Summary: We use persistent homology in order to define a family of fractal dimensions, denoted \(\dim_{\text{PH}}^i(\mu)\) for each homological dimension \(i\geq 0\), assigned to a probability measure \(\mu\) on a metric space. The case of zero-dimensional homology \((i = 0)\) relates to work by J. M. Steele [Ann. Probab. 16, No. 4, 1767–1787 (1988; Zbl 0655.60023)] 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 \(m \geq 2\), then Steele’s work implies that \(\dim_{\text{PH}}^0(\mu )=m\) if the absolutely continuous part of \(\mu\) has positive mass, and otherwise \(\dim_{\text{PH}}^0(\mu )<m\). 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 zero-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.
For the entire collection see [Zbl 1448.62008].