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 an i.i.d. Gaussian source. 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.