Approximating loops in a shortest homology basis 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 loops from a point data that presumably sample a smooth manifold M ⊂ Rd. These loops approximate a shortest basis of the one dimensional homology group H1(M) over coefficients in finite field Z2. 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 our result, we also present a polynomial time algorithm for computing a shortest basis of H1 (Κ) for any finite simplicial complex Κ whose edges have non-negative weights.