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

• URL: http://arxiv.org/abs/2609.05641v1
• Date: Fri, 04 Sep 2026 18:22:01 GMT
• Title: A Nuclear-Norm Lower Bound for Dithered Scalar Quantization of Matrix Products
• Abstract summary: We consider the problem of minimizing error in quantized matrix multiplication $C=AB$.<n>We optimize over product-preserving transformations that alter the factor ranges and grid steps without changing $C$.<n>For power-of-two $K$, conditional expectations deterministically select the Hadamard signs in $O((m+n))$ exact-real operations.
• Score: 0.14680035572775532
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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