A greedy, adaptive approach to learning geometry of nonlinear manifolds

Abstract

In this paper, we address the problem of learning the geometry of a non-linear manifold in the ambient Euclidean space into which the manifold is embedded. We propose a bottom-up approach to manifold approximation using tangent planes where the number of planes is adaptive to manifold curvature. Also, we exploit the local linearity of the manifold to subsample the manifold data before using it to learn the manifold geometry with negligible loss of approximation accuracy. In our experiments, our proposed Geometry Preserving Union-of-Affine Subspaces algorithm shows more than a 100-times decrease in the learning time when compared to state-ofthe-art manifold learning algorithm, while achieving similar approximation accuracy.