Encoding mismatch in lossy source compression

A source sequence is described by the index of the codeword in some codebook C that is closest to it according to the distortion measure d/sub 0/(x,x/spl circ//sub 1/). We study the distortion associated with reconstructing the source sequence so as to minimize the distortion measured by d/sub 1/(x,x/spl circ//sub 1/), which is in general different from d/sub 0/(x,x/spl circ//sub 0/). Using a random coding argument we derive an upper bound on the resulting distortion. Applying this bound to blocks of source symbols we construct a sequence of bounds which are shown to converge to the least distortion achievable in this setup. This solves the rate distortion dual of an open problem related to the mismatched capacity of channels. Addressing a different kind of mismatch, we also study mean-squared-error description of non-Gaussian sources with Gaussian codebooks. It is shown that the use of a Gaussian codebook to compress any ergodic source results in an average distortion which depends on the source only via its second moment. The hardest source to describe is the memoryless Gaussian source, and it is best described using a Gaussian codebook. Once a Gaussian codebook is used, we show that all sources of a given second moment become equally hard to describe.