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Constructive sampling for patch-based embedding

To process high-dimensional big data, we assume that sufficiently small patches (or neighborhoods) of the data are approximately linear. These patches represent the tangent spaces of an underlying manifold structure from which we assume the data is sampled. We use these tangent spaces to extend the scalar relations that are used by many kernel methods to matrix relations, which encompass multidimensional similarities between local neighborhoods in the data. The incorporation of these matrix relations improves the utilization of kernelbased data analysis methodologies. However, they also result in a larger kernel and a higher computational cost of its spectral decomposition. We propose a dictionary construction that approximates the oversized kernel in this case and its associated patch-to-tensor embedding. The performance of the proposed dictionary construction is demonstrated on a superkernel example that utilizes the Diffusion Maps methodology together with linear-projection operators between tangent spaces in the manifold.