Optimal one-bit quantization

We consider the problem of finding the optimal one-bit quantizer for symmetric source distributions, with the Euclidean norm as the measure of distortion. For fixed rate quantizers, we prove that for (symmetric) monotonically decreasing source distributions with ellipsoidal level curves, the centroids of the optimal 1-bit quantizer must be on the major axis of the ellipsoids. Under the same assumptions on the source distribution, the centroids of the optimal one-bit variable-rate quantizer lie on one of the axes of the ellipsoid. If further, the source distribution f(x) is log-concave in x, the optimal 1-bit fixed-rate quantizer is unique and symmetric about the origin. (The Gaussian is an example of a distribution that satisfies all these conditions.) Under a further set of conditions on the source distributions, we show that there is a threshold below which the optimal fixed rate and variable rate quantizer are the same.