Computing persistent Stiefel-Whitney classes of line bundles

TL;DR

This paper introduces a new method for computing persistent Stiefel-Whitney classes of line bundles from data, providing stable, consistent, and practical algorithms with applications in image analysis.

Contribution

It defines persistent Stiefel-Whitney classes for vector bundle filtrations and develops an effective algorithm for their computation from finite datasets.

Findings

The construction of Cech bundle filtrations is stable and consistent.

An effective algorithm for computing persistent Stiefel-Whitney classes is proposed.

Application to datasets in image analysis demonstrates practical utility.

Abstract

We propose a definition of persistent Stiefel-Whitney classes of vector bundle filtrations. It relies on seeing vector bundles as subsets of some Euclidean spaces. The usual \v{C}ech filtration of such a subset can be endowed with a vector bundle structure, that we call a \v{C}ech bundle filtration. We show that this construction is stable and consistent. When the dataset is a finite sample of a line bundle, we implement an effective algorithm to compute its persistent Stiefel-Whitney classes. In order to use simplicial approximation techniques in practice, we develop a notion of weak simplicial approximation. As a theoretical example, we give an in-depth study of the normal bundle of the circle, which reduces to understanding the persistent cohomology of the torus knot (1,2). We illustrate our method on several datasets inspired by image analysis.