Persistence Diagrams and the Heat Equation Homotopy

TL;DR

This paper introduces a method using persistence diagrams and the heat equation to analyze topological differences between functions on various manifolds, revealing how these features evolve over a homotopy.

Contribution

It formulates a heat equation-based homotopy for functions and stacks persistence diagrams into vineyards to study topological changes across different topologies.

Findings

Persistence diagrams change systematically over the homotopy.

Topological features evolve differently on various manifolds.

Heat equation homotopy reveals topological differences clearly.

Abstract

Persistence homology is a tool used to measure topological features that are present in data sets and functions. Persistence pairs births and deaths of these features as we iterate through the sublevel sets of the data or function of interest. I am concerned with using persistence to characterize the difference between two functions f, g : M -> R, where M is a topological space. Furthermore, I formulate a homotopy from g to f by applying the heat equation to the difference function g-f. By stacking the persistence diagrams associated with this homotopy, we create a vineyard of curves that connect the points in the diagram for f with the points in the diagram for g. I look at the diagrams where M is a square, a sphere, a torus, and a Klein bottle. Looking at these four topologies, we notice trends (and differences) as the persistence diagrams change with respect to time.