The Gaussian Rate-Distortion Function of Sparse Regression Codes with Optimal Encoding

We study the rate-distortion performance of Sparse Regression Codes where the codewords are linear combinations of subsets of columns of a design matrix. It is shown that with minimum-distance encoding and squared error distortion, these codes achieve $R^*(D)$, the Shannon rate-distortion function for i.i.d. Gaussian sources. This completes a previous result which showed that $R^*(D)$ was achievable for distortions below a certain threshold. The proof 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. We first identify the reason behind the failure of the vanilla second moment method for this problem, and then introduce a refinement to show that $R^*(D)$ is achievable for all distortions.