Barcoding Invariants and Their Equivalent Discriminating Power

TL;DR

This paper formalizes barcoding invariants in persistent homology, compares their discriminating power, and shows that invariants with the same basis are equally powerful, highlighting limitations in current invariant design.

Contribution

It introduces a formal framework for barcoding invariants, proves their equivalence in discriminating power when sharing the same basis, and discusses implications for developing new invariants.

Findings

All barcoding invariants with the same basis have equivalent discriminating power.

Introducing a new invariant with the same basis does not improve discriminating power.

Generalizes recent results on invariants for poset representations.

Abstract

The persistence barcode (equivalently, the persistence diagram), which can be obtained from the interval decomposition of a persistence module, plays a pivotal role in applications of persistent homology. For multi-parameter persistent homology, which lacks a complete discrete invariant, and where persistence modules are no longer always interval decomposable, many alternative invariants have been proposed. Many of these invariants are akin to persistence barcodes, in that they assign (possibly signed) multisets of intervals. Furthermore, to any interval decomposable module, those invariants assign the multiset of intervals that correspond to its summands. Naturally, identifying the relationships among invariants of this type, or ordering them by their discriminating power, is a fundamental question. To address this, we formalize the notion of barcoding invariants and compare their discriminating powers. Notably, this formalization enables us to prove that all barcoding invariants with the same basis possess equivalent discriminating power. One implication of our result is that introducing a new barcoding invariant does not add any value in terms of its generic discriminating power, even if the new invariant is distinct from the existing barcoding invariants. This suggests the need for a more flexible and adaptable comparison framework for barcoding invariants. Along the way, we generalize several recent results on the discriminative power of invariants for poset representations within our unified framework.