On testing trained vector quantizer codebooks

This paper discusses a criterion for testing a vector quantizer (VQ) codebook that is obtained by "training". When a VQ codebook is designed by a clustering algorithm using a training set, "time-average" distortion, which is called the training-set-distortion (TSD), is usually calculated in each iteration of the algorithm, since the input probability function is unknown in general and cumbersome to deal with. The algorithm stops when the TSD ceases to significantly decrease. In order to test the resultant codebook, validating-set-distortion (VSD) is calculated on a separate validating set (VS). Codebooks that yield small difference between the TSD and the VSD are regarded as good ones. However, the difference VSD-TSD is not necessarily a desirable criterion for testing a trained codebook unless certain conditions are satisfied. A condition that is previously assumed to be important is that the VS has to be quite large to well approximate the source distribution. This condition implies greater computational burden of testing a codebook. In this paper, we first discuss the condition under which the difference VSD-TSD is a meaningful codebook testing criterion. Then, convergence properties of the VSD, a time-average quantity, are investigated. Finally we show that for large codebooks, a VS size as small as the size of the codebook is sufficient to evaluate the VSD. This paper consequently presents a simple method to test trained codebooks for VQ's. Experimental results on synthetic data and real images supporting the analysis are also provided and discussed.