Lossy compression via sparse linear regression: Computationally efficient encoding and decoding

We propose computationally efficient encoders and decoders for lossy compression using a Sparse Regression Code. Codewords are structured linear combinations of columns of a design 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 with squared-error distortion. 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 is: computational resource (space or time) per source sample of O((n/ log n)²) and probability of excess distortion decaying exponentially in n/ log n, where n is the block length. 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 very good empirical performance, especially at low and moderate rates.