Related papers: Interleaving and Gromov-Hausdorff distance
Pairwise comparison of graphs is key to many applications in Machine learning ranging from clustering, kernel-based classification/regression and more recently supervised graph prediction. Distances between graphs usually rely on…
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 Gromov-Hausdorff distance is a dissimilarity metric capturing how far two spaces are from being isometric. The Gromov-Prokhorov distance is a similar notion for metric measure spaces. In this paper, we study the topological dimension of…
The Gromov-Hausdorff (GH) distance is traditionally used for measuring distances between metric spaces. It is defined as the minimal distortion of embedding one surface into the other, while the optimal correspondence can be described as…
This paper employs techniques from metric geometry and optimal transport theory to address questions related to the analysis of functional data on metric or metric-measure spaces, which we refer to as fields. Formally, fields are viewed as…
In the present paper we investigate the properties of the Hausdorff mapping $\mathcal{H}$, which takes each compact metric space to the space of its nonempty closed subspaces. It is shown that this mapping is nonexpanding (Lipschitz mapping…
This thesis develops the theory of sheaves and cosheaves with an eye towards applications in science and engineering. To provide a theory that is computable, we focus on a combinatorial version of sheaves and cosheaves called cellular…
Herein we present open problems and survey examples and theorems concerning sequences of Riemannian manifolds with uniform lower bounds on scalar curvature and their limit spaces. Examples of Gromov and of Ilmanen which naturally ought to…
We prove fixed point theorems in a space with a distance function that takes values in a partially ordered monoid. On the one hand, such an approach allows one to generalize some fixed point theorems in a broad class of spaces, including…
We develop analysis of H\"older continuous mappings with applications to geometry and topology of the Heisenberg groups. We cover the theory of distributional Jacobians of H\"older continuous mappings and pullbacks of differential forms…
The purpose of this article is to demonstrate the connection between the properties of the Gromov--Hausdorff distance and the Borsuk conjecture. The Borsuk number of a given bounded metric space $X$ is the infimum of cardinal numbers $n$…
In this paper, we study the Gromov-Hausdorff stability and continuous dependence of the inertial manifolds under perturbations of the domain and equation. More precisely, we use the Gromov-Hausdorff distances between two inertial manifolds…
We show that attractors are semicontinuous for closed relations on compact Hausdorff spaces. Semicontinuity is what guarantees that small changes to a system do not result in massive growth of certain features, notably attractors. That is,…
In this note, we prove optimal extension results for roughly isometric relations between metric (R-)trees and injective metric spaces. This yields sharp stability estimates, in terms of the Gromov-Hausdorff (GH) distance, for certain metric…
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…
The aim of this paper is to demonstrate relations between Gromov-Hausdorff distance properties and the Borsuk Conjecture. The Borsuk number of a given bounded metric space $X$ is the infimum of cardinal numbers $n$ such that $X$ can be…
We introduce several geometric notions, including the width of a homology class, to the theory of persistent homology. These ideas provide geometric interpretations of persistence diagrams. Indeed, we give quantitative and geometric…
This paper surveys various distance measures for networks and graphs that were introduced in persistent homology. The scope of the paper is limited to network distances that were actually used in brain networks but the methods can be easily…
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…
In this work, a metric is presented on the set of boundedly-compact pointed metric spaces that generates the Gromov-Hausdorff topology. A similar metric is defined for measured metric spaces that generates the Gromov-Hausdorff-Prokhorov…