Fractal Dimension Estimation with Persistent Homology

Jonathan Jaquette
Brandeis University

Persistent homology quantifies the shape of a geometric object in terms of how its topology changes as it is thickened. Over the past decade there has been a surge of interest in applications of persistent homology. We propose that the recently defined persistent homology dimensions are a practical tool for fractal dimension estimation of point samples. We implement an algorithm to estimate the persistent homology dimension, and compare its performance to classical methods (e.g. the correlation and box-counting dimensions) in examples of self-similar fractals, chaotic attractors, and an empirical dataset.