Optimal Non-Linear Models for Sparsity and Sampling

Details Optimal Non-Linear Models for Sparsity and Sampling Optimal Non-Linear Models for Sparsity and Sampling

2007-07-13

arxiv.org/abs/0707.2008v2

Mathematics

Classical Analysis and ODEs

20 pages, 2 figures. Final version of the paper. To appear in Journal of Fourier Analysis and Applications

Scientific paper

Given a set of vectors (the data) in a Hilbert space H, we prove the existence of an optimal collection of subspaces minimizing the sum of the square of the distances between each vector and its closest subspace in the collection. This collection of subspaces gives the best sparse representation for the given data, in a sense defined in the paper, and provides an optimal model for sampling in union of subspaces. The results are proved in a general setting and then applied to the case of low dimensional subspaces of R^N and to infinite dimensional shift-invariant spaces in L^2(R^d). We also present an iterative search algorithm for finding the solution subspaces. These results are tightly connected to the new emergent theories of compressed sensing and dictionary design, signal models for signals with finite rate of innovation, and the subspace segmentation problem.

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