A note on the existence of optimal entropy-constrained vector quantizers

Chou and Betts (1998) showed that for any source distribution with sufficiently light tails the optimal entropy-constrained vector quantizer (ECVQ) has a finite number of code points. This result addressed those optimal vector quantizers that achieve the lower convex hull of the operational distortion-rate function, i.e., quantizers Q that minimize the Lagrangian performance D(Q)+λH(Q) for some λ>0, where D(Q) is the quantizer´s expected distortion and H(Q) is its output entropy. This Lagrangian optimality criterion also matches the Lloyd-type algorithm used to optimize ECVQs, and corresponds to the fixed-slope approach in variable-rate lossy source coding. The existence of Lagrangian-optimal quantizers, however, was an open question. In this paper we show that optimal ECVQs minimizing the Lagrangian performance always exist under general conditions on the source and the distortion measure. A related result showing the existence of a mean-square optimal ECVQ in the class of regular ECVQs is also presented.