High-Resolution Source Coding for Non-Difference Distortion Measures: The Rate-Distortion Function

The problem of asymptotic (i.e., low-distortion) behavior of the rate-distortion function of a random vector is investigated for a class of non-difference distortion measures. The main result is an asymptotically tight expression which parallels the Shannon lower bound for difference distortion measures. For example, for an input-weighted squared error distortion measure d(x; y) = kW (x)(y x)k2; y; x 2 n, the asymptotic expression for the rate-distortion function of X 2 n at distortion level D equals h(X) n 2 log (2 eD=n) +E log jdetW (X)j where h(X) is the differential entropy of X . Extensions to staionary sources and to high-resolution remote (“noisy”) source coding are also given. In a companion paper in this issue these results are applied to develop a high-resolution quantization theory for non-difference distortion measures.