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Article Dans Une Revue Journal of Approximation Theory Année : 2021

Optimal dual quantizers of 1D log-concave distributions: uniqueness and Lloyd like algorithm

Résumé

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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hal-03918420 , version 1 (24-04-2023)

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Benjamin Jourdain, Gilles Pagès. Optimal dual quantizers of 1D log-concave distributions: uniqueness and Lloyd like algorithm. Journal of Approximation Theory, 2021, 267 (105581), pp.105581. ⟨10.1016/J.JAT.2021.105581⟩. ⟨hal-03918420⟩
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