Optimal dual quantizers of $1D$ $\log$-concave distributions: uniqueness and Lloyd like algorithm
Abstract
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.
- Publication:
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arXiv e-prints
- Pub Date:
- October 2020
- DOI:
- 10.48550/arXiv.2010.10816
- arXiv:
- arXiv:2010.10816
- Bibcode:
- 2020arXiv201010816J
- Keywords:
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- Mathematics - Probability;
- Mathematics - Numerical Analysis