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Persistent homology is a way of determining the topological properties of a data set. It is well known that each persistence module admits the structure of a representation of a finite totally ordered set. In previous work, the authors…
Throughout this paper, a persistence diagram ${\cal P}$ is composed of a set $P$ of planar points (each corresponding to a topological feature) above the line $Y=X$, as well as the line $Y=X$ itself, i.e., ${\cal P}=P\cup\{(x,y)|y=x\}$.…
In many matching markets--such as athlete recruitment or academic admissions--participants on one side are evaluated by attribute vectors known to the other side, which in turn applies individual \emph{salience vectors} to assign relative…
In this paper, we study pointwise finite-dimensional (p.f.d.) $2$-parameter persistence modules where each module admits a finite convex isotopy subdivision. We show that a p.f.d. $2$-parameter persistence module $M$ (with a finite convex…
Persistent homology analysis provides means to capture the connectivity structure of data sets in various dimensions. On the mathematical level, by defining a metric between the objects that persistence attaches to data sets, we can…
An important problem in the field of Topological Data Analysis is defining topological summaries which can be combined with traditional data analytic tools. In recent work Bubenik introduced the persistence landscape, a stable…
The pruning distance recently introduced by Bjerkevik compares persistence modules using approximate decompositions called prunings. Bjerkevik conjectures that this distance is Lipschitz equivalent to the classical interleaving distance on…
The classical persistence algorithm computes the unique decomposition of a persistence module implicitly given by an input simplicial filtration. Based on matrix reduction, this algorithm is a cornerstone of the emergent area of topological…
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…
In topological data analysis (TDA), persistence diagrams have been a succesful tool. To compare them, Wasserstein and Bottleneck distances are commonly used. We address the shortcomings of these metrics and show a way to investigate them in…
Persistent homology has become an important tool for extracting geometric and topological features from data, whose multi-scale features are summarized in a persistence diagram. From a statistical perspective, however, persistence diagrams…
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…
Local conditions for the direct summands of a persistence module to belong to a certain class of indecomposables have been proposed in the 2-parameter setting, notably for the class of indecomposables called block modules, which plays a…
Motivated by the need to relate the biparameter persistence module induced by a pair of scalar functions with the monoparameter persistence modules induced by each function separately, we introduce a construction that defines a kind of…
As well-known, inner functions play an important role in the study of bounded analytic function theory. In recent years, persistence module theory, as a main tool applied to Topological Data Analysis, has received widespread attention. In…
In multi-parameter persistence, the matching distance is defined as the supremum of weighted bottleneck distances on the barcodes given by the restriction of persistence modules to lines with a positive slope. In the case of finitely…
In persistent homology analysis, interval modules play a central role in describing the birth and death of topological features across a filtration. In this work, we extend this setting, and propose the use of bipath persistent homology,…
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 consider the degree-Rips construction from topological data analysis, which provides a density-sensitive, multiparameter hierarchical clustering algorithm. We analyze its stability to perturbations of the input data using the…
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…