A Theory of Sub-Barcodes

TL;DR

This paper introduces sub-barcodes as a new mathematical framework in topological data analysis, enabling stronger guarantees about unknown barcodes using bounds, and establishing a functorial relationship with factorizations of persistence modules.

Contribution

It develops the theory of sub-barcodes, providing a functor from factorizations of persistence module homomorphisms to a poset of barcodes, offering a looser but more robust alternative to traditional matchings.

Findings

Sub-barcodes can be constructed from bounds of functions.

Sub-barcodes form subobjects in a functor category.

The theory offers strong guarantees without interleavings.

Abstract

From the work of Bauer and Lesnick, it is known that there is no functor from the category of pointwise finite-dimensional persistence modules to the category of barcodes and overlap matchings. In this work, we introduce sub-barcodes and show that there is a functor from the category of factorizations of persistence module homomorphisms to a poset of barcodes ordered by the sub-barcode relation. Sub-barcodes and factorizations provide a looser alternative to bottleneck matchings and interleavings that can give strong guarantees in a number of settings that arise naturally in topological data analysis. The main use of sub-barcodes is to make strong claims about an unknown barcode in the absence of an interleaving. For example, given only upper and lower bounds $g\geq f\geq \ell$ of an unknown real-valued function $f$, a sub-barcode associated with $f$ can be constructed from $\ell$ and $g$ alone. We propose a theory of sub-barcodes and observe that the subobjects in the category of functors from intervals to matchings naturally correspond to sub-barcodes.