Approximating cycles in a shortest basis of the first homology group 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 cycles from a point data that presumably sample a smooth manifold M ⊂ ℝ d . These cycles approximate a shortest basis of the first homology group H 1 (M) over coefficients in the finite field ℤ 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 at our result, we also present a polynomial time algorithm for computing a shortest basis of H 1 (K) for any finite simplicial complex K whose edges have non-negative weights.