# A Nuclear-Norm Lower Bound for Dithered Scalar Quantization of Matrix Products

arXiv: https://arxiv.org/abs/2609.05641 · HTML: https://mlsanity.com/paper/2609.05641/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 18 references that are in the map.
- First posted on arXiv 4 Sep 2026.

## Builds on (most cited first, 8 of 8)

- [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)
- [AffineQuant: Affine Transformation Quantization for Large Language Models](https://mlsanity.com/paper/2403.12544/) (2024)
- [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)
- [High-Rate Quantized Matrix Multiplication I](https://mlsanity.com/paper/2601.17187/) (2026)
- [High-Rate Quantized Matrix Multiplication II](https://mlsanity.com/paper/2605.13768/) (2026)
- [Contraction-Gauge Preconditioning for Quantized Matrix Multiplication](https://mlsanity.com/paper/2607.18745/) (2026)

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

- [Product-Aware Deterministic Rounding for Quantized Matrix Multiplication](https://mlsanity.com/paper/2609.31641/) (2026)

## Abstract (arXiv)

We consider the problem of minimizing error in quantized matrix multiplication C=AB. Scalar quantization of the factors introduces rounding errors whose scale depends on the maximum absolute entries -- the ranges -- of their rows and columns. These ranges determine the quantization grid steps. To reduce the error, we optimize over product-preserving transformations that alter the factor ranges and grid steps without changing C. Specifically, we seek the smallest leading expected squared error over invertible inner changes of basis and orthogonal outer rotations. Under independent, zero-mean subtractive dither noise on an unbounded lattice, we prove the output-only bound Erm lead ≥ (cA+cB)/K Vert ABVert_*2, where K is the inner dimension, cA and cB are normalized noise variances, and Vert ABVert_* is the nuclear norm. The bound is tight: an SVD-aligned Hadamard construction attains the infimum whenever a Hadamard matrix of order K exists, including every power of two, while an SVD-aligned DCT construction is within a factor of two for every K. Without outer rotations, Gram-matrix balancing minimizes factorization energy, and finite-set flattening achieves the bound within Clog(K(m+n)). For power-of-two K, conditional expectations deterministically select the Hadamard signs in O((m+n)K2) exact-real operations. Synthetic experiments verify both constructions and illustrate the tradeoff between regularization and conditioning. These results characterize the full-gauge optimum and quantify the cost of preserving row and column indices.