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)=||W(x)(y-x)||²,y,x∈Rⁿ, the asymptotic expression for the rate-distortion function of X∈Rⁿ at distortion level D equals h(X)-₂ /ⁿlog(2πeD/n)+Elog|detW(X)| where h(X) is the differential entropy of X. Extensions to stationary sources and to high-resolution remote (“noisy”) source coding are also given