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

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. ).Communicated by M. Elad, Associate Editor for Signal Processing. 1 The error exponent of a compression code measures how fast the probability of excess distortion decays to zero with growing block length.The Sparse Regression codebook is constructed based on the statistical framework of high-dimensional linear regression, and was proposed recently by Barron and Joseph for communication over the AWGN channel at rates approaching the channel capacity [4], [5]. The codewords are sparse linear combinations of columns of an n × N design matrix or 'dictionary', where n is the block-length and N is a loworder polynomial in n. This structure enables the design of computationally efficient encoders based on sparse approximation ideas (e.g., [6], [7]). We propose one such encoding algorithm and analyze it performance.

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“…The dictionary design is heavily inspired by SPARC, presented in [14]. The structure of the dictionary in SPARC can be used in the design of computationally efficient encoders [20]. An additional feature of SPARC is, that it has a low memory requirement, since only one section of the dictionary needs to be stored in the memory for every iteration.…”

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“…We consider a dataset composed of M = 1000 binary sequences of length n = 512, with p = q = 0.5. As the fixed-length lossy compression algorithm, we use a binary-Hamming version of the successive refinement compression scheme [13].…”

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“…It was shown in [11] that any ergodic source can be compressed to the Gaussian distortion-rate function…”

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