Efficient Simplicial Reconstructions of Manifolds from Their Samples

An algorithm for manifold learning is presented. Given only samples of a finite-dimensional differentiable manifold and no a priori knowledge of the manifold's geometry or topology except for its dimension, the goal is to find a description of the manifold. The learned manifold must approximate the true manifold well, both geometrically and topologically, when the sampling density is sufficiently high. The proposed algorithm constructs a simplicial complex based on approximations to the tangent bundle of the manifold. An important property of the algorithm is that its complexity depends on the dimension of the manifold, rather than that of the embedding space. Successful examples are presented in the cases of learning curves in the plane, curves in space, and surfaces in space; in addition, a case when the algorithm fails is analyzed.