A Fractal Dimension for Measures via Persistent Homology

We use persistent homology in order to define a family of fractal dimensions, denoted dim(PH)(i)(mu) for each homological dimension i >= 0, assigned to a probability measure mu on a metric space. The case of zero-dimensional homology (i = 0) relates to work by Steele (Ann Probab 16( 4): 1767-1787, 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 R-m for m >= 2, then Steele's work implies that dim(PH)(0)(mu) = m if the absolutely continuous part of mu has positive mass, and otherwise dim(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 zerodimensional 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.