# Product-Aware Deterministic Rounding for Quantized Matrix Multiplication

arXiv: https://arxiv.org/abs/2609.31641 · HTML: https://mlsanity.com/paper/2609.31641/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 32 references that are in the map.
- First posted on arXiv 11 Sep 2026.

## Builds on (most cited first, 11 of 11)

- [GPTQ: Accurate Post-Training Quantization for Generative Pre-trained Transformers](https://mlsanity.com/paper/2210.17323/) (2022)
- [Up or Down? Adaptive Rounding for Post-Training Quantization](https://mlsanity.com/paper/2004.10568/) (2020)
- [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 Gram-Schmidt Walk: A Cure for the Banaszczyk Blues](https://mlsanity.com/paper/1708.01079/) (2017)
- [The Lattice Geometry of Neural Network Quantization -- A Short Equivalence Proof of GPTQ and Babai's Algorithm](https://mlsanity.com/paper/2508.01077/) (2025)
- [DiscQuant: A Quantization Method for Neural Networks Inspired by Discrepancy Theory](https://mlsanity.com/paper/2501.06417/) (2025)
- [Optimal Scalar Quantization for Matrix Multiplication: Closed-Form Density and Phase Transition](https://mlsanity.com/paper/2603.19559/) (2026)
- [Contraction-Gauge Preconditioning for Quantized Matrix Multiplication](https://mlsanity.com/paper/2607.18745/) (2026)
- [A Nuclear-Norm Lower Bound for Dithered Scalar Quantization of Matrix Products](https://mlsanity.com/paper/2609.05641/) (2026)

## Abstract (arXiv)

Scalar rounding decisions interact through matrix multiplication. We study deterministic product-aware rounding after scales, clipping bounds, and grids are fixed, with each active scalar choosing between adjacent levels. For dynamic activation rounding, null-space reduction preserves the relaxed product while leaving at most r fractional decisions, where r is the rank of the active gap-weighted weight block. Conditional-expectation completion gives a deterministic polynomial-time algorithm with squared product error at most  OPTdyn+rnumax2/4, where OPTdyn is the best admissible error and numax is the largest row norm of that block. For reusable static weights, the exact expected product-loss metric is the uncentered input second moment with fixed output bias; free bias recalibration yields the centered covariance. Exact optimization is NP-hard even at rank one. In balanced blocks with K=1024 and r=16, conditional- expectation completion attains a dither-normalized median error of 0.010, compared with 0.899 for round-to-nearest. Clipping-aware initialization reduces median normalized error by a factor of 43.4 at ten-percent clipping. Held-out Digits experiments show that retaining the input mean or correcting the output bias improves median product error over round-to-nearest in all four tested bit-width and calibration-size settings.