The minimax distortion redundancy in empirical quantizer design

Bartlett, Peter; Linder, Tamas; Lugosi, Gábor

The minimax distortion redundancy in empirical quantizer design

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IEEE Transactions on Information Theory, 44, (1998), pp. 1802-1813

Abstract

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 $\Omega (n^{-1/2})$ away from the optimal distortion for some distribution on a bounded subset of ${\cal R}^d$. 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.