On the stability of multigraded Betti numbers and Hilbert functions

Oudot, Steve; Scoccola, Luis

doi:10.48550/arxiv.2112.11901

Cited by 3 publications

(3 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…After the first version of this manuscript was posted to arXiv, Oudot and Scoccola proved a stability result for certain invariants coming from exact structures [OS,Theorem 26]. Their result is proven with respect to the "Bottleneck dissimilarity function" for finitely presented R n -persistence modules.…”

Section: Generalized Betti Numbers and G-vectorsmentioning

confidence: 99%

Homological approximations in persistence theory

BLANCHETTE¹,

Brüstle²,

Hanson³

2022

Can. J. Math.-J. Can. Math.

View full text Add to dashboard Cite

We define a class of invariants, which we call homological invariants, for persistence modules over a finite poset. Informally, a homological invariant is one that respects some homological data and takes values in the free abelian group generated by a finite set of indecomposable modules. We focus in particular on groups generated by "spread modules", which are sometimes called "interval modules" in the persistence theory literature. We show that both the dimension vector and rank invariant are equivalent to homological invariants taking values in groups generated by spread modules. We also show that the free abelian group generated by the "single-source" spread modules gives rise to a new invariant which is finer than the rank invariant. They are also thankful to an anonymous referee for their thorough reading of this paper and suggestions for improvement. Contents 1. Introduction 1 2. Notation and Terminology 4 3. Motivation and related invariants 6 4. Relative homological algebra 9 5. Classes and morphisms of spread modules. 14 6. Homological invariants in persistence theory 17 7. Examples and comparison to other invariants 19 References 23

show abstract

Section: Generalized Betti Numbers and G-vectorsmentioning

confidence: 99%

Homological approximations in persistence theory

BLANCHETTE¹,

Brüstle²,

Hanson³

2022

Can. J. Math.-J. Can. Math.

View full text Add to dashboard Cite

show abstract

“…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.…”

Section: Upset Decomposable Modulesmentioning

confidence: 99%

On the Stability of Interval Decomposable Persistence Modules

Bjerkevik

2021

Discrete Comput Geom

View full text Add to dashboard Cite

The algebraic stability theorem for persistence modules is a central result in the theory of stability for persistent homology. We introduce a new proof technique which we use to prove a stability theorem for n-dimensional rectangle decomposable persistence modules up to a constant $$2n-1$$ 2 n - 1 that generalizes the algebraic stability theorem, and give an example showing that the bound cannot be improved for $$n=2$$ n = 2 . We then apply the technique to prove stability for block decomposable modules, from which novel results for zigzag modules and Reeb graphs follow. These results are improvements on weaker bounds in previous work, and the bounds we obtain are optimal.

show abstract

“…To do so, the TDA community has been mostly using module-theoretic notions, such as the rankinvariant [15,16], the Hilbert function or the graded Betti numbers [24,5,35,31]. From a sheaf-theoretic perspective, a natural numerical invariant to consider is the local Euler characteristic.…”

Section: Introductionmentioning

confidence: 99%

Persistence and the Sheaf-Function Correspondence

Berkouk

2023

Forum of Mathematics, Sigma

View full text Add to dashboard Cite

The sheaf-function correspondence identifies the group of constructible functions on a real analytic manifold M with the Grothendieck group of constructible sheaves on M. When M is a finite dimensional real vector space, Kashiwara-Schapira have recently introduced the convolution distance between sheaves of $\mathbf {k}$ -vector spaces on M. In this paper, we characterize distances on the group of constructible functions on a real finite dimensional vector space that can be controlled by the convolution distance through the sheaf-function correspondence. Our main result asserts that such distances are almost trivial: they vanish as soon as two constructible functions have the same Euler integral. We formulate consequences of our result for Topological Data Analysis: there cannot exist nontrivial additive invariants of persistence modules that are continuous for the interleaving distance.

show abstract

On the stability of multigraded Betti numbers and Hilbert functions

Cited by 3 publications

References 12 publications

Homological approximations in persistence theory

Homological approximations in persistence theory

On the Stability of Interval Decomposable Persistence Modules

Persistence and the Sheaf-Function Correspondence

Contact Info

Product

Resources

About