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Optimal vector quantization in terms of Wasserstein distance

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  • Kreitmeier, Wolfgang

Abstract

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 Rényi-[alpha]-entropy as a complexity constraint, which includes the special case of (Shannon-)entropy constrained ([alpha]=1) and memory-size constrained ([alpha]=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 György and Linder (2002, 2003)Â [11] and [12].

Suggested Citation

  • Kreitmeier, Wolfgang, 2011. "Optimal vector quantization in terms of Wasserstein distance," Journal of Multivariate Analysis, Elsevier, vol. 102(8), pages 1225-1239, September.
    • Handle: RePEc:eee:jmvana:v:102:y:2011:i:8:p:1225-1239
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