Manifold Learning: Generalization Ability and Tangent Proximity
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Alexander V. Bernstein,Alexander P. Kuleshov. Manifold Learning: Generalization Ability and Tangent Proximity. International Journal of Software and Informatics, 2013,7(3):359~390
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Fund:This work is partially supported by Laboratory for Structural Methods of Data Analysis in Predictive Modeling, MIPT, RF Government grant, ag. 11.G34.31.0073; and RFBR, research projects MIPT, RF Government grant, ag. 11.G34.31.0073; and RFBR, research projects 13-01-12447 and 13-07-12111.
Abstract: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 DM 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.
keywords:dimensionality reduction  manifold learning  generalization ability  tangent spaces  tangent bundle manifoldlearning  Grassmann manifold  Stiefel manifold
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