On the support of MSE-optimal, fixed-rate, scalar quantizers

This paper determines how the support regions of optimal and asymptotically optimal fixed-rate scalar quantizers (with respect to mean-squared error) depend on the number of quantization points N and the probability density of the variable being quantized. It shows that for asymptotic optimality it is necessary and sufficient that the support region grow fast enough that the outer (or overload) distortion decreases as o(1/N-2). Formulas are derived for the minimal support of asymptotically optimal quantizers for generalized gamma densities, including Gaussian and Laplacian. Interestingly, these turn out to be essentially the same as for the support of optimal fixed-rate uniform scalar quantizers. Heuristic arguments are then used to find closed-form estimates for the support of truly optimal quantizers for generalized gamma densities. These are found to be more accurate than the best prior estimates, as computed by numerical algorithms. They demonstrate that the support of an optimal quantizer is larger than the minimal asymptotically optimal support by a factor depending on the density but not N, and that the outer distortion of optimal quantizers decreases as 1/N-3.