Technical Report CIS-2009-05

Title: Nonlinear Dimensionality Reduction by Topologically Constrained Isometric Embedding
Authors: Guy Rosman, Alexander M. Bronstein, Michael M. Bronstein, and Ron Kimmel
Abstract: Many manifold learning procedures try to embed a given feature data into a flat space of low dimensionality while preserving as much as possible the metric in the natural feature space. The flat embedding process usually relies on distances between neighboring features, mainly since distances between features that are far apart from each other often provide an unreliable estimation of the true distance on the feature manifold due to non-convexity of the given manifold. Presented is a framework for nonlinear dimensionality reduction that uses both local and global distances in order to learn the intrinsic geometry of locally flat manifolds with boundaries. The main idea is to filter out potentially problematic distances between distant feature points based on the properties of the geodesics connecting those points and their relative distance to the edge of the feature manifold. Since the proposed algorithm matches non-local structures, it is robust to strong noise. We show experimental results demonstrating the advantages of the proposed approach over conventional dimensionality reduction techniques.

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