Lossy Compression via Sparse Linear Regression: Performance under Minimum-distance Encoding

TL;DR

This paper introduces sparse regression codes for lossy compression that achieve Shannon's rate-distortion limit with optimal error exponents, demonstrating robustness and efficiency in high-dimensional settings.

Contribution

It proposes a new class of codes based on high-dimensional linear regression that attain optimal compression performance and are robust across different source variances.

Findings

Achieves Shannon rate-distortion function for Gaussian sources.

Attains the optimal error exponent for minimum-distance encoding.

Robust to source variance variations, maintaining compression quality.

Abstract

We study a new class of codes for lossy compression with the squared-error distortion criterion, designed using the statistical framework of high-dimensional linear regression. Codewords are linear combinations of subsets of columns of a design matrix. Called a Sparse Superposition or Sparse Regression codebook, this structure is motivated by an analogous construction proposed recently by Barron and Joseph for communication over an AWGN channel. For i.i.d Gaussian sources and minimum-distance encoding, we show that such a code can attain the Shannon rate-distortion function with the optimal error exponent, for all distortions below a specified value. It is also shown that sparse regression codes are robust in the following sense: a codebook designed to compress an i.i.d Gaussian source of variance $\sigma^2$ with (squared-error) distortion $D$ can compress any ergodic source of variance less than $\sigma^2$ to within distortion $D$. Thus the sparse regression ensemble retains many of the good covering properties of the i.i.d random Gaussian ensemble, while having having a compact representation in terms of a matrix whose size is a low-order polynomial in the block-length.