The rate loss in the Wyner-Ziv problem

The rate-distortion function for source coding with side information at the decoder (the \Wyner-Ziv problem") is given in terms of an auxiliary random variable, which forms a Markov chain with the source and the side information. This Markov chain structure, typical to the solution of multiterminal source coding problems, corresponds to a loss in coding rate with respect to the conditional rate-distortion function, i.e., to the case where the encoder is fully informed. We show that for diierence (or balanced) distortion measures, this loss is bounded by a universal constant, which is the minimax capacity of a suitable additive noise channel. Furthermore, in the worst case this loss is equal to the maximin redundancy over the rate distortion function of the additive noise \test" channel. For example, the loss in the Wyner-Ziv problem is less than 0:5 bit per sample in the squared-error distortion case, and it is less than 0:22 bits for a binary source with Hamming-distance. These results have implications also in universal quantization with side information, and in more general multiterminal source coding problems.