Related papers: Metrics for generalized persistence modules
The interleaving distance is arguably the most widely used metric in topological data analysis (TDA) due to its applicability to a wide array of inputs of interest, such as (multiparameter) persistence modules, Reeb graphs, merge trees, and…
The persistence diagram of Cohen-Steiner, Edelsbrunner, and Harer was recently generalized by Patel to the case of constructible persistence modules with values in a symmetric monoidal category with images. Patel also introduced a distance…
In persistent topology, q-tame modules appear as a natural and large class of persistence modules indexed over the real line for which a persistence diagram is definable. However, unlike persistence modules indexed over a totally ordered…
Multiparameter persistent homology has been largely neglected as an input to machine learning algorithms. We consider the use of lattice-based convolutional neural network layers as a tool for the analysis of features arising from…
We introduce topological conditions on a broad class of functionals that ensure that the persistent homology modules of their associated sublevel set filtration admit persistence diagrams, which, in particular, implies that they satisfy…
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…
Characterizing the dynamics of time-evolving data within the framework of topological data analysis (TDA) has been attracting increasingly more attention. Popular instances of time-evolving data include flocking/swarming behaviors in…
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…
We show that computing the interleaving distance between two multi-graded persistence modules is NP-hard. More precisely, we show that deciding whether two modules are $1$-interleaved is NP-complete, already for bigraded, interval…
We introduce and investigate the notions of expansiveness, topological stability and persistence for Borel measures with respect to time varying bi-measurable maps on metric spaces. We prove that expansive persistent measures are…
We consider different notions of equivalence for Morse functions on the sphere in the context of persistent homology, and introduce new invariants to study these equivalence classes. These new invariants are as simple, but more discerning…
We show that a persistence module (for a totally ordered indexing set) consisting of finite-dimensional vector spaces is a direct sum of interval modules. The result extends to persistence modules with the descending chain condition on…
Multiparameter persistence modules come up naturally in topological data analysis and topological robotics. Given a metric graph $(X,\delta)$, the second configuration space of $(X,\delta)$ with proximity parameters (for example, the…
For any persistence module $M$ over a finite poset $\mathbf{P}$, and any interval $I$ of $\mathbf{P}$, we give a formula for the multiplicity $d_M(V_I)$ of the interval module $V_I$ in the indecomposable decomposition of $M$ in terms of the…
In this paper, we study properties and patterns on permutations of multisets whose multivariate generating functions are symmetric. We interpret this phenomenon through the lens of group actions and define such a property or pattern as…
Let $\mathbf{k}$ be a field and let $V: \mathscr{C} \to \mathbf{k}\textup{-Mod}$ be a point-wise finite dimensional persistence modules, where $\mathscr{C}$ is a small category. Assume that for all local Artinian $\mathbf{k}$-algebras $R$…
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…
The use of topological persistence in contemporary data analysis has provided considerable impetus for investigations into the geometric and functional-analytic structure of the space of persistence modules. In this paper, we isolate a…
We introduce persistence matching diagrams induced by set mappings of metric spaces, based on 0-persistent homology of Vietoris-Rips filtrations. Also, we present a geometric definition of the persistence matching diagram that is more…
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…