Patch-Based Data Analysis Using Linear-Projection Diffusion

Abstract

To process massive high-dimensional datasets, we utilize the underlying assumption that data on a manifold is approximately linear in sufficiently small patches (or neighborhoods of points) that are sampled with sufficient density from the manifold. Under this assumption, each patch can be represented by a tangent space of the manifold in its area and the tangential point of this tangent space. We use these tangent spaces, and the relations between them, to extend the scalar relations that are used by many kernel methods to matrix relations, which can encompass multidimensional similarities between local neighborhoods of points on the manifold. The properties of the presented construction are explored and its spectral decomposition is utilized to embed the patches of the manifold into a tensor space in which the relations between them are revealed. We present two applications that utilize the patch-to-tensor embedding framework: data classification and data clustering for image segmentation.