Approximation by interval-decomposables and interval resolutions of persistence modules

Abstract

In topological data analysis, two-parameter persistence can be studied using the representation theory of the 2d commutative grid, the tensor product of two Dynkin quivers of type A. In a previous work, we defined interval approximations using restrictions to essential vertices of intervals together with Mobius inversion. In this work, we consider homological approximations using interval resolutions, and show that the interval resolution global dimension is finite for finite posets, and that it is equal to the maximum of the interval dimensions of the Auslander-Reiten translates of the interval representations. Furthermore, in the commutative ladder case, by a suitable modification of our interval approximation, we provide a formula linking the two conceptions of approximation.

On the Stability of Interval Decomposable Persistence Modules

“…Using Lemma 5.6, one can see that there are well-defined nonzero morphisms from I i to I j (1) whenever j ∈ {i, i + 1}. Thus, the morphism f : I → I (1) given by e i → e i+1 for 1 ≤ i ≤ 2n − 1 and e 2n → e 2n is welldefined. One can also check that f restricts to a well-defined morphism M → M (1): It is enough to check that f (M ) ⊆ M (1), and for this, it is enough to check f applied to the generators.…”

“…There is a constant c ≥ 1 such that for any ǫ-interleaved upset decomposable modules M and N , there is a cǫ-matching between their barcodes. This conjecture is of independent interest, as there is a number of recent papers [1,10,13,14,28] applying relative homological algebra to multiparameter persistence, where one describes a module in terms of interval modules. To show that invariants obtained this way are stable, one often ends up needing a statement analogous to Conjecture 6.5 (see e.g.…”

“…One can also check that f restricts to a well-defined morphism M → M (1): It is enough to check that f (M ) ⊆ M (1), and for this, it is enough to check f applied to the generators. For i ≤ 2n − 1, we have f qi (e i ) = e i+1 ∈ I qi+ (1,1) , which is in M , as there is a generator e i+1 ∈ M (1) qi+1− (1,1) , and q i+1 − (1, 1) ≤ q i . We leave the almost identical check of the other generators of M to the reader.…”

Homological approximations in persistence theory

“…Remark 6.6 After a preliminary version of this paper was posted on the arXiv, Asashiba, Escolar, Nakashima, and Yoshiwaki provided a positive answer to Question 6.5 in [AENYa]. Their proof is based on a result of Iyama [Iya03] (see also [Rin10]) which shows that for any finite-dimensional algebra Λ and any generator-cogenerator T ∈ modΛ, then there exists T ′ ∈ modΛ such that (i) End Λ (T ⊕ T ′ ) o p has finite global dimension and (ii) every indecomposable direct summand of T ′ is a submodule of an indecomposable direct summand of T. The positive answer to Question 6.5 then comes from showing that if X is the set of all connected spreads, then every submodule of an object in X lies in add(X) (see [AENYa,Lemma 4.4]).…”