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Inference of topological and geometric attributes of a hidden manifold fromits point data is a fundamental problem arising in many scientific studies andengineering applications. In this paper we present an algorithm to compute aset of loops from a point data that presumably sample a smooth manifold$M\subset \mathbb{R}^d$. These loops approximate a {\em shortest} basis of theone dimensional homology group $H_1(M)$ over coefficients in finite field$\mathbb{Z}_2$. Previous results addressed the issue of computing the rank ofthe homology groups from point data, but there is no result on approximatingthe 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$H_1(K)$ for any finite {\em simplicial complex} $K$ whose edges havenon-negative weights.

Approximating Loops in a Shortest Homology Basis from Point Data