  Topological analysis of population activity in visual cortex. , 2008, Journal of vision.

[30]  David Cohen-Steiner,et al.  Stability of Persistence Diagrams , 2005, Discret. Comput. Geom..

[31]  Sebastian Thrun,et al.  SCAPE: shape completion and animation of people , 2005, SIGGRAPH 2005.

[32]  A. Cuevas,et al.  On boundary estimation , 2004, Advances in Applied Probability.

[33]  H. Edelsbrunner.  The union of balls and its dual shape , 1995 .

[34]  Larry A. Wasserman,et al.  Manifold Estimation and Singular Deconvolution Under Hausdorff Loss , 2011, ArXiv.

[35]  Stephen Smale,et al.  Finding the Homology of Submanifolds with High Confidence from Random Samples , 2008, Discret. Comput. Geom..

[36]  Larry A. Wasserman,et al.  Minimax Manifold Estimation , 2010, J. Mach. Learn. Res..

[37]  Leonidas J. Guibas,et al.  BIOINFORMATICS ORIGINAL PAPER doi:10.1093/bioinformatics/btm250 Structural bioinformatics Persistent voids: a new structural metric for membrane fusion , 2022 .

[38]  Peter Bubenik,et al.  Statistical topology using persistence landscapes , 2012, ArXiv.

[39]  A. Tsybakov,et al.  Efficient Estimation of Monotone Boundaries , 1995 .

[40]  E. D. Vito,et al.  Learning Sets with Separating Kernels , 2012, 1204.3573.

[41]  Alberto Rodríguez Casal,et al.  Set estimation under convexity type assumptions , 2007 .

[42]  Steve Oudot,et al.  Persistence stability for geometric complexes , 2012, ArXiv.

[43]  R. Ho.  Algebraic Topology , 2022 .