A Framework for Differential Calculus on Persistence Barcodes

We define notions of differentiability for maps from and to the space of persistence barcodes. Inspired by the theory of diffeological spaces, the proposed framework uses lifts to the space of ordered barcodes, from which derivatives can be computed. The two derived notions of differentiability (respectively, from and to the space of barcodes) combine together naturally to produce a chain rule that enables the use of gradient descent for objective functions factoring through the space of barcodes. We illustrate the versatility of this framework by showing how it can be used to analyze the smoothness of various parametrized families of filtrations arising in topological data analysis.

[1]  Bernd Sturmfels,et al.  Learning algebraic varieties from samples , 2018, Revista Matemática Complutense.

[2]  Henry Adams,et al.  Persistence Images: A Stable Vector Representation of Persistent Homology , 2015, J. Mach. Learn. Res..

[3]  Vincent Divol,et al.  Understanding the topology and the geometry of the space of persistence diagrams via optimal partial transport , 2019, Journal of Applied and Computational Topology.

[4]  H. Fédérer.  Geometric Measure Theory , 1969 .

[5]  Steve Oudot,et al.  Sliced Wasserstein Kernel for Persistence Diagrams , 2017, ICML.

[6]  Julien Rabin,et al.  Wasserstein Barycenter and Its Application to Texture Mixing , 2011, SSVM.

[7]  John Milnor,et al.  Morse Theory. (Am-51), Volume 51 , 1963 .

[8]  R. Robinson.  Structural Stability Theorems , 1976 .

[9]  Peter Bubenik,et al.  Statistical topological data analysis using persistence landscapes , 2012, J. Mach. Learn. Res..

[10]  M. Golubitsky,et al.  Stable mappings and their singularities , 1973 .

[11]  Steve Oudot,et al.  The Structure and Stability of Persistence Modules , 2012, Springer Briefs in Mathematics.

[12]  David Cohen-Steiner,et al.  Stability of Persistence Diagrams , 2007, Discret. Comput. Geom..

[13]  P. Michor,et al.  The Convenient Setting of Global Analysis , 1997 .

[14]  Leonidas J. Guibas,et al.  A Topology Layer for Machine Learning , 2019, AISTATS.

[15]  J. Mather.  Notes on Topological Stability , 2012 .

[16]  Maks Ovsjanikov,et al.  Topological Function Optimization for Continuous Shape Matching , 2018, Comput. Graph. Forum.

[17]  Marcio Gameiro,et al.  Continuation of Point Clouds via Persistence Diagrams , 2015, ArXiv.

[18]  D. Trotman.  Stability of transversality to a stratification implies Whitney (a)-regularity , 1978 .

[19]  Alfred Frölicher,et al.  Linear Spaces And Differentiation Theory , 1988 .

[20]  Afra Zomorodian,et al.  Computing Persistent Homology , 2005, Discret. Comput. Geom..

[21]  Vin de Silva,et al.  The observable structure of persistence modules , 2014, 1405.5644.

[22]  Steve Oudot,et al.  Eurographics Symposium on Geometry Processing 2015 Stable Topological Signatures for Points on 3d Shapes , 2022 .

[23]  Diffeology of Manifolds with Boundary , 2007 .

[24]  W. Crawley-Boevey.  Decomposition of pointwise finite-dimensional persistence modules , 2012, 1210.0819.

[25]  L. Nicolaescu.  An Invitation to Morse Theory , 2007 .

[26]  Rickard Brüel Gabrielsson,et al.  Topology‐Aware Surface Reconstruction for Point Clouds , 2018, Comput. Graph. Forum.

[27]  M. Goresky,et al.  Stratified Morse theory , 1988 .

[28]  Jacob Leygonie,et al.  Optimization of Spectral Wavelets for Persistence-Based Graph Classification , 2021, Frontiers in Applied Mathematics and Statistics.

[29]  Marc Niethammer,et al.  Graph Filtration Learning , 2019, ICML.

[30]  Herbert Edelsbrunner,et al.  Topological Persistence and Simplification , 2000, Proceedings 41st Annual Symposium on Foundations of Computer Science.

[31]  Massimo Ferri,et al.  Comparing Persistence Diagrams Through Complex Vectors , 2015, ICIAP.

[32]  Ulrich Bauer,et al.  Induced matchings and the algebraic stability of persistence barcodes , 2013, J. Comput. Geom..

[33]  Frédéric Chazal,et al.  Stochastic Convergence of Persistence Landscapes and Silhouettes , 2013, J. Comput. Geom..

[34]  David Cohen-Steiner,et al.  Lipschitz Functions Have Lp-Stable Persistence , 2010, Found. Comput. Math..

[35]  Frédéric Chazal,et al.  The density of expected persistence diagrams and its kernel based estimation , 2018, SoCG.

[36]  Sara Kalisnik,et al.  Tropical Coordinates on the Space of Persistence Barcodes , 2019, Found. Comput. Math..

[37]  D. Morozov,et al.  Persistence-sensitive simplication of functions on surfaces in linear time , 2009 .

[38]  Dmitriy Drusvyatskiy,et al.  Stochastic Subgradient Method Converges on Tame Functions , 2018, Foundations of Computational Mathematics.

[39]  Sara Kališnik Verovšek.  Tropical Coordinates on the Space of Persistence Barcodes , 2016, 1604.00113.

[40]  P. Pansu,et al.  Métriques de Carnot-Carthéodory et quasiisométries des espaces symétriques de rang un , 1989 .

[41]  Chao Chen,et al.  A Topological Regularizer for Classifiers via Persistent Homology , 2019, AISTATS.