Optimal dual quantizers of 1D log-concave distributions: uniqueness and Lloyd like algorithm
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We establish for dual quantization the counterpart of Kieffer's uniqueness result for compactly supported one dimensional probability distributions having a log-concave density (also called strongly unimodal): for such distributions, L^r-optimal dual quantizers are unique at each level N, the optimal grid being the unique critical point of the quantization error. An example of non-strongly unimodal distribution for which uniqueness of critical points fails is exhibited. In the quadratic r=2 case, we propose an algorithm to compute the unique optimal dual quantizer. It provides a counterpart of Lloyd's method I algorithm in a Voronoi framework. Finally semi-closed forms of L^r-optimal dual quantizers are established for power distributions on compacts intervals and truncated exponential distributions.
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