An on-line universal lossy data compression algorithm via continuous codebook refinement .III. Redundancy analysis

For pt.II see ibid., vol.42, p.822-36 (1996). The Gold-washing data compression algorithm is an adaptive vector quantization algorithm with vector dimension n. In this paper, a redundancy problem of the Gold-washing data compression algorithm is considered. It is demonstrated that for any memoryless source with finite alphabet A and generic distribution p and for any R>0, the redundancy of the Gold-washing data compression algorithm with dimension n (defined as the difference between the average performance of the algorithm and the distortion-rate function D(p,R) of p) is upper-bounded by |/sub /spl delta/R///sup /spl delta//D(p,R)|((|A|+2/spl xi/+4 log n)/2n)+/spl sigma/(logn/n) where /sub /spl delta/R///sup /spl delta//D(p,R) is the partial derivative of D(p,R) with respect to R, |A| is the cardinality of A, and /spl xi/>0 is a parameter used to control the threshold in the Gold-washing algorithm. In connection with the results of Zhang, Yang, and Wei (see ibid., vol.43, no.1, p.71-91, 1997) on the redundancy of lossy source coding, this shows that the Gold-washing algorithm has the optimal convergence rate among all adaptive finite-state vector quantizers.