Optimal solution of a Monge-Kantorovitch transportation problem
Journal of Computational and Applied Mathematics
Optimal Quantizer Performance and the Wasserstein Distortion
DCC '05 Proceedings of the Data Compression Conference
IEEE Transactions on Information Theory
Optimal entropy-constrained scalar quantization of a uniform source
IEEE Transactions on Information Theory
On the structure of optimal entropy-constrained scalar quantizers
IEEE Transactions on Information Theory
Lagrangian empirical design of variable-rate vector quantizers: consistency and convergence rates
IEEE Transactions on Information Theory
Codecell convexity in optimal entropy-constrained vector quantization
IEEE Transactions on Information Theory
High-Resolution Scalar Quantization With Rényi Entropy Constraint
IEEE Transactions on Information Theory
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The optimal quantizer in memory-size constrained vector quantization induces a quantization error which is equal to a Wasserstein distortion. However, for the optimal (Shannon-)entropy constrained quantization error a proof for a similar identity is still missing. Relying on principal results of the optimal mass transportation theory, we will prove that the optimal quantization error is equal to a Wasserstein distance. Since we will state the quantization problem in a very general setting, our approach includes the Renyi-@a-entropy as a complexity constraint, which includes the special case of (Shannon-)entropy constrained (@a=1) and memory-size constrained (@a=0) quantization. Additionally, we will derive for certain distance functions codecell convexity for quantizers with a finite codebook. Using other methods, this regularity in codecell geometry has already been proved earlier by Gyorgy and Linder (2002, 2003) [11,12].