Minimum Distortion Quantization with Specified Output Distribution
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Abstract (arXiv)
We derive the optimal quantizer of a real-valued random variable W with distribution PW such that 1) the distribution of the quantization output X that can take k values follows any specified distribution PX over 1,ldots,k, and 2) the minimum mean squared error (MMSE) of estimating W from X is minimized. It is shown that the optimal quantizer takes the form X=σbig(F_σ-1(X)-1(FW(W))big), where σ is the optimal permutation of 1,ldots,k among all permutations to minimize the MMSE, and F is the cumulative distribution function. When PW is uniform over an interval or PX is uniform over 1,ldots,k, the quantizer takes a simple form X=FX-1(FW(W)). The concept of majorization plays a key role in the optimality proof. Specifying the output distribution is useful for designing quantizers with explicitly controlled output entropy, maximized mutual information between input and output, tailored output distribution to match channel input requirements for communication, and data anonymization.
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References · 6
- 2021A Survey of Quantization Methods for Efficient Neural Network Inferencecited by 798
- 2024Evaluating Quantized Large Language Modelscited by 75
- 2020Minimum Excess Risk in Bayesian Learningcited by 35
- 2014Optimal interval clustering: Application to Bregman clustering and statistical mixture learningcited by 3
- 2017Distribution-Preserving k-Anonymitycited by 1
- 2026Optimal Feedback Communication with Information Maximization and Distortion Minimizationcited by 1
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