Distortion bounds for vector quantizers with finite codebook size

R. Meir, V. Maiorov
1999 IEEE Transactions on Information Theory  
Upper and lower bounds are presented for the distortion of the optimal N-point vector quantizer applied to k-dimensional signals. Under certain smoothness conditions on the source distribution, the bounds are shown to hold for each and every value of N, the codebook size. These results extend bounds derived in the high-resolution limit, which assumes that the number of code vectors is arbitrarily large. Two approaches to the upper bound are presented. The first, constructive construction,
more » ... construction, achieves the correct asymptotic rate of convergence as well as the correct dependence on the source density, although leading to an inferior value for the constant. The second construction, based on a random coding argument, is shown to additionally achieve a value of the constant which is much closer to the best known result derived within the asymptotic theory. Lower bound results derived in the correspondence are again shown to possess the correct asymptotic form and yield a constant which is almost indistinguishable from the best value achieved in the asymptotic regime. Finally, application of the results to the problem of source coding yields upper bounds on the distortion rate function for a wide class of processes. Index Terms-Finite codebook, lower bounds, upper bounds, vector quantization. k ! 1, yielding the value lim k!1 k 0r=2 J k;r = (2e) 0r=2 : Note that similar upper and lower bounds can be shown to hold for the more general k 1 k norm, with slightly modified constants. As pointed out by Bucklew [2], a key assumption in the derivations has been that the density p(x x x) may be assumed to be constant over small bounded sets. However, since in this work we will be interested in results which hold for finite values of N, the assumption that the quantization cells are small does not hold, and, consequently, 0018-9448/99$10.00
doi:10.1109/18.771232 fatcat:npvzom5ox5bnhca4bs5algl2iy