A fractal dimension for measures via persistent homology

Abstract

We use persistent homology in order to define a family of fractal dimensions, denoted ${\dim }_{\mathrm{PH}}^i(\mu )$ for each homological dimension $i\ge 0$, assigned to a probability measure $\mu$ on a metric space. The case of $0$-dimensional homology ($i=0$) relates to work by Michael J Steele (1988) studying the total length of a minimal spanning tree on a random sampling of points. Indeed, if $\mu$ is supported on a compact subset of Euclidean space ${\mathbb{R}}^m$ for $m\ge 2$, then Steele's work implies that ${\dim }_{\mathrm{PH}}^0(\mu )=m$ if the absolutely continuous part of $\mu$ has positive mass, and otherwise ${\dim }_{\mathrm{PH}}^0(\mu )<m$. Experiments suggest that similar results may be true for higher-dimensional homology $0<i<m$, though this is an open question. Our fractal dimension is defined by considering a limit, as the number of points $n$ goes to infinity, of the total sum of the $i$-dimensional persistent homology interval lengths for $n$ random points selected from $\mu$ in an i.i.d. fashion. To some measures $\mu ,$ we are able to assign a finer invariant, a curve measuring the limiting distribution of persistent homology interval lengths as the number of points goes to infinity. We prove this limiting curve exists in the case of $0$-dimensional homology when $\mu$ is the uniform distribution over the unit interval, and conjecture that it exists when $\mu$ is the rescaled probability measure for a compact set in Euclidean space with positive Lebesgue measure.