Gaussian Rate-Distortion via Sparse Regression over Compact Dictionaries

We study a class of codes for compressing memoryless Gaussian sources, designed using the statistical framework of high-dimensional linear regression. Codewords are linear combinations of subsets of columns of a design matrix. With maximum-likelihood encoding we show that such a codebook can attain the rate-distortion function with the optimal error-exponent, for all distortions below a specified value. The structure of the codebook is motivated by an analogous construction proposed recently by Barron and Joseph for communication over an AWGN channel.