document: 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
doc_bio_tags: O B I O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O B I O O O O O O O O O B I O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O
extractive_keyphrases: simplicial reconstructions manifold learning learned manifold true manifold simplicial complex sampling density
abstractive_keyphrases: finite-dimensional differentiable manifold