On the Convexity of the MSE Distortion of Symmetric Uniform Scalar Quantization

This paper investigates the convexity of the mean squared-error distortion of symmetric uniform scalar quantization with respect to step size. The principal results include proofs for odd numbers of levels that distortion is not convex for any symmetric density and that it is convex for even numbers of levels for densities, such as Gaussian, Laplacian, and gamma, but is not, in general for two-sided Rayleigh. For the latter case, an interval is derived that includes the optimal step size and over which the distortion is convex. The proofs of convexity use the Euler–Maclaurin formula applied to the second derivative of distortion, with upper bounds on the remainder term. These results imply that a zero of the derivative of the distortion for these densities, which has been previously conjectured optimal, is indeed the optimal step size, because the distortion is convex either globally or locally over a sufficiently wide interval to ensure a global minimizer.