Approximating cycles in a shortest basis of the first homology group from point data

Abstract

Inference of topological and geometric attributes of a hidden manifold from its point data is a fundamental problem arising in many scientific studies and engineering applications. In this paper, we present an algorithm to compute a set of cycles from a point data that presumably sample a smooth manifold . These cycles approximate a shortest basis of the first homology group over coefficients in the finite field . Previous results addressed the issue of computing the rank of the homology groups from point data, but there is no result on approximating the shortest basis of a manifold from its point sample. In arriving at our result, we also present a polynomial time algorithm for computing a shortest basis of for any finite simplicial complex whose edges have non-negative weights.