Patch-to-tensor embedding

A popular approach to deal with the “curse of dimensionality” in relation with high-dimensional data analysis is to assume that points in these datasets lie on a low-dimensional manifold immersed in a high-dimensional ambient space. Kernel methods operate on this assumption and introduce the notion of local affinities between data points via the construction of a suitable kernel. Spectral analysis of this kernel provides a global, preferably low-dimensional, coordinate system that preserves the qualities of the manifold. In this paper, we extend the scalar relations used in this framework to matrix relations, which can encompass multidimensional similarities between local neighborhoods of points on the manifold. We utilize the diffusion maps methodology together with linear-projection operators between tangent spaces of the manifold to construct a super-kernel that represents these relations. The properties of the presented super-kernels are explored and their spectral decompositions are 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.