Lossy Compression via Sparse Linear Regression: Performance Under Minimum-Distance Encoding

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 Additive White Gaussian Noise channel. For independent identically distributed (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 σ² with (squared-error) distortion D can compress any ergodic source of variance less than σ² 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 a compact representation in terms of a matrix whose size is a low-order polynomial in the block-length.