Improved Approximate Rips Filtrations with Shifted Integer Lattices

06/22/2017

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by   Aruni Choudhary, et al.

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Rips complexes are important structures for analyzing topological features of metric spaces. Unfortunately, generating these complexes constitutes an expensive task because of a combinatorial explosion in the complex size. For n points in R^d, we present a scheme to construct a 3√(2)-approximation of the multi-scale filtration of the L_∞-Rips complex, which extends to a O(d^0.25)-approximation of the Rips filtration for the Euclidean case. The k-skeleton of the resulting approximation has a total size of n2^O(d k). The scheme is based on the integer lattice and on the barycentric subdivision of the d-cube.