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Persistent Homology and the Upper Box Dimension
Published 2 Feb 2018 in math.MG, cs.CG, math.AT, and math.CO | (1802.00533v7)
Abstract: We introduce a fractal dimension for a metric space defined in terms of the persistent homology of extremal subsets of that space. We exhibit hypotheses under which this dimension is comparable to the upper box dimension; in particular, the dimensions coincide for subsets of $\mathbb{R}2$ whose upper box dimension exceeds $1.5.$ These results are related to extremal questions about the number of persistent homology intervals of a set of $n$ points in a metric space.
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