Ephemeral persistence features and the stability of filtered chain complexes

TL;DR

This paper introduces verbose barcodes, an enhanced invariant for persistent homology that incorporates ephemeral points, providing greater discriminative power and stability compared to traditional barcodes.

Contribution

It extends the stability theorem for Vietoris-Rips persistent homology by incorporating ephemeral points into filtered chain complexes, resulting in a more discriminative invariant called verbose barcode.

Findings

Verbose barcodes are more discriminative than standard barcodes.

Verbose barcodes are stable under certain sensitive metrics.

Examples show verbose barcodes distinguish spaces with identical standard barcodes.

Abstract

We strengthen the usual stability theorem for Vietoris-Rips (VR) persistent homology of finite metric spaces by building upon constructions due to Usher and Zhang in the context of filtered chain complexes. The information present at the level of filtered chain complexes includes points with zero persistence which provide additional information to that present at homology level. The resulting invariant, called verbose barcode, which has a stronger discriminating power than the usual barcode, is proved to be stable under certain metrics that are sensitive to these ephemeral points. In some situations, we provide ways to compute such metrics between verbose barcodes. We also exhibit several examples of finite metric spaces with identical (standard) VR barcodes yet with different verbose VR barcodes thus confirming that these ephemeral points strengthen the standard VR barcode.