Divol, Vincent

Topological data analysis (or TDA for short) consists in a set of methods aiming to extract topological and geometric information from complex nonlinear datasets. This field is here tackled from two different perspectives.First, we consider techniques from geometric inference, whose goal is to reconstruct geometric invariants of a manifold thanks t...

Barannikov, Serguei Korotin, Alexander Oganesyan, Dmitry Emtsev, Daniil Burnaev, Evgeny

We apply the canonical forms (barcodes) of gradient Morse complexes to explore topology of loss surfaces. We present a new algorithm for calculations of the objective function's barcodes of minima. Our experiments confirm two principal observations: (1) the barcodes of minima are located in a small lower part of the range of values of loss function...

Harker, Shaun Kramár, Miroslav Levanger, Rachel Mischaikow, Konstantin
Published in
Journal of applied and computational topology

We present a generalization of the induced matching theorem of as reported by Bauer and Lesnick (in: Proceedings of the thirtieth annual symposium computational geometry 2014 ) and use it to prove a generalization of the algebraic stability theorem for ℝ -indexed pointwise finite-dimensional persistence modules. Via numerous examples, we show how t...

Barannikov, S.

There is canonical partition of set of critical values of smooth function into pairs "birth-death" and a separate set representing basis in homology, as was shown in S.Barannikov "Framed Morse complex and its invariants"(1994). This partition arises from bringing filtered complex, defined by gradient trajectories of the function, to so called "cano...

Angeli, Alessia Ferri, Massimo Monti, Eleonora Tomba, Ivan

This is the report of a preliminary study, in which a new coding of persistence diagrams and two relevance feedback methods, designed for use with persistent homology, are combined. The coding consists in substituting persistence diagrams with complex polynomials; these are “shortened”, in the sense that only the first few coefficients are used. Th...

Turner, Katharine Mileyko, Yuriy Mukherjee, Sayan Harer, John
Published in
Discrete & Computational Geometry

Given a distribution ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document} on persistence diagrams and observations X1,…,Xn∼iidρ\documentclass[12pt]{mi...

Cerri, Andrea Di Fabio, Barbara Jabłoński, Grzegorz Medri, Filippo
Published in
Computer Vision and Image Understanding

Two of the main ingredients of topological persistence for shape comparison are persistence diagrams and the matching distance. Persistence diagrams are signatures capturing meaningful properties of shapes, while the matching distance can be used to stably compare them. From the application viewpoint, one drawback of these tools is the computationa...

Chazal, Frédéric Cohen-Steiner, David Glisse, Marc Guibas, Leonidas Oudot, Steve

Topological persistence has proven to be a key concept for the study of real-valued functions defined over topological spaces. Its validity relies on the fundamental property that the persistence diagrams of nearby functions are close. However, existing stability results are restricted to the case of continuous functions defined over triangulable s...

Chazal, Frédéric Cohen-Steiner, David Glisse, Marc Guibas, Leonidas J. Oudot, Steve

Topological persistence has proven to be a key concept for the study of real-valued functions defined over topological spaces. Its validity relies on the fundamental property that the persistence diagrams of nearby functions are close. However, existing stability results are restricted to the case of continuous functions defined over triangulable s...