相关论文: Erosion distance for generalized persistence modul…

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…

Recently, $p$-presentation distances for $p\in [1,\infty]$ were introduced for merge trees and multiparameter persistence modules as more sensitive variations of the respective interleaving distances ($p=\infty)$. It is well-known that…

Persistent homology has been recently studied with the tools of sheaf theory in the derived setting by Kashiwara and Schapira, after J. Curry has made the first link between persistent homology and sheaves. We prove the isometry theorem in…

We study the multi-dimensional persistence of Carlsson and Zomorodian and obtain a finer classification based upon the higher tor-modules of a persistence module. We propose a variety structure on the set of isomorphism classes of these…

Let P be a distribution with support S. The salient features of S can be quantified with persistent homology, which summarizes topological features of the sublevel sets of the distance function (the distance of any point x to S). Given a…

Topological data analysis is becoming a popular way to study high dimensional feature spaces without any contextual clues or assumptions. This paper concerns itself with one popular topological feature, which is the number of…

Adapting a definition given by Bjerkevik and Lesnick for multiparameter persistence modules, we introduce an $\ell^p$-type extension of the interleaving distance on merge trees. We show that our distance is a metric, and that it…

Persistent homology is a topological data analysis tool that has been widely generalized, extending its scope beyond the field of topology. Among its extensions, steady and ranging persistence were developed to study a wide variety of graph…

In this paper we study a new metric for comparing Betti numbers functions in bidimensional persistent homology, based on coherent matchings, i.e. families of matchings that vary in a continuous way. We prove some new results about this…

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…

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 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…

In this article, we consider extended tame persistence commutative differential graded algebras (CDGAs) associated with relative Sullivan algebras. In particular, if the relative Sullivan algebra is a model for a map between spaces, then…

Persistence diagrams offer a way to summarize topological and geometric properties latent in datasets. While several methods have been developed that utilize persistence diagrams in statistical inference, a full Bayesian treatment remains…

Persistent homology is a fundamental tool in Topological Data Analysis. The associated algebraic structure is the persistence module, a sequence of vector spaces connected by linear maps. Persistence modules admit a complete and…

The distance function to a compact set plays a crucial role in the paradigm of topological data analysis. In particular, the sublevel sets of the distance function are used in the computation of persistent homology -- a backbone of the…

This paper brings together three distinct theories with the goal of quantifying shape textures with complex morphologies. Distance fields are central objects in shape representation, while topological data analysis uses algebraic topology…

We introduce the concept of weighted persistence diagrams and develop a functorial pipeline for constructing them from finite metric measure spaces. This builds upon an existing functorial framework for generating classical persistence…

Persistent homology maps a simplicial complex filtered by elements in $\mathbb R$ to finite formal sums of elements of $\mathbb R_{\leq}^{2} = \{ (b,d) \in \mathbb R^2 \cup \{ \infty \} \mid b < d \}$ called (finite) persistence diagrams.…

We consider a smooth closed orientable submanifold $M \subset \mathbb{R}^D$ with narrow cycles. We embed $M$ into a scaled oriented Grassmannian bundle via the Gauss map in order to enlarge the scale of these cycles. Under mild assumptions,…