# Fast Matrix Multiplication in fp8: Certified Coefficient Optimization and Measured Error

arXiv: https://arxiv.org/abs/2609.26077 · HTML: https://mlsanity.com/paper/2609.26077/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 31 references that are in the map.
- First posted on arXiv 4 Aug 2026.

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

- [Think you have Solved Question Answering? Try ARC, the AI2 Reasoning Challenge](https://mlsanity.com/paper/1803.05457/) (2018)
- [HellaSwag: Can a Machine Really Finish Your Sentence?](https://mlsanity.com/paper/1905.07830/) (2019)
- [Pointer Sentinel Mixture Models](https://mlsanity.com/paper/1609.07843/) (2016)
- [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)
- [Optimal Quantization for Matrix Multiplication](https://mlsanity.com/paper/2410.13780/) (2024)
- [Towards a theory of non-commutative optimization: geodesic first and second order methods for moment maps and polytopes](https://mlsanity.com/paper/1910.12375/) (2019)
- [High-Rate Quantized Matrix Multiplication I](https://mlsanity.com/paper/2601.17187/) (2026)
- [Symmetries of matrix multiplication algorithms. I](https://mlsanity.com/paper/1508.01110/) (2015)
- [FalconGEMM: Surpassing Hardware Peaks with Lower-Complexity Matrix Multiplication](https://mlsanity.com/paper/2605.06057/) (2026)

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

- [Beyond Accuracy: Prefix-Invariant Realizations of Low-Precision Fast Matrix Multiplication](https://mlsanity.com/paper/2609.39816/) (2026)

## Abstract (arXiv)

A Strassen-type algorithm has many realizations with the same exact product and multiplication count yet different fp8 error because basis changes reshape coefficient geometry, posing the question of which to run. No current account settles this: classical stability controls worst-case ℓ1 growth, not the expected-error magnitude, and the Dumas--Pernet--Sedoglavic optimizer could only be called probably optimal, its global optimality unproved. To settle this, we attach to each realization a coefficient functional Phi, a scalar summary of its coefficient geometry, which we minimize over the change-of-basis orbit. This Kempf--Ness problem on a Hadamard manifold lets us certify the global Phi optimum rather than merely search for it: an exact moment-map zero fixes Phimin = 200/9, and de Groote's classification extends that optimality to every exact real rank-7 2×2 decomposition. Every exact real rank-7 realization therefore has a Phi-predicted RMS constant at least 5/3 times that of the cubic algorithm, at fixed noise coefficient. We then introduce an explicit block-scaled e4m3 model in which Phi is the leading-order coefficient of relative expected mean-squared error, and we test the resulting Phi-predicted ordering against realized fp8 error. Ordering and re-basing experiments support that prediction within tested fused block-scaled regimes, and on real matmul tiles from four architecture families the Phi-optimal realization falls in the fp8 low-error region. Across two ~70B models on real deepgemm kernels, the same realization removes 10 to 55% of classic Strassen's excess NLL over the clean model. Algorithm realization thus becomes a mathematically certified design problem rather than a tuning choice: an independent low-precision axis with a global Phi optimum and measured fp8 relevance.