Sparse Filtered Nerves

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.