A Geometric Perspective on Sparse Filtrations

TL;DR

This paper introduces a geometric approach to sparse filtrations in topological data analysis, simplifying proofs and extending applicability to various filtrations and metrics, along with an efficient algorithm for their construction.

Contribution

It provides a new geometric perspective that simplifies proofs, generalizes to different filtrations and metrics, and offers an algorithm for constructing sparse filtrations.

Findings

Simplified proofs for sparse filtrations

Generalization to Rips and Cech filtrations with convex metrics

Efficient algorithm for simplex construction and vertex removal

Abstract

We present a geometric perspective on sparse filtrations used in topological data analysis. This new perspective leads to much simpler proofs, while also being more general, applying equally to Rips filtrations and Cech filtrations for any convex metric. We also give an algorithm for finding the simplices in such a filtration and prove that the vertex removal can be implemented as a sequence of elementary edge collapses.