Nonlinear Approximation and Image Representation using Wavelets

Abstract

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 $ell_p$ distances under a wide variety of compactly-supported wavelet bases.  More specifically, given a vector $f in mathbb{R}^n$, a compactly-supported wavelet basis, a sparsity constraint $B in mathbb{Z}$, and $pin[1,infty]$, our algorithm returns a $B$-term representation (a linear combination of $B$ vectors from the given basis) whose $ell_p$ distance from $f$ is a $O(log n)$ factor away from that of the optimal such representation of $f$.   Our algorithm applies in the one-pass sublinear-space data streaming model of computation, and it generalize to weighted $p$-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 $B(log n)^2$ terms and gives a $O(log n)$-approximation under all $p$-norms simultaneously.