Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Dynamical Persistent Homology via Wasserstein Gradient Flow

Minghua Wang, Jinhui Xu

TL;DR

This work tackles the challenge of translating dynamical changes in persistence diagrams, as computed by topological data analysis, back into actionable data-space modifications. It introduces two frameworks: (i) dynamical persistence via McCann interpolation that steers PDs along Wasserstein geodesics toward a target diagram using Benamou–Brenier dynamics and OT plans, and (ii) dynamical persistence via an energy functional that drives PD evolution without a predefined target through the JKO scheme. The methodologies enable controlled alteration of topological features by manipulating filtrations and data representations, validated on circle denoising and circle emergence tasks with Rips filtrations. The work advances the integration of optimal transport, differentiability in persistence, and gradient-flow ideas into practical dynamical TDA, while outlining limitations and avenues for future theory and efficiency improvements.

Abstract

In this study, we introduce novel methodologies designed to adapt original data in response to the dynamics of persistence diagrams along Wasserstein gradient flows. Our research focuses on the development of algorithms that translate variations in persistence diagrams back into the data space. This advancement enables direct manipulation of the data, guided by observed changes in persistence diagrams, offering a powerful tool for data analysis and interpretation in the context of topological data analysis.

Dynamical Persistent Homology via Wasserstein Gradient Flow

TL;DR

This work tackles the challenge of translating dynamical changes in persistence diagrams, as computed by topological data analysis, back into actionable data-space modifications. It introduces two frameworks: (i) dynamical persistence via McCann interpolation that steers PDs along Wasserstein geodesics toward a target diagram using Benamou–Brenier dynamics and OT plans, and (ii) dynamical persistence via an energy functional that drives PD evolution without a predefined target through the JKO scheme. The methodologies enable controlled alteration of topological features by manipulating filtrations and data representations, validated on circle denoising and circle emergence tasks with Rips filtrations. The work advances the integration of optimal transport, differentiability in persistence, and gradient-flow ideas into practical dynamical TDA, while outlining limitations and avenues for future theory and efficiency improvements.

Abstract

In this study, we introduce novel methodologies designed to adapt original data in response to the dynamics of persistence diagrams along Wasserstein gradient flows. Our research focuses on the development of algorithms that translate variations in persistence diagrams back into the data space. This advancement enables direct manipulation of the data, guided by observed changes in persistence diagrams, offering a powerful tool for data analysis and interpretation in the context of topological data analysis.

Paper Structure

This paper contains 19 sections, 2 theorems, 18 equations, 10 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Gradient Flow in Hilbert Space
  5. Proximal Map in Hilbert Space
  6. Gradient Flow in Wasserstein Space
  7. JKO Scheme
  8. Differentiability on Persistence Diagrams
  9. Methodology
  10. Dynamical Persistent Homology via McCann Interpolation
  11. Dynamical Persistent Homology via Energy Functional
  12. Case Illustrations
  13. Circle Denoising
  14. Repulsion Loss
  15. Circle Denoising via McCann Interpolation on 0th Persistence Diagram
  16. ...and 4 more sections

Key Result

Theorem 1

Let $\mu_0$ and $\mu_1$ be two probability measures on $\mathbb{R}^d$ with finite second moments. The squared Wasserstein distance $W_2^2(\mu_0, \mu_1)$ between $\mu_0$ and $\mu_1$ can be expressed as: subject to the continuity constraints:

Figures (10)

  • Figure 1: The input data is a set of points in a 2D plane, sampled from a circle with added Gaussian noise.
  • Figure 2: The evolution of the noisy circle data towards the target circle persistence diagram using McCann interpolation on the 0th persistence diagram and a denoising algorithm nigmetov2024topological on the 1st persistence diagram.
  • Figure 3: The evolution of the 0th persistence diagram.
  • Figure 4: The evolution of the 1st persistence diagram.
  • Figure 5: The evolution of the noisy circle data towards the target circle persistence diagram using McCann interpolation on the 0th persistence diagram and energy functional on the 1st persistence diagram.
  • ...and 5 more figures

Theorems & Definitions (3)

  • Theorem 1: Benamou-Brenier benamou2000computational
  • Theorem 2: McCann Interpolation villani2021topics
  • Remark 3: McCann Interpolation is a WGF ambrosio2008gradient