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- W2894892703 abstract "Given a point cloud $P$ in Euclidean space and a positive parameter $t$ we can consider the $t$-neighborhood $P^{t}$ of $P$ consisting of points at distance less than $t$ to $P$. Homology of $P^{t}$ gives information about components, holes, voids etc. in $P^{t}$. The idea of persistent homology is that it may happen that we are interested in some of holes in the spaces $P^t$ 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 $alpha$-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 ${P^{t}}_{t ge 0}$. 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 ${P^{t}}_{t ge 0}$ and more generally to sparsification of filtered Dowker nerves. To our best knowledge, this is the first approach to sparsification of general Dowker nerves." @default.
- W2894892703 created "2018-10-12" @default.
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- W2894892703 date "2018-10-04" @default.
- W2894892703 modified "2023-09-27" @default.
- W2894892703 title "Sparse Filtered Nerves" @default.
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