On 𝜖 approximations of persistence diagrams

Abstract

Biological and physical systems often exhibit distinct structures at different spatial/temporal scales. Persistent homology is an algebraic tool that provides a mathematical framework for analyzing the multi-scale structures frequently observed in nature. In this paper a theoretical framework for the algorithmic computation of an arbitrarily good approximation of the persistent homology is developed. We study the filtrations generated by sub-level sets of a function f : X → R f \colon X \to \mathbb {R} , where X X is a CW-complex. In the special case X = [ 0 , 1 ] N X = [0,1]^N , N ∈ N N \in \mathbb {N} , we discuss implementation of the proposed algorithms. We also investigate a priori and a posteriori bounds of the approximation error introduced by our method.