ar X iv : 0 70 7 . 20 08 v 1 [ m at h . C A ] 1 3 Ju l 2 00 7 OPTIMAL NON - LINEAR MODELS FOR SPARSITY AND SAMPLING

Given a set of functions, we prove the existence of an optimal collection of subspaces minimizing the sum of the square of the distances between each function 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 algorithm to find the solution subspaces. These results are tightly connected to the new emergent theories of compressed sensing and dictionary finding, signal models for signals with finite rate of innovation, and the subspace segmentation problem.