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Nonlinear Approximation and Image Representation using Wavelets
We address the problem of finding sparse wavelet representations of high-dimensional vectors. We present a lower-bounding technique and use it to develop an algorithm for computing provably-approximate instance-specific representations minimizing general distances under a wide variety of compactly-supported wavelet bases. More specifically, given a vector , a compactly-supported wavelet basis, a sparsity constraint , and , our algorithm returns a -term representation (a linear combination of vectors from the given basis) whose distance from is a factor away from that of the optimal such representation of . Our algorithm applies in the one-pass sublinear-space data streaming model of computation, and it generalize to weighted -norms and multidimensional signals. Our technique also generalizes to a version of the problem where we are given a bit-budget rather than a term-budget. Furthermore, we use it to construct a emph{universal representation} that consists of at most terms and gives a -approximation under all -norms simultaneously