Manifold Learning: Generalization Ability and Tangent Proximity
One of the ultimate goals of Manifold Learning (ML) is to reconstruct an unknown nonlinear low-dimensional Data Manifold (DM) embedded in a high-dimensional observation space from a given set of data points sampled from the manifold. We derive asymptotic expansion and local lower and upper bounds for the maximum reconstruction error in a small neighborhood of an arbitrary point. The expansion and bounds are defined in terms of the distance between tangent spaces to the original Data manifold and the Reconstructed Manifold (RM) at the selected point and its reconstructed value, respectively. We propose an amplification of the ML, called Tangent Bundle ML, in which proximity is required not only between the DM and RM but also between their tangent spaces. We present a new geometrically motivated Grassman&Stiefel Eigenmaps algorithm that solves this problem and gives a new solution for the ML also.