aiqbench analysis
This research establishes a theoretical lower bound for error in dithered scalar quantization of matrix products, offering new mathematical limits for low-precision linear algebra operations. By optimizing over product-preserving transformations like SVD-aligned Hadamard constructions, the study provides concrete methods to minimize rounding errors without altering the final result. These findings are critical for advancing the efficiency and accuracy of large-scale AI models that rely heavily on quantized matrix multiplications.
A Nuclear-Norm Lower Bound for Dithered Scalar Quantization of Matrix Products
arXiv:2609.05641v1 Announce Type: cross Abstract: 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 $E_{\rm lead} \ge (c_A+c_B)/K \Vert AB\Vert_*^2$, where $K$ is the inner dimension, $c_A$ and $c_B$ are normalized noise variances, and $\Vert AB\Vert_*$ 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 $C\log(K(m+n))$. For power-of-two $K$, conditional expectations deterministically select the Hadamard signs in $O((m+n)K^2)$ 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.
Abstract: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 $E_{\rm lead} \ge (c_A+c_B)/K \Vert AB\Vert_*^2$, where $K$ is the inner dimension, $c_A$ and $c_B$ are normalized noise variances, and $\Vert AB\Vert_*$ 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 $C\log(K(m+n))$. For power-of-two $K$, conditional expectations deterministically select the Hadamard signs in $O((m+n)K^2)$ 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.
17 pages, 2 figures. Code: this https URL
Information Theory (cs.IT); Machine Learning (cs.LG); Numerical Analysis (math.NA)
G.1.3; E.4; F.2.1; H.1.1; I.2.6
arXiv:2609.05641 [cs.IT]
(or arXiv:2609.05641v1 [cs.IT] for this version)
https://doi.org/10.48550/arXiv.2609.05641 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.
Full text under the publisher's open licence · 334 words
Related coverage
Productionizing TurboQuant on AMD GPUs for KV-Cache-Bound LLM Inference
AMD · Jun 11, 2026
Convergence of Practical Muon with Finite Newton-Schulz Iterations and Nesterov Momentum
arXiv · Oct 1, 2026
Product-Aware Deterministic Rounding for Quantized Matrix Multiplication
arXiv · Sep 30, 2026
When Does Compensating Inherited Quantization Error Help? Mean-Square Guarantees for Layer-Wise Quantization and Their Limits
arXiv · Oct 15, 2026