Continuation of Point Clouds via Persistence Diagrams

TL;DR

This paper introduces a mathematical and algorithmic framework for evolving point clouds to match target persistence diagrams using a differentiable persistence map and Newton-Raphson continuation, enabling inverse problems in topological data analysis.

Contribution

It presents a novel method leveraging differentiability of the persistence map to perform point cloud continuation towards desired persistence diagrams.

Findings

Successfully applied the method to various inverse problems in topological data analysis.

Demonstrated convergence of the Newton-Raphson continuation in this setting.

Provides a new tool for reconstructing point clouds from persistence diagrams.

Abstract

In this paper, we present a mathematical and algorithmic framework for the continuation of point clouds by persistence diagrams. A key property used in the method is that the persistence map, which assigns a persistence diagram to a point cloud, is differentiable. This allows us to apply the Newton-Raphson continuation method in this setting. Given an original point cloud $P$, its persistence diagram $D$, and a target persistence diagram $D'$, we gradually move from $D$ to $D'$, by successively computing intermediate point clouds until we finally find a point cloud $P'$ having $D'$ as its persistence diagram. Our method can be applied to a wide variety of situations in topological data analysis where it is necessary to solve an inverse problem, from persistence diagrams to point cloud data.