The persistent homology of a sampled map: From a viewpoint of quiver representations

TL;DR

This paper introduces a new filtration analysis method for sampled maps using persistent homology and quiver representations, enabling map reconstruction without prior eigenvalue information, and proves stability under dense sampling.

Contribution

It extends homology induced maps via quiver representations and provides a stable, eigenvalue-free approach for analyzing sampled maps with practical numerical demonstrations.

Findings

Persistent homology of sampled maps matches that of the underlying map with dense sampling.

The method does not require prior eigenvalue information.

Numerical examples demonstrate effectiveness of the approach.

Abstract

This paper aims to introduce a filtration analysis of sampled maps based on persistent homology, providing a new method for reconstructing the underlying maps. The key idea is to extend the definition of homology induced maps of correspondences using the framework of quiver representations. Our definition of homology induced maps is given by most persistent direct summands of representations, and the direct summands uniquely determine a persistent homology. We provide stability theorems of this process and show that the output persistent homology of the sampled map is the same as that of the underlying map if the sample is dense enough. Compared to existing methods using eigenspace functors, our filtration analysis has an advantage that no prior information on the eigenvalues of the underlying map is required. Some numerical examples are given to illustrate the effectiveness of our method.