Randomized Quantization and Source Coding with Constrained Output Distribution

TL;DR

This paper explores fixed-rate randomized vector quantization with a specified output distribution, establishing existence and optimality conditions using optimal transport theory and deriving a rate-distortion theorem for stationary sources.

Contribution

It introduces a general representation of randomized quantizers and proves the existence of optimal quantizers with given output distributions, including a rate-distortion characterization.

Findings

Optimal randomized quantizers exist under various conditions.

Convex codecells suffice for optimal quantizers with density sources.

A rate-distortion theorem for stationary sources is established.

Abstract

This paper studies fixed-rate randomized vector quantization under the constraint that the quantizer's output has a given fixed probability distribution. A general representation of randomized quantizers that includes the common models in the literature is introduced via appropriate mixtures of joint probability measures on the product of the source and reproduction alphabets. Using this representation and results from optimal transport theory, the existence of an optimal (minimum distortion) randomized quantizer having a given output distribution is shown under various conditions. For sources with densities and the mean square distortion measure, it is shown that this optimum can be attained by randomizing quantizers having convex codecells. For stationary and memoryless source and output distributions a rate-distortion theorem is proved, providing a single-letter expression for the optimum distortion in the limit of large block-lengths.