Related papers: Is magnitude 'generically continuous' for finite m…
The magnitude of metric spaces does not appear to possess a simple, convenient continuity property, and previous studies have presented affirmative results under additional constraints or weaker notions, as well as counterexamples. In this…
Magnitude is an isometric invariant of metric spaces introduced by Leinster. Since its inception, it has inspired active research into its connections with integral geometry, geometric measure theory, fractal dimensions, persistent…
Magnitude is an isometric invariant of metric spaces introduced by Leinster. Although magnitude is nowhere continuous on the Gromov-Hausdorff space of finite metric spaces, continuity results are possible if we restrict the ambient space.…
Magnitude is a real-valued invariant of metric spaces, analogous to the Euler characteristic of topological spaces and the cardinality of sets. The definition of magnitude is a special case of a general categorical definition that clarifies…
The magnitude of a metric space is a real-valued function whose parameter controls the scale of the metric. A metric space is said to have the one-point property if its magnitude converges to 1 as the space is scaled down to a point. Not…
Magnitude is a numerical isometric invariant of metric spaces, whose definition arises from a precise analogy between categories and metric spaces. Despite this exotic provenance, magnitude turns out to encode many invariants from integral…
Magnitude homology is an emerging framework that captures the intrinsic topological and geometric features of metric spaces, demonstrating significant potential for topoplogical data analysis and geometric data analysis. This work…
Magnitude is a numerical invariant of compact metric spaces. Its theory is most mature for spaces satisfying the classical condition of being of negative type, and the magnitude of such a space lies in the interval $[1, \infty]$. Until now,…
The magnitude of a metric space is a novel invariant that provides a measure of the 'effective size' of a space across multiple scales, while also capturing numerous geometrical properties, such as curvature, density, or entropy. We develop…
Magnitude is a numerical invariant of finite metric spaces, recently introduced by T. Leinster, which is analogous in precise senses to the cardinality of finite sets or the Euler characteristic of topological spaces. It has been extended…
Magnitude is a numerical invariant of compact metric spaces, originally inspired by category theory and now known to be related to myriad other geometric quantities. Generalizing earlier results in $\ell_1^n$ and Euclidean space, we prove…
Deep learning models have seen significant successes in numerous applications, but their inner workings remain elusive. The purpose of this work is to quantify the learning process of deep neural networks through the lens of a novel…
We give sufficient conditions for a finite metric space to be determined by the magnitude function. In particular, a generic finite metric space such that the distances between the points are rationally independent is determined by the…
The paper is devoted to the study of Gromov-Hausdorff convergence and stability of irreversible metric-measure spaces, both in the compact and noncompact cases. While the compact setting is mostly similar to the reversible case developed by…
The Gromov--Hausdorff distance (hereinafter referred to as the GH-distance) is a measure of non-isometricity of metric spaces. In this paper, we study a modification of this distance that also takes topological differences into account. The…
In this paper we introduce the persistent magnitude, a new numerical invariant of (sufficiently nice) graded persistence modules. It is a weighted and signed count of the bars of the persistence module, in which a bar of the form $[a,b)$ in…
The main result of this article is a geometric interpretation of magnitude, a real-valued invariant of metric spaces. We introduce a Euclidean embedding of a (suitable) finite metric space $X$ such that the magnitude of $X$ can be expressed…
Magnitude is a numerical invariant of metric spaces introduced by Leinster, motivated by considerations from category theory. This paper extends the original definition for finite spaces to compact spaces, in an equivalent but more natural…
Techniques from metric geometry have become fundamental tools in modern mathematical data science, providing principled methods for comparing datasets modeled as finite metric spaces. Two of the central tools in this area are the…
Magnitude is a numerical invariant of enriched categories, including in particular metric spaces as $[0,\infty)$-enriched categories. We show that in many cases magnitude can be categorified to a homology theory for enriched categories,…