The minimax distortion redundancy in empirical quantizer design

We obtain minimax lower and upper bounds for the expected distortion redundancy of empirically designed vector quantizers. We show that the mean-squared distortion of a vector quantizer designed from n independent and identically distributed (i.i.d.) data points using any design algorithm is at least Ω(n^-1/2) away from the optimal distortion for some distribution on a bounded subset of ℛ ^d. Together with existing upper bounds this result shows that the minimax distortion redundancy for empirical quantizer design, as a function of the size of the training data, is asymptotically on the order of n^-1/2. We also derive a new upper bound for the performance of the empirically optimal quantizer