Related papers: Gromov-Hausdorff Approximation of Metric Spaces wi…
The paper is devoted to geometrical investigation of the Gromov-Hausdorff distance on the classes of all metric spaces and of all bounded metric spaces. The main attention is paid to pass connectivity questions. The pass connected…
In this paper we introduce a pruning of the medial axis called the $(\lambda,\alpha)$-medial axis ($\textrm{ax}_\lambda^\alpha $). We prove that the $(\lambda,\alpha)$-medial axis of a set $K$ is stable in a Gromov-Hausdorff sense under…
We study the subsets of metric spaces that are negligible for the infimal length of connecting curves; such sets are called metrically removable. In particular, we show that every totally disconnected set with finite Hausdorff measure of…
Data consisting of a graph with a function mapping into $\mathbb{R}^d$ arise in many data applications, encompassing structures such as Reeb graphs, geometric graphs, and knot embeddings. As such, the ability to compare and cluster such…
In this paper we study the geometry of graph spaces endowed with a special class of graph edit distances. The focus is on geometrical results useful for statistical pattern recognition. The main result is the Graph Representation Theorem.…
Some results on the approximation of functions from the Sobolev spaces on metric graphs by step functions are obtained. The estimates are uniform with respect to all graphs of a given finite length, and the constant factors in the…
In this paper, we investigate compact ultrametric measure spaces which form a subset $\mathcal{U}^w$ of the collection of all metric measure spaces $\mathcal{M}^w$. Similar as for the ultrametric Gromov-Hausdorff distance on the collection…
This work investigates minimal parametric networks in hyperspaces of closed subsets of metric spaces endowed with the Hausdorff distance. It is shown that the problems of finding such networks are nontrivial only within finiteness classes,…
In the present paper we study the original Gromov-Hausdorff distance between real normed spaces. In the first part of the paper we prove that two finite-dimensional real normed spaces on a finite Gromov-Hausdorff distance are isometric to…
Fused Gromov-Wasserstein (FGW) distances provide a principled framework for comparing objects by jointly aligning structure and node features. However, existing FGW formulations treat all features uniformly, which limits interpretability…
In the recent paper "Reparametrizations of continuous paths", J. Homotopy Relat. Struct. 2 (2007), 93-117 (with U. Fahrenberg), we performed a systematic investigation of reparametrizations of continuous paths in a Hausdorff space that…
Context. Filamentary structures appear to be ubiquitous in the interstellar medium. Being able to detect and characterize them is the first step toward understanding their origin, their evolution, and their role in the Galactic cycle of…
The surge of research in image segmentation has yielded remarkable performance gains but also exposed a reproducibility crisis. A major contributor is performance evaluation, where both selection and implementation of metrics play critical…
In this paper we prove the existence of isoperimetric regions of any volume in Riemannian manifolds with Ricci bounded below assuming Gromov--Hausdorff asymptoticity to the suitable simply connected model of constant sectional curvature.…
The Gromov-Wasserstein distance is a notable extension of optimal transport. In contrast to the classic Wasserstein distance, it solves a quadratic assignment problem that minimizes the pair-wise distance distortion under the transportation…
Given a pointed metric space $(X,\mathsf{dist}, w)$ on $n$ points, its Gromov's approximating tree is a 0-hyperbolic pseudo-metric space $(X,\mathsf{dist}_T)$ such that $\mathsf{dist}(x,w)=\mathsf{dist}_T(x,w)$ and $\mathsf{dist}(x, y)-2…
Statistical graph models aim at modeling graphs as random realization among a set of possible graphs. One issue is to evaluate whether or not a graph is likely to have been generated by one particular model. In this paper we introduce the…
Graphs are versatile tools for representing structured data. As a result, a variety of machine learning methods have been studied for graph data analysis. Although many such learning methods depend on the measurement of differences between…
A common approach to implementing similarity search applications is the usage of distance functions, where small distances indicate high similarity. In the case of metric distance functions, metric index structures can be used to accelerate…
We introduce a theoretical framework for performing statistical tasks---including, but not limited to, averaging and principal component analysis---on the space of (possibly asymmetric) matrices with arbitrary entries and sizes. This is…