Related papers: Invariants of persistence modules defined by order…
In the persistent homology of filtrations, the indecomposable decompositions provide the persistence diagrams. However, in almost all cases of multidimensional persistence, the classification of all indecomposable modules is known to be a…
Persistent homology was shown by Carlsson and Zomorodian to be homology of graded chain complexes with coefficients in the graded ring $\kk[t]$. As such, the behavior of persistence modules -- graded modules over $\kk[t]$ is an important…
Persistence diagrams are important descriptors in Topological Data Analysis. Due to the nonlinearity of the space of persistence diagrams equipped with their {\em diagram distances}, most of the recent attempts at using persistence diagrams…
Topological invariants have played a fundamental role in the advancement of theoretical high energy physics. Physicists have used several kinematic techniques to distinguish new physics predictions from the Standard Model (SM) of particle…
We study pointwise free and finitely-generated persistence modules over a principal ideal domain, indexed by a (possibly infinite) totally-ordered poset category. We show that such persistence modules admit interval decompositions if and…
We discuss applications of exact structures and relative homological algebra to the study of invariants of multiparameter persistence modules. This paper is mostly expository, but does contain a pair of novel results. Over finite posets,…
Modern representation learning increasingly relies on unsupervised and self-supervised methods trained on large-scale unlabeled data. While these approaches achieve impressive generalization across tasks and domains, evaluating embedding…
We study the structure of invariant measures for continuous automorphisms of compact metrizable abelian groups satisfying the descending chain condition. We show that the finitely supported invariant measures are weak-* dense in the space…
This paper addresses the stability analysis of infinite-dimensional sampled-data systems under unbounded perturbations. We present two classes of unbounded perturbations preserving the exponential stability of sampled-data systems. To this…
We introduce a formal definition of a pattern poset which encompasses several previously studied posets in the literature. Using this definition we present some general results on the M\"obius function and topology of such pattern posets.…
The Isometry Theorem of Chazal et al. and Lesnick is a fundamental result in persistence theory, which states that the interleaving distance between two one-parameter persistence modules is equal to the bottleneck distance between their…
Dey and Xin (J.Appl.Comput.Top., 2022, arXiv:1904.03766) describe an algorithm to decompose finitely presented multiparameter persistence modules using a matrix reduction algorithm. Their algorithm only works for modules whose generators…
Persistence has proved to be a valuable tool to analyze real world data robustly. Several approaches to persistence have been attempted over time, some topological in flavor, based on the vector space-valued homology functor, other…
Appropriately representing elements in a database so that queries may be accurately matched is a central task in information retrieval; recently, this has been achieved by embedding the graphical structure of the database into a manifold in…
The output of persistent homology is an algebraic object called a persistence module. This object admits a decomposition into a direct sum of interval persistence modules described entirely by the barcode invariant. In this paper we…
We study the computational complexity of decomposing finite discrete dynamical systems (FDDSs) in terms of the semiring operations of alternative and synchronous execution, which is useful for the analysis of discrete phenomena in science…
The use of persistent homology in applications is justified by the validity of certain stability results. At the core of such results is a notion of distance between the invariants that one associates with data sets. Here we introduce a…
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…
Barcodes form a complete set of invariants for interval decomposable persistence modules and are an important summary in topological data analysis. The set of barcodes is equipped with a canonical one-parameter family of metrics, the…
In many applications concerning the comparison of data expressed by $\mathbb{R}^m$-valued functions defined on a topological space $X$, the invariance with respect to a given group $G$ of self-homeomorphisms of $X$ is required. While…