A new construction for sublevel set persistence

TL;DR

This paper introduces a novel method for constructing filtered simplicial complexes from data using higher order Voronoi cells, enabling new insights into sublevel set persistence in topological data analysis.

Contribution

It presents a new construction of filtered simplicial complexes based on constrained optimization over Voronoi cells, linking it to classical sublevel set persistence under certain conditions.

Findings

The construction produces isomorphic homology groups under specified conditions.

It yields meaningful topological summaries in example datasets.

The method extends existing approaches by incorporating higher order Voronoi partitions.

Abstract

We construct a filtered simplicial complex $(X_L,f_L)$ associated to a subset $X\subset \mathbb{R}^d$, a function $f:X\rightarrow \mathbb{R}$ with compactly supported sublevel sets, and a collection of landmark points $L\subset \mathbb{R}^d$. The persistence values $f_L(\Delta)$ are defined as the minimizing values of a family of constrained optimization problems, whose domains are certain higher order Voronoi cells associated to $L$. We prove that $H_k^{a,b}(X_L)\cong H^{a,b}_k(X)$ provided that $f$ is the restriction of a smooth function, the landmarks are sufficiently dense, and $a<b$ are generic, and we show that the construction produces desirable results in some examples.