Individual convergence rates in empirical vector quantizer design

We consider the rate of convergence of the expected distortion redundancy of empirically optimal vector quantizers. Earlier results show that the mean-squared distortion of an empirically optimal quantizer designed from n independent and identically distributed (i.i.d.) source samples converges uniformly to the optimum at a rate of O(1/√n), and that this rate is sharp in the minimax sense. We prove that for any fixed distribution supported on a given finite set the convergence rate is O(1/n) (faster than the minimax lower bound), where the corresponding constant depends on the source distribution. For more general source distributions we provide conditions implying a little bit worse O(logn/n) rate of convergence. Although these conditions, in general, are hard to verify, we show that sources with continuous densities satisfying certain regularity properties (similar to the ones of Pollard that were used to prove a central limit theorem for the code points of the empirically optimal quantizers) are included in the scope of this result. In particular, scalar distributions with strictly log-concave densities with bounded support (such as the truncated Gaussian distribution) satisfy these conditions.