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Patch-to-Tensor Embedding Guy Wolf Moshe Salhov Amir Averbuch

Summary: Patch-to-Tensor Embedding
Guy Wolf Moshe Salhov Amir Averbuch
January 19, 2011
A popular approach to deal with the "curse of dimensionality" in relation with the
analysis of high-dimensional datasets, 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 be-
tween 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
method together with linear-projection operators between tangent spaces of the man-
ifold 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.
1 Introduction
High-dimensional datasets have become increasingly common in many areas, due to high


Source: Averbuch, Amir - School of Computer Science, Tel Aviv University


Collections: Computer Technologies and Information Sciences