Estimation error bounds for denoising by sparse approximation

If a signal is known to have a sparse representation with respect to a given frame, thc signal can he estimated from a noise-corrupted observation of the signal by finding the hest sparse approximation to the observation. ‘The ability to remove noise in this manner depends on the frame heing designed to efficiently represent the signal while it ine@cienf/v represents the noise. ‘This paper gives hounds to show how inefficiently white Gaussian noise is represented by sparse linear combinations of frame vectors. Combined with knowledge of the approximation efficiency of a given family of frames for a given signal class, this work leads lo a better understanding of the merits of frame denoising. 1. DENOISING BY SPARSE APPROXIMATION Consider the problem of estimating an unknown signal 2: E R” from the noisy observation y = z + d where d E R” has the N ( o , f f 2 / N ) distribution, If z is known to lie in a given K-dimensional subspace of R”, the situation can immediately he improved hy projecting y to the given suhspace; since the noise distribution is spherically symmetric this leaves only K / N fraction ofthe original noise. Further information ahout the distribution of z could he exploited to remove even more noise. In many scenarios in which one attempts to estimate z from y, to know with certainty the K-dimensional subspace that contains z is too much to ask. However, it may he reasonable to know a set of &‘-dimensional subspaces such that the union contains or can closely approximate :c. In other words, a basis may be known such that a K-term nonlinear approximation is very accurate [I ] . (Wavelet bases have this property for piecewise smooth signals.) At high SNR (small u2), the selection of the I<-dimensional suhspace is unperturbed by d. The approximation error is approximately KO* + f.v(K), where f , w ( h ) is a decreasing function of I< that represents the error of the optimal h-term nonlinear approximation of z. Now suppose U = {u~,. . . , un,} c RN is chosen so that the class of signals of interest can he sparsely repreKannun Ramchandrun Universi ty of California, Berkeley [email protected]