Operational duality between lossy compression and channel coding: Channel decoders as lossy compressors

We explore the duality between lossy compression and channel coding in the operational sense: whether a capacity-achieving encoder-decoder sequence achieves the rate distortion function of the dual problem when the channel decoder [encoder] is the source compressor [decompressor, resp]. We show that, if used as a lossy compressor, the maximum-likelihood channel decoder of a randomly chosen capacity-achieving codebook achieves the rate-distortion function almost surely. However, operational duality does not hold for every capacity achieving encoder-decoder sequence. We show that there exist optimal channel coding schemes, which operate far from the rate-distortion function when used for the dual lossy compression problem.