A new construction for sublevel set persistence

Abstract

We construct a filtered simplicial complex $(X_L,f_L)$ associated to a subset $X\subset {\mathbb{R}}^d$, a function $f:X\to \mathbb{R}$ 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 $H_k^{a,b}(X_L)\cong H_k^{a,b}(X)$ 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.