On the persistent homology of almost surely C0 stochastic processes

The persistent homology of random processes has been a question of interest in the TDA community. This paper aims to extend a result of Chazal and Divol and to reconcile the probabilistic approach to the study of the topology of superlevel sets of random processes in dimension 1. We provide explicit descriptions of the barcode for Brownian motion and the Brownian bridge. Additionally, we use these results on almost surely $C^0$-processes to describe the barcodes of sequences which admit these objects as limits, up to some quantifiable error.

[1]  Thomas Duquesne,et al.  Random Trees, Levy Processes and Spatial Branching Processes , 2002 .

[2]  Justin Curry,et al.  The fiber of the persistence map for functions on the interval , 2017, Journal of Applied and Computational Topology.

[3]  G. Parisi.  Brownian motion , 2005, Nature.

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

[5]  Spherical and Hyperbolic Fractional Brownian Motion , 2005 .

[6]  J. Neveu,et al.  Renewal property of the extrema and tree property of the excursion of a one-dimensional brownian motion , 1989 .

[7]  Steve Oudot,et al.  Persistence Theory - From Quiver Representations to Data Analysis , 2015, Mathematical surveys and monographs.

[8]  Thomas Kühn,et al.  Optimal series representation of fractional Brownian sheets , 2002 .

[9]  Sayan Mukherjee,et al.  Fréchet Means for Distributions of Persistence Diagrams , 2012, Discrete & Computational Geometry.

[10]  R. Adler,et al.  Random Fields and Geometry , 2007 .

[11]  J. Kahane.  Some Random Series of Functions , 1985 .

[12]  J. Picard.  A tree approach to $p$-variation and to integration , 2007, 0705.2128.

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

[14]  W. Feller.  The Asymptotic Distribution of the Range of Sums of Independent Random Variables , 1951 .

[15]  Kari Karhunen,et al.  Über lineare Methoden in der Wahrscheinlichkeitsrechnung , 1947 .

[16]  S. Evans.  Probability and Real Trees , 2008 .

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

[18]  E. Iglói.  A Rate-Optimal Trigonometric Series Expansion of the Fractional Brownian Motion , 2005 .

[19]  A. Borodin,et al.  Handbook of Brownian Motion - Facts and Formulae , 1996 .

[20]  John Roe,et al.  Lectures on coarse geometry , 2003 .

[21]  J. L. Gall,et al.  Brownian Motion, Martingales, and Stochastic Calculus , 2016 .

[22]  Théo Lacombe,et al.  Understanding the Topology and the Geometry of the Persistence Diagram Space via Optimal Partial Transport , 2019, ArXiv.

[23]  P. Massart,et al.  HUNGARIAN CONSTRUCTIONS FROM THE NONASYMPTOTIC VIEWPOINT , 1989 .

[24]  J. L. Gall,et al.  The Brownian cactus I. Scaling limits of discrete cactuses , 2011, 1102.4177.

[25]  Peter Orbanz.  Probability Theory II , 2011 .

[26]  S. Rice.  Mathematical analysis of random noise , 1944 .

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

[28]  D. P. Kennedy,et al.  The distribution of the maximum Brownian excursion , 1976, Journal of Applied Probability.

[29]  P. Major,et al.  An approximation of partial sums of independent RV'-s, and the sample DF. I , 1975 .

[30]  J. Azaïs,et al.  Level Sets and Extrema of Random Processes and Fields , 2009 .

[31]  J. L. Gall,et al.  Probabilistic and fractal aspects of Lévy trees , 2005, math/0501079.

[32]  W. Vervaat,et al.  A Relation between Brownian Bridge and Brownian Excursion , 1979 .

[33]  Daniel Perez.  On C0-persistent homology and trees , 2020, ArXiv.

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

[35]  yuliy baryshnikov.  Time Series, Persistent Homology and Chirality , 2019, 1909.09846.

[36]  H. McKean,et al.  Diffusion processes and their sample paths , 1996 .