Related papers: Persistent Cost of Lipschitz Maps
We consider the category of all locally Lipschitz contractible metric spaces and all locally Lipschitz maps, which is a wide class of metric spaces, including all finite dimensional Alexandrov spaces and all CAT spaces. We also consider the…
This paper is connected with the problem of describing path metric spaces that are homeomorphic to manifolds and biLipschitz homogeneous, i.e., whose biLipschitz homeomorphism group acts transitively. Our main result is the following. Let…
We study the persistent homology of both functional data on compact topological spaces and structural data presented as compact metric measure spaces. One of our goals is to define persistent homology so as to capture primarily properties…
A labeled metric space is intuitively speaking a metric space together with a special set of points to be understood as the geometric boundary of the space. We study basic properties of a recently introduced labeled Gromov-Hausdorff…
Swarm and constellation reconfiguration can be viewed as motion of an unordered point configuration in an ambient space. Here, we provide persistence-stable, symmetry-invariant geometric representations for comparing and monitoring…
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…
Persistence modules that decompose into interval modules are important in topological data analysis because we can interpret such intervals as the lifetime of topological features in the data. We can classify the settings in which…
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…
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…
We use persistence modules and their corresponding barcodes to quantitatively distinguish between different fiberwise star-shaped domains in the cotangent bundle of a fixed manifold. The distance between two fiberwise star-shaped domains is…
This paper establishes a general topological condition under which the semilocal stability of a set-valued mapping can be exactly determined by its local stability properties. Specifically, we investigate the relationship between the…
The study of persistent homology has contributed new insights and perspectives into a variety of interesting problems in science and engineering. Work in this domain relies on the result that any finitely-indexed persistence module of…
n this paper, we establish the sharp estimate of the Lipschitz continuity with respect to the Bergman metric. The obtained results are the improvement and generalization of the corresponding results of Ghatage, Yan and Zheng (Proc. Amer.…
Persistence homology is a tool used to measure topological features that are present in data sets and functions. Persistence pairs births and deaths of these features as we iterate through the sublevel sets of the data or function of…
In the present paper we investigate geometric characteristics of compact metric spaces, which can be described in terms of Gromov-Hausdorff distances to simplexes, i.e., to finite metric spaces such that all their nonzero distances are…
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…
We study the geometry of the space of measures of a compact ultrametric space X, endowed with the L^p Wasserstein distance from optimal transportation. We show that the power p of this distance makes this Wasserstein space affinely…
We work within the framework of a program aimed at exploring various extended versions for theorems from a class containing Borsuk-Ulam type theorems, some fixed point theorems, the KKM lemma, Radon, Tverberg, and Helly theorems. In this…
In this article, we consider the H\"{o}lder continuity of injective maps in Orlicz-Sobolev classes defined on the unit ball. Under certain conditions on the growth of dilatations, we obtain the H\"{o}lder continuity of the indicated class…
Vietoris-Rips and degree Rips complexes are represented as homotopy types by their underlying posets of simplices, and basic homotopy stability theorems are recast in these terms. These homotopy types are viewed as systems (or functors),…