A Universal Random Coding Ensemble for Sample-Wise Lossy Compression

Abstract

We propose a universal ensemble for the random selection of rate–distortion codes which is asymptotically optimal in a sample-wise sense. According to this ensemble, each reproduction vector, x^, is selected independently at random under the probability distribution that is proportional to 2−LZ(x^), where LZ(x^) is the code length of x^ pertaining to the 1978 version of the Lempel–Ziv (LZ) algorithm. We show that, with high probability, the resulting codebook gives rise to an asymptotically optimal variable-rate lossy compression scheme under an arbitrary distortion measure, in the sense that a matching converse theorem also holds. According to the converse theorem, even if the decoder knew the ℓ-th order type of source vector in advance (ℓ being a large but fixed positive integer), the performance of the above-mentioned code could not have been improved essentially for the vast majority of codewords pertaining to source vectors in the same type. Finally, we present a discussion of our results, which includes among other things, a clear indication that our coding scheme outperforms the one that selects the reproduction vector with the shortest LZ code length among all vectors that are within the allowed distortion from the source vector.