Rigorous cubical approximation and persistent homology of continuous functions

TL;DR

This paper presents a method for discretizing vector-valued functions on Euclidean spaces to compute their persistent homology with guaranteed error bounds, requiring minimal smoothness assumptions.

Contribution

It introduces a rigorous cubical approximation technique that allows for verified computation of persistent homology of continuous functions with explicit error control.

Findings

Provides a discretization scheme with bounded error for vector-valued functions.

Enables rigorous computation of persistent homology with minimal smoothness requirements.

Demonstrates applicability to computational homology problems.

Abstract

The interaction between discrete and continuous mathematics lies at the heart of many fundamental problems in applied mathematics and computational sciences. In this paper we discuss the problem of discretizing vector-valued functions defined on finite-dimensional Euclidean spaces in such a way that the discretization error is bounded by a pre-specified small constant. While the approximation scheme has a number of potential applications, we consider its usefulness in the context of computational homology. More precisely, we demonstrate that our approximation procedure can be used to rigorously compute the persistent homology of the original continuous function on a compact domain, up to small explicitly known and verified errors. In contrast to other work in this area, our approach requires minimal smoothness assumptions on the underlying function.