# Minimum Distortion Quantization with Specified Output Distribution

arXiv: https://arxiv.org/abs/2606.10458 · HTML: https://mlsanity.com/paper/2606.10458/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 23 references that are in the map.
- First posted on arXiv 9 Jun 2026.

## Builds on (most cited first, 6 of 6)

- [A Survey of Quantization Methods for Efficient Neural Network Inference](https://mlsanity.com/paper/2103.13630/) (2021)
- [Evaluating Quantized Large Language Models](https://mlsanity.com/paper/2402.18158/) (2024)
- [Minimum Excess Risk in Bayesian Learning](https://mlsanity.com/paper/2012.14868/) (2020)
- [Optimal interval clustering: Application to Bregman clustering and statistical mixture learning](https://mlsanity.com/paper/1403.2485/) (2014)
- [Distribution-Preserving k-Anonymity](https://mlsanity.com/paper/1711.01514/) (2017)
- [Optimal Feedback Communication with Information Maximization and Distortion Minimization](https://mlsanity.com/paper/2606.09698/) (2026)

## 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.