Persistent Homology and the Upper Box Dimension

2018/02/02 by Schweinhart, Benjamin · 3 citations
#Algebraic Topology (math.AT) #Combinatorics (math.CO) #Computational Geometry (cs.CG) #FOS: Computer and information sciences #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1802.00533

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 ℝ² 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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