Minimum Distortion Quantization with Specified Output Distribution

arXiv:2606.10458v1 Announce Type: cross Abstract: We derive the optimal quantizer of a real-valued random variable W with distribution P_W such that 1) the distribution of the quantization output X that can take k values follows any specified distribution P_X over {1,ldots,k}, and 2) the minimum mean squared error (MMSE) of estimating W from X is minimized. It is shown that the optimal quantizer takes the form X=sigmaig(F_{sigma^{-1}(X)}^{-1}(F_W(W))ig), where sigma is the optimal permutation of {1,ldots,k} among all permutations to minimize the MMSE, and F is the cumulative distribution function. When P_W is uniform over an interval or P_X is uniform over {1,ldots,k}, the quantizer takes a simple form X=F_{X}^{-1}(F_W(W)). The concept of majorization plays a key role in the optimality proof. Specifying the output distribution is useful for designing quantizers with explicitly controlled output entropy, maximized mutual information between input and output, tailored output distribution to match channel input requirements for communication, and data anonymization.

Source: arXiv cs.AI | 2026-06-10