Papers › Sparse Filtered Nerves
Sparse Filtered Nerves
Nello Blaser, Morten Brun
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Given a point cloud P in Euclidean space and a positive parameter t we can consider the t-neighborhood Pᵗ of P consisting of points at distance less than t to P. Homology of Pᵗ gives information about components, holes, voids etc. in Pᵗ. The idea of persistent homology is that it may happen that we are interested in some of holes in the spaces Pᵗ that are not detected simultaneously in homology for a single value of t, but where each of these holes is detected for t in a wide range. When the dimension of the ambient Euclidean space is small, persistent homology is efficiently computed by the α-complex. For dimension bigger than three this becomes resource consuming. Don Sheehy discovered that there exists a filtered simplicial complex whose size depends linearly on the cardinality of P and whose persistent homology is an approximation of the persistent homology of the filtered topological space {Pᵗ}_(t ≥0). In this paper we pursue Sheehy's sparsification approach and give a more general approach to sparsification of filtered simplicial complexes computing the homology of filtered spaces of the form {Pᵗ}_(t ≥0) and more generally to sparsification of filtered Dowker nerves. To our best knowledge, this is the first approach to sparsification of general Dowker nerves.
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