Approximating loops in a shortest homology basis from point data

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 <i>M</i> ⊂ <b>R</b>^<i>d</i>. These loops approximate a <i>shortest</i> basis of the one dimensional homology group H₁(<i>M</i>) over coefficients in finite field <b>Z</b>². 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 <i>simplicial complex Κ</i> whose edges have non-negative weights.