Lossy Compression via Sparse Linear Regression: Computationally Efficient Encoding and Decoding

TL;DR

This paper introduces a computationally efficient lossy compression method using Sparse Regression Codes, achieving optimal distortion-rate performance for Gaussian sources with practical encoding complexity and robustness across ergodic sources.

Contribution

It presents a novel encoding algorithm based on sparse linear regression that is both computationally efficient and achieves optimal distortion-rate trade-offs for Gaussian sources.

Findings

Achieves optimal distortion-rate function for i.i.d Gaussian sources.

Provides a flexible trade-off between distortion and encoding complexity.

Demonstrates good empirical performance at low and moderate rates.

Abstract

We propose computationally efficient encoders and decoders for lossy compression using a Sparse Regression Code. The codebook is defined by a design matrix and codewords are structured linear combinations of columns of this matrix. The proposed encoding algorithm sequentially chooses columns of the design matrix to successively approximate the source sequence. It is shown to achieve the optimal distortion-rate function for i.i.d Gaussian sources under the squared-error distortion criterion. For a given rate, the parameters of the design matrix can be varied to trade off distortion performance with encoding complexity. An example of such a trade-off as a function of the block length n is the following. With computational resource (space or time) per source sample of O((n/\log n)^2), for a fixed distortion-level above the Gaussian distortion-rate function, the probability of excess distortion decays exponentially in n. The Sparse Regression Code is robust in the following sense: for any ergodic source, the proposed encoder achieves the optimal distortion-rate function of an i.i.d Gaussian source with the same variance. Simulations show that the encoder has good empirical performance, especially at low and moderate rates.