On interval decomposability of 2D persistence modules

Asashiba, Hideto; Buchet, Mickaël; Escolar, Emerson G.; Nakashima, Ken; Yoshiwaki, Michio

doi:10.1016/j.comgeo.2022.101879

Cited by 16 publications

(10 citation statements)

References 17 publications

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“…If a persistence module does decompose into a direct sum of interval modules, then one can define the persistence diagram as the multiset of interval summands as in the one parameter case. Asashiba et al [1] developed a criterion for determining if an n-parameter persistence module is interval decomposable, as well as providing an algorithm to check this criterion in the n = 2 setting. In [2], Ashashiba et al provide an approach to approximate a 2-parameter persistence module by an interval decomposable module, with the approximation retaining the same dimension function and rank function.…”

Section: Related Workmentioning

confidence: 99%

“…If M is a persistence module over Z 2 , or a finite subset thereof, and I is a finite interval, we can use Dey et al's aforementioned algorithm to compute rank M (I). Even for a finite, small poset P ⊂ Z 2 , the size of Int(P ) can grow rapidly, see [1]. As a result, computing rank M (I) over all I ∈ Int(P ) is generally not practical.…”

Section: Our Contributionmentioning

confidence: 99%

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The discriminating power of the generalized rank invariant

Clause¹,

Kim²,

Mémoli³

2022

Preprint

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It is a well-known fact that there is no complete and discrete invariant on the collection of all multiparameter persistence modules. Nonetheless, many invariants have been proposed in the literature to study multiparameter persistence modules, though each invariant will lose some amount of information. One such invariant is the generalized rank invariant. This invariant is known to be complete on the class of interval decomposable persistence modules in general, under mild assumptions on the indexing poset P .There is often a trade-off, where the stronger an invariant is, the more expensive it is to compute in practice. The generalized rank invariant on its own is difficult to compute, whereas the standard rank invariant is readily computable through software implementations such as RIVET. We can interpolate between these two to induce new invariants via restricting the domain of the generalized rank invariant, and this family exhibits the aforementioned trade-off. This work studies the tension which exists between computational efficiency and retaining strength when restricting the domain of the generalized rank invariant. We provide a characterization result on where such restrictions are complete invariants in the setting where P is finite, and furthermore show that such restricted generalized rank invariants are stable.

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Section: Related Workmentioning

confidence: 99%

Section: Our Contributionmentioning

confidence: 99%

The discriminating power of the generalized rank invariant

Clause¹,

Kim²,

Mémoli³

2022

Preprint

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“…Note that [2] if part (2) of the definition is weakend to require only isomorphism, then the definition is weakened for modules that cannot be embedded in N m for m > 2.…”

Section: Multiparameter Persistence Modulesmentioning

confidence: 99%

On the complexity of zero-dimensional multiparameter persistence

Brodzki,

Burfitt,

Pirashvili

2020

Preprint

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Multiparameter persistence is a natural extension of the well-known persistent homology, which has attracted a lot of interest. However, there are major theoretical obstacles preventing the full development of this promising theory.In this paper we consider the interesting special case of multiparameter persistence in zero dimensions which can be regarded as a form of multiparameter clustering. In particular, we consider the multiparameter persistence modules of the zero-dimensional homology of filtered topological spaces when they are finitely generated. Under certain assumptions, we characterize such modules and study their decompositions. In particular we identify a natural class of representations that decompose and can be extended back to form zero-dimensional multiparameter persistence modules.Our study of this set of representations concludes that despite the restrictions, there are still infinitely many classes of indecomposables in this set.

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“….}. 1 Note that this definition is opposite to the usual definition of the incidence algebra [23]. We adopt this multiplication to make it match up to the usual interpretation of composition of functions from the left.…”

Section: Introductionmentioning

confidence: 99%

“…By the discussion in Section 4.1 of [1], each interval I in I( #-G m,n ) has a "staircase" form, which was denoted by:…”

Section: Introductionmentioning

confidence: 99%

Approximation by interval-decomposables and interval resolutions of persistence modules

Asashiba¹,

Escolar²,

Nakashima³

et al. 2022

Preprint

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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.

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On interval decomposability of 2D persistence modules

Cited by 16 publications

References 17 publications

The discriminating power of the generalized rank invariant

The discriminating power of the generalized rank invariant

On the complexity of zero-dimensional multiparameter persistence

Approximation by interval-decomposables and interval resolutions of persistence modules

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