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 i.i.d. data points using any design algorithm is at least n 1=2 away from the optimal distortion for some distribution on a bounded subset of Rd. 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. We also derive a new upper bound for the performance of the empirically optimal quantizer. P. Bartlett is with the Department of Systems Engineering, Research School of Information Sciences and Engineering, Australian National University, Canberra 0200, Australia (email: [email protected]). T. Linder is with the Department of Electrical and Computer Engineering, University of California, San Diego, California, on leave from the Technical University of Budapest, Hungary (email: [email protected]). G. Lugosi is with the Department of Economics, Pompeu Fabra University, Ramon Trias Fargas 25-27, 08005 Barcelona, Spain (email: [email protected]). This work was partially supported by OTKA Grant F 014174.