Barcoding Invariants and Their Equivalent Discriminating Power

TL;DR

The paper develops a general, representation-theoretic framework for barcoding invariants in multi-parameter persistence by treating them as additive invariants on $K_0^{sp}(\mathcal{D})$ with a fixed basis. It proves that any two barcoding invariants sharing the same basis have equivalent discriminating power, meaning they induce isomorphic kernels, and thus new invariants add no generic advantage. It also shows that Möbius inversion preserves equivalence among invariants and that the generalized persistence diagram attains a universal role under suitable conditions, linking it with other interval-based invariants. The results unify and extend prior findings about interval barcodes, generalized ranks, and dim-hom invariants across finite and infinite posets, while suggesting that data-dependent methods are needed to leverage differences in practical settings.

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.

Figures (1)

• Figure 1: Illustration for Theorem \ref{['thm:classifying_interval-barcoding_invariants']}, for any finite poset $P$ for which the $2\times 3$ commutative grid is embedded. Inside the box is the Hasse diagram of the poset of equivalence classes of interval-barcoding invariants on $P$. The collection of interval Betti numbers $\{\beta_i^\mathrm{Int}\}_{i=0}^\infty$ it is an invariant on $\mathop{\mathrm{fp\text{-}rep}}\nolimits P$ which determines both $\chi^\mathrm{Int}$ and $\beta_0^\mathrm{Int}$. The equivalence $\mathrm{rk}_M^\mathrm{Int}\sim c^\xi$ follows from Remark \ref{['rem:essentially_covers']} and that $\mathrm{rk}^\mathrm{Int}\sim\mathrm{dgm}^\mathrm{Int}$ (cf. Example \ref{['example:gpd']}).

Theorems & Definitions (78)

• Definition 2.1
• Proposition 2.2: botnan2018algebraic
• Definition 2.3
• Remark 2.4
• Proposition 2.5
• proof
• Lemma 2.6
• Definition 2.8: Split Grothendieck group
• Lemma 2.9: e.g. lu2013algebraic
• Definition 2.10