# Price of metric universality in vector quantization is at most 0.11 bit

arXiv: https://arxiv.org/abs/2602.05790 · HTML: https://mlsanity.com/paper/2602.05790/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 28 references that are in the map.
- First posted on arXiv 5 Feb 2026.

## Builds on (most cited first, 13 of 13)

- [GPTQ: Accurate Post-Training Quantization for Generative Pre-trained Transformers](https://mlsanity.com/paper/2210.17323/) (2022)
- [SmoothQuant: Accurate and Efficient Post-Training Quantization for Large Language Models](https://mlsanity.com/paper/2211.10438/) (2022)
- [QuaRot: Outlier-Free 4-Bit Inference in Rotated LLMs](https://mlsanity.com/paper/2404.00456/) (2024)
- [SpinQuant: LLM quantization with learned rotations](https://mlsanity.com/paper/2405.16406/) (2024)
- [QuIP: 2-Bit Quantization of Large Language Models With Guarantees](https://mlsanity.com/paper/2307.13304/) (2023)
- [QuIP#: Even Better LLM Quantization with Hadamard Incoherence and Lattice Codebooks](https://mlsanity.com/paper/2402.04396/) (2024)
- [QTIP: Quantization with Trellises and Incoherence Processing](https://mlsanity.com/paper/2406.11235/) (2024)
- [Pretraining Large Language Models with NVFP4](https://mlsanity.com/paper/2509.25149/) (2025)
- [NestQuant: Nested Lattice Quantization for Matrix Products and LLMs](https://mlsanity.com/paper/2502.09720/) (2025)
- [The Geometry of LLM Quantization: GPTQ as Babai's Nearest Plane Algorithm](https://mlsanity.com/paper/2507.18553/) (2025)
- [WUSH: Near-Optimal Adaptive Transforms for LLM Quantization](https://mlsanity.com/paper/2512.00956/) (2025)
- [Optimal Quantization for Matrix Multiplication](https://mlsanity.com/paper/2410.13780/) (2024)
- [The Lattice Geometry of Neural Network Quantization -- A Short Equivalence Proof of GPTQ and Babai's Algorithm](https://mlsanity.com/paper/2508.01077/) (2025)

## Built on by (most cited first, 1 of 1)

- [High-Rate Quantized Matrix Multiplication II](https://mlsanity.com/paper/2605.13768/) (2026)

## Abstract (arXiv)

Fast computation of a matrix product W^top X is a workhorse of modern LLMs. To make their deployment more efficient, a popular approach is that of using a low-precision approximation widehat W in place of true W (``weight-only quantization''). Information theory demonstrates that an optimal algorithm for reducing precision of W depends on the (second order) statistics of X and requires a careful alignment of vector quantization codebook with PCA directions of X (a process known as ``waterfilling allocation''). Dependence of the codebook on statistics of X, however, is highly impractical. This paper proves that there exist a universal codebook that is simultaneously near-optimal for all possible statistics of X, in the sense of being at least as good as an X-adapted waterfilling codebook with rate reduced by 0.11 bit per dimension in the case when W is Gaussian. Such universal codebook would be an ideal candidate for the low-precision storage format, a topic of active modern research, but alas the existence proof is non-constructive. Equivalently, our result shows existence of a net in ℝn that is a nearly-optimal covering of a sphere simultaneously with respect to all Hilbert norms.