Related papers: A Comparison Framework for Interleaved Persistence…
The extended persistence diagram is an invariant of piecewise linear functions, which is known to be stable under perturbations of functions with respect to the bottleneck distance as introduced by Cohen-Steiner, Edelsbrunner, and Harer. We…
Computation of the interleaving distance between persistence modules is a central task in topological data analysis. For $1$-parameter persistence modules, thanks to the isometry theorem, this can be done by computing the bottleneck…
For a $P$-indexed persistence module ${\sf M}$, the (generalized) rank of ${\sf M}$ is defined as the rank of the limit-to-colimit map for the diagram of vector spaces of ${\sf M}$ over the poset $P$. For $2$-parameter persistence modules,…
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…
While decomposition of one-parameter persistence modules behaves nicely, as demonstrated by the algebraic stability theorem, decomposition of multiparameter modules is known to be unstable in a certain precise sense. Until now, it has not…
We generalize the persistence diagram of Cohen-Steiner, Edelsbrunner, and Harer to the setting of constructible persistence modules valued in a symmetric monoidal category. We call this the type A persistence diagram of a persistence…
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…
In this paper, we consider a matroid generalization of the stable matching problem. In particular, we consider the setting where preferences may contain ties. For this generalization, we propose a polynomial-time algorithm for the problem…
Recently, bipath persistent homology has been proposed as an extension of standard persistent homology, along with its visualization (bipath persistence diagram) and computational methods. In the setting of standard persistent homology, the…
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 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…
Since persistence diagrams do not admit an inner product structure, a map into a Hilbert space is needed in order to use kernel methods. It is natural to ask if such maps necessarily distort the metric on persistence diagrams. We show that…
The notion of generalized rank invariant in the context of multiparameter persistence has become an important ingredient for defining interesting homological structures such as generalized persistence diagrams. Naturally, computing these…
In this paper, we are interested in investigating notions of stability for generalized linear differential equations (GLDEs). Initially, we propose and revisit several definitions of stability and provide a complete characterisation of them…
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…
The concept of edit distance, which dates back to the 1960s in the context of comparing word strings, has since found numerous applications with various adaptations in computer science, computational biology, and applied topology. By…
We introduce persistence with an emphasis on its algebraic foundations, using the representation theory of posets. Linear representations of posets arise in several areas of mathematics, including the representation theory of quivers and…
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…
When a category $\mathcal{C}$ satisfies certain conditions, we define the notion of rank invariant for arbitrary poset-indexed functors $F:\mathbf{P} \rightarrow \mathcal{C}$ from a category theory perspective. This generalizes the standard…
Persistence modules are representations of products of totally ordered sets in the category of vector spaces. They appear naturally in the representation theory of algebras, but in recent years they have also found applications in other…