Distributionally Robust Scalar Quantization Using the Wasserstein Distance

We study the scalar quantization of random information sources with unknown probability distributions through the lens of distributional robustness. Assuming the true source distribution resides within an ambiguity set of plausible distributions, we formulate the problem of distributionally robust scalar quantization (DRQ) as finding a minimax optimal quantizer that minimizes the mean-squared quantization error (MSQE) under the least favorable distribution in the ambiguity set. Specifically, we define the ambiguity set using the Wasserstein-2 ($\mathrm{W}_{2}$) distance, a statistical measure of distribution shift, as a statistical ball centered around a nominal source distribution. We derive a tractable dual reformulation and establish the necessary optimality conditions for $\mathrm{W}_{2}$-DRQ. Building on these results, we propose a novel fixedpoint iteration algorithm, inspired by the classical Lloyd-Max algorithm, to compute locally optimal DR quantizers. We prove the monotonic convergence of the algorithm and demonstrate its effectiveness through numerical simulations. Our results highlight the performance advantages of the proposed approach under distributional uncertainty, particularly in comparison to the classical Lloyd-Max algorithm.