Related papers: Algebraic Stability of Zigzag Persistence Modules

We show how to formulate some recent results from homological stability of algebras in Graham and Lehrer's language of cellular algebras. The aim is to begin to connect the new results from topology to well-established representation…

The theory of zigzag persistence is a substantial extension of persistent homology, and its development has enabled the investigation of several unexplored avenues in the area of topological data analysis. In this paper, we discuss three…

We develop some aspects of the homological algebra of persistence modules, in both the one-parameter and multi-parameter settings, considered as either sheaves or graded modules. The two theories are different. We consider the graded module…

This paper develops the idea of homology for 1-parameter families of topological spaces. We express parametrized homology as a collection of real intervals with each corresponding to a homological feature supported over that interval or,…

The theory of persistence modules is an emerging field of algebraic topology which originated in topological data analysis. In these notes we provide a concise introduction into this field and give an account on some of its interactions…

In this article we establish two fundamental results for the sublevel set persistent homology for stationary processes indexed by the positive integers. The first is a strong law of large numbers for the persistence diagram (treated as a…

The stability of persistence diagrams is among the most important results in applied and computational topology. Most results in the literature phrase stability in terms of the bottleneck distance between diagrams and the $\infty$-norm of…

We define a class of multiparameter persistence modules that arise from a one-parameter family of functions on a topological space and prove that these persistence modules are stable. We show that this construction can produce…

One-dimensional persistent homology is arguably the most important and heavily used computational tool in topological data analysis. Additional information can be extracted from datasets by studying multi-dimensional persistence modules and…

Multigraded Betti numbers are one of the simplest invariants of multiparameter persistence modules. This invariant is useful in theory -- it completely determines the Hilbert function of the module and the isomorphism type of the free…

We prove a general representation stability result for polynomial coefficient systems which lets us prove representation stability and secondary homological stability for many families of groups with polynomial coefficients. This gives two…

In recent work, generalized persistence modules have proved useful in distinguishing noise from the legitimate topological features of a data set. Algebraically, generalized persistence modules can be viewed as representations for the poset…

We study distances on zigzag persistence modules from the viewpoint of derived categories and Auslander--Reiten quivers. The derived category of ordinary persistence modules is derived equivalent to that of arbitrary zigzag persistence…

In this paper we introduce the notion of the stability of a sequence of modules over Hecke algebras. We prove that a finitely generated consistent sequence associated with Hecke algebras is representation stable.

We show that the homology of modules for Hurwitz spaces stabilizes and compute its stable value. As one consequence, we compute the moments of Selmer groups in quadratic twist families of abelian varieties over suitably large function…

We discuss the algebra behind the matrix reduction algorithm for persistent homology, as presented in the paper ''Computing Persistent Homology'' by Afra Zomorodian and Gunnar Carlsson, in the lens of the more modern characterization of…

Persistent homology is a popular and useful tool for analysing finite metric spaces, revealing features that can be used to distinguish sets of unlabeled points and as input into machine learning pipelines. The famous stability theorem of…

A method to apply and visualize persistent homology of time series is proposed. The method captures persistent features in space and time, in contrast to the existing procedures, where one usually chooses one while keeping the other fixed.…

We prove a new kind of homological stability theorem for automorphism groups of finitely-generated projective modules over Dedekind domains, which takes into account all possible stabilisation maps between these, rather than only…

We prove a homological stability theorem for the moduli spaces of manifolds diffeomorphic to g(S^n x S^n), provided n > 2. This generalises Harer's stability theorem for the homology of mapping class groups. Combined with previous work of…