Related papers: A method of induction the distances with Hilbert s…
The distance metric plays an important role in nearest neighbor (NN) classification. Usually the Euclidean distance metric is assumed or a Mahalanobis distance metric is optimized to improve the NN performance. In this paper, we study the…
We study the metric structure of walks on graphs, understood as Lipschitz sequences. To this end, a weighted metric is introduced to handle sequences, enabling the definition of distances between walks based on stepwise vertex distances and…
We obtain a general concept of triplet of Hilbert spaces with closed (unbounded) embeddings instead of continuous (bounded) ones. The construction starts with a positive selfadjoint operator $H$, that is called the Hamiltonian of the…
We introduce a distance in the space of fully-supported probability measures on one-dimensional symbolic spaces. We compare this distance to the $\bar{d}$-distance and we prove that in general they are not comparable. Our projective…
The question in the title is discussed briefly, with emphasis on a few basic examples and their properties.
Some examples and basic properties of ultrametric spaces are briefly discussed.
It is shown that the Hilbert metric on the interior of a convex polytope is bilipschitz to a normed vector space of the same dimension.
The Gromov-Hausdorff distance measures the similarity between two metric spaces by isometrically embedding them into an ambient metric space. We introduce an analogue of this distance for metric spaces endowed with directed structures. The…
The connection between several hyperbolic type metrics is studied in subdomains of the Euclidean space. In particular, a new metric is introduced and compared to the distance ratio metric.
The paper introduces the class of O-metric spaces, a novel generalization of metric-type spaces, classifying almost all possible metric types into upward and downward O-metrics. We list some topologies arising from O-metrics and discuss…
This report introduces and investigates a family of metrics on sets of pointed Kripke models. The metrics are generalizations of the Hamming distance applicable to countably infinite binary strings and, by extension, logical theories or…
The paper presents a systematic strategy for implementing Hilbert's space filling curve for use in online exploration tasks and addresses its application in scenarios wherein the space to be searched obstacles (or holes) whose locations 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 construct an isometric embedding of a bounded set in a Euclidean space into the Gromov-Hausdorff space. In particular, we can embed a bounded and connected Riemannian manifold into the Gromov-Hausdorff space by a bilipschitz map.
We introduce strings in metric spaces and define string complexes of metric spaces. We describe the class of 2-dimensional topological spaces which arise in this way from finite metric spaces.
A key technique of machine learning and computer vision is to embed discrete weighted graphs into continuous spaces for further downstream processing. Embedding discrete hierarchical structures in hyperbolic geometry has proven very…
An ultrametric topology formalizes the notion of hierarchical structure. An ultrametric embedding, referred to here as ultrametricity, is implied by a natural hierarchical embedding. Such hierarchical structure can be global in the data…
We study the relation between metric entropy and escape of mass for the Hilbert modular spaces with the action of a diagonal element.
We provide a concise and accessible introduction to (geometric) string structures, highlighting their connection to loop spaces and outlining relationships with neighboring topics.
Learning the embedding space, where semantically similar objects are located close together and dissimilar objects far apart, is a cornerstone of many computer vision applications. Existing approaches usually learn a single metric in the…