Approximate Cech Complexes in Low and High Dimensions

Abstract: \v{C}ech complexes reveal valuable topological information about point sets at a certain scale in arbitrary dimensions, but the sheer size of these complexes limits their practical impact. While recent work introduced approximation techniques for filtrations of (Vietoris-)Rips complexes, a coarser version of \v{C}ech complexes, we propose the approximation of \v{C}ech filtrations directly. For fixed dimensional point set $S$, we present an approximation of the \v{C}ech filtration of $S$ by a sequence of complexes of size linear in the number of points. We generalize well-separated pair decompositions (WSPD) to well-separated simplicial decomposition (WSSD) in which every simplex defined on $S$ is covered by some element of WSSD. We give an efficient algorithm to compute a linear-sized WSSD in fixed dimensional spaces. Using a WSSD, we then present a linear-sized approximation of the filtration of \v{C}ech complex of $S$. We also present a generalization of the known fact that the Rips complex approximates the \v{C}ech complex by a factor of $\sqrt{2}$. We define a class of complexes that interpolate between \v{C}ech and Rips complexes and that, given any parameter $\eps > 0$, approximate the \v{C}ech complex by a factor $(1+\eps)$. Our complex can be represented by roughly $O(n^{\lceil 1/2\eps\rceil})$ simplices without any hidden dependence on the ambient dimension of the point set. Our results are based on an interesting link between \v{C}ech complex and coresets for minimum enclosing ball of high-dimensional point sets. As a consequence of our analysis, we show improved bounds on coresets that approximate the radius of the minimum enclosing ball.