Related papers: Magnitude of metric measure spaces and integrals o…

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…

In this paper we define the magnitude of metric spaces using measures rather than finite subsets as had been done previously and show that this agrees with earlier work with Leinster in arXiv:0908.1582. An explicit formula for the magnitude…

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…

This paper studies the magnitude homology groups of geodesic metric spaces. We start with a description of the second magnitude homology of a general metric space in terms of the zeroth homology groups of certain simplicial complexes. Then,…

Magnitude of a finite metric space and the related notion of magnitude functions on metric spaces is an active area of research in algebraic topology. Magnitude originally arose in the context of biology, where it represents the number of…

Magnitude homology is an $\mathbf{R}^+$-graded homology theory of metric spaces that captures information on the complexity of geodesics. Here we address the question: when are two metric spaces magnitude homology equivalent, in the sense…

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…

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…

The magnitude of a finite metric space has recently emerged as a novel invariant quantity, allowing to measure the effective size of a metric space. Despite encouraging first results demonstrating the descriptive abilities of the magnitude,…

Magnitude is a numerical invariant of metric spaces and graphs, analogous, in a precise sense, to Euler characteristic. Magnitude homology is an algebraic invariant constructed to categorify magnitude. Among the important features of 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,…

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…

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…

Magnitude is a measure of size defined for certain classes of metric spaces; it arose from ideas in category theory. In particular, magnitude is defined for compact subsets of Euclidean space and, in arXiv:1507.02502, Barcel\'o and Carbery…

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 Euler characteristic is an invariant of a topological space that in a precise sense captures its canonical notion of size, akin to the cardinality of a set. The Euler characteristic is closely related to the homology of a space, as it…

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…

Metric space magnitude, an active subject of research in algebraic topology, originally arose in the context of biology, where it was used to represent the effective number of distinct species in an environment. In a more general setting,…

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…