Persistent homology of semi-algebraic sets

TL;DR

This paper introduces the first singly exponential algorithm for computing persistent homology barcodes of semi-algebraic sets filtered by polynomial sub-level sets, extending previous Betti number computation methods.

Contribution

It presents a novel algorithm with singly exponential complexity for persistent homology of semi-algebraic sets, generalizing Betti number computation to filtered sets.

Findings

First singly exponential algorithm for persistent homology of semi-algebraic sets

Generalizes Betti number algorithms to filtration-based topological features

Efficient computation up to fixed dimension ll

Abstract

We give an algorithm with singly exponential complexity for computing the barcodes up to dimension $\ell$ (for any fixed $\ell \geq 0$) of the filtration of a given semi-algebraic set by the sub-level sets of a given polynomial. Our algorithm is the first algorithm for this problem with singly exponential complexity, and generalizes the corresponding results for computing the Betti numbers up to dimension $\ell$ of semi-algebraic sets with no filtration present.