Computing Persistence Diagram Bundles

TL;DR

This paper introduces an algorithm for computing piecewise-linear persistence diagram bundles, extending the concept of vineyards to more complex parameter spaces, with detailed methods for triangulated surfaces.

Contribution

The paper presents a novel algorithm for computing piecewise-linear PD bundles, including detailed methods for triangulated surfaces and generalizations to higher dimensions.

Findings

Algorithm effectively computes PD bundles for triangulated surfaces.

Extends vineyard concepts to broader classes of parameter spaces.

Provides a framework for practical computation of complex persistent homology structures.

Abstract

Persistence diagram (PD) bundles, a generalization of vineyards, were introduced as a way to study the persistent homology of a set of filtrations parameterized by a topological space $B$. In this paper, we present an algorithm for computing piecewise-linear PD bundles, a wide class that includes many of the PD bundles that one may encounter in practice. Full details are given for the case in which $B$ is a triangulated surface, and we outline the generalization to higher dimensions and other cases.