The Rate-Distortion Function and Excess-Distortion Exponent of Sparse Regression Codes with Optimal Encoding

Abstract: This paper studies the performance of sparse regression codes for lossy compression with the squared-error distortion criterion. In a sparse regression code, codewords are linear combinations of subsets of columns of a design matrix. It is shown that with minimum-distance encoding, sparse regression codes achieve the Shannon rate-distortion function for i.i.d. Gaussian sources $R<sup>*(D)$ as well as the optimal excess-distortion exponent. This completes a previous result which showed that $R<sup>*(D)$ and the optimal exponent were achievable for distortions below a certain threshold. The proof of the rate-distortion result is based on the second moment method, a popular technique to show that a non-negative random variable $X$ is strictly positive with high probability. In our context, $X$ is the number of codewords within target distortion $D$ of the source sequence. We first identify the reason behind the failure of the standard second moment method for certain distortions, and illustrate the different failure modes via a stylized example. We then use a refinement of the second moment method to show that $R<sup>*(D)$ is achievable for all distortion values. Finally, the refinement technique is applied to Suen's correlation inequality to prove the achievability of the optimal Gaussian excess-distortion exponent.