Barcoding Invariants and Their Comparison

The persistence barcode, 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 (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 them by comparing their kernels, which are taken as a measure of their (in-)discriminating power. We show that any two different barcoding invariants $f$ and $g$ with the same basis are incomparable; i.e. one cannot be strictly finer than the other. Furthermore, we identify what we call a transfer isomorphism between the kernels of $f$ and $g$, implying that, given any pair of persistence modules that are not distinguishable via $f$ but are via $g$, one can generate another pair of persistence modules that are so via $f$, but not via $g$. One implication of the existence of the transfer isomorphism is that introducing a new barcoding invariant does not add any value in terms of its generic discriminating power, even if it is distinct from the existing barcoding invariants. Another implication is a novel characterization of the generalized persistence diagram without involving M\"obius inversion. Along the way, we generalize several recent results on the discriminative power of invariants for poset representations within our unified framework.