Optimal Quantization Method Derived for Specified Output Distributions
A new preprint on arXiv presents a mathematical framework for designing quantizers that simultaneously enforce a specified output probability distribution and minimize mean squared error in estimating the original signal. The work shows the optimal quantizer can be expressed in closed form using cumulative distribution functions and an optimal permutation, with majorization theory underpinning the optimality proof. The result has practical implications for communications, data compression, and privacy-preserving data anonymization.
Researchers have derived the optimal quantizer for a real-valued random variable under the constraint that the quantization output follows any pre-specified distribution over a finite set of values, while simultaneously minimizing the minimum mean squared error (MMSE) of reconstructing the original variable from the quantized output. The optimal quantizer is expressed as a composition of cumulative distribution functions and an optimal permutation of output labels, reducing to a simpler closed form when either the input distribution is uniform over an interval or the output distribution is uniform. Majorization—a mathematical ordering concept from linear algebra and probability—plays a central role in proving the optimality of the proposed solution. The framework is broadly applicable: it enables quantizer design with explicitly controlled output entropy, maximization of mutual information between input and output, matching of output statistics to channel input requirements in communications systems, and controlled data anonymization. The paper was submitted to arXiv on June 9, 2026, and covers intersecting areas of information theory, artificial intelligence, optimization, and statistics.
What's missing
As a preprint, this work has not yet undergone formal peer review, so the theoretical claims and proofs have not been independently validated by journal referees. The paper does not appear to include empirical benchmarks comparing the proposed quantizer's performance against existing methods on real-world datasets, leaving practical performance gains unquantified. Computational complexity of finding the optimal permutation σ for large k is not discussed in the abstract, which may be a significant practical limitation.
What different sources said
- arXiv cs.AICenter
Minimum Distortion Quantization with Specified Output Distribution
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