Related papers: An isometry theorem for persistent homology of cir…
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…
Persistence diagrams, combining geometry and topology for an effective shape description used in pattern recognition, have already proven to be an effective tool for shape representation with respect to a certainfiltering function.…
The stability theorem for persistent homology is a central result in topological data analysis. While the original formulation of the result concerns the persistence barcodes of $\mathbb{R}$-valued functions, the result was later cast in a…
The $\gamma$-linear projected barcode was recently introduced as an alternative to the well-known fibered barcode for multiparameter persistence, in which restrictions of the modules to lines are replaced by pushforwards of the modules…
The aim of applied topology is to use and develop topological methods for applied mathematics, science and engineering. One of the main tools is persistent homology, an adaptation of classical homology, which assigns a barcode, i.e. a…
The Isometry Theorem for continuous quiver of type $A$ plays an important role in persistent homology. In this paper, we shall generalize Isometry Theorem to continuous quiver of type $\tilde{A}$.
Persistent homology is a popular and useful tool for analysing finite metric spaces, revealing features that can be used to distinguish sets of unlabeled points and as input into machine learning pipelines. The famous stability theorem of…
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…
This paper constructs a persistence module of Floer cohomology groups associated to a contactomorphism of the ideal boundary of a Liouville manifold. The barcode (or, bottleneck) distance between the persistence modules is bounded from…
We study circle valued maps and consider the persistence of the homology of their fibers. The outcome is a finite collection of computable invariants which answer the basic questions on persistence and in addition encode the topology of the…
Relying on sheaf theory, we introduce the notions of projected barcodes and projected distances for multi-parameter persistence modules. Projected barcodes are defined as derived pushforward of persistence modules onto $\mathbb{R}$.…
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…
The persistence barcode (equivalently, the persistence diagram), which can be obtained from the interval decomposition of a persistence module, plays a pivotal role in applications of persistent homology. For multi-parameter persistent…
We provide a definition of ephemeral multi-persistent modules and prove that the quotient of persistent modules by the ephemeral ones is equivalent to the category of $\gamma$-sheaves. In the case of one-dimensional persistence, our…
The natural pseudo-distance of spaces endowed with filtering functions is precious for shape classification and retrieval; its optimal estimate coming from persistence diagrams is the bottleneck distance, which unfortunately suffers from…
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…
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…
The theory of persistence, which arises from topological data analysis, has been intensively studied in the one-parameter case both theoretically and in its applications. However, its extension to the multi-parameter case raises numerous…
The interleaving distance is arguably the most prominent distance measure in topological data analysis. In this paper, we provide bounds on the computational complexity of determining the interleaving distance in several settings. We show…
The Quillen-McCord theorem (aka Quillen fiber lemma) gives a sufficient condition on a map between classifying spaces of posetal categories to be a homotopy equivalence. Jonathan Ariel Barmak in his paper [arXiv:1005.0538] gives an…