Alpha magnitude

TL;DR

This paper introduces alpha magnitude, a new, computationally efficient invariant for metric spaces that may help estimate fractal dimensions of data, building on the concept of magnitude and its variants.

Contribution

We propose alpha magnitude, a novel invariant inspired by persistent magnitude, with properties and conjectured links to Minkowski dimension, offering practical advantages for data analysis.

Findings

Alpha magnitude is computationally more efficient than existing invariants.

It shows potential for estimating fractal dimensions of real-world data.

Conjectured relationship with Minkowski dimension suggests theoretical significance.

Abstract

Magnitude is an isometric invariant for metric spaces that was introduced by Leinster around 2010, and is currently the object of intense research, since it has been shown to encode many known invariants of metric spaces. In recent work, Govc and Hepworth introduced persistent magnitude, a numerical invariant of a filtered simplicial complex associated to a metric space. Inspired by Govc and Hepworth's definition, we introduce alpha magnitude and investigate some of its key properties. Heuristic observations lead us to conjecture a relationship with the Minkowski dimension of compact subspaces of Euclidean space. Finally, alpha magnitude presents computational advantages over both magnitude as well as Rips magnitude, and we thus propose it as a new measure for the estimation of fractal dimensions of real-world data sets that is easily computable.