发表机构
Oak Ridge National Laboratory（橡树岭国家实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对量化矩阵乘法误差最小化问题，通过保积变换优化因子范围，证明输出误差的核范数下界，并给出达到该界的SVD对齐Hadamard和DCT构造，刻画了全规范最优。
AI 中文摘要
我们考虑量化矩阵乘法 $C=AB$ 中的误差最小化问题。因子的标量量化引入了舍入误差，其大小取决于其行和列的最大绝对条目（即范围）。这些范围决定了量化网格步长。为了减少误差，我们优化保积变换，该变换改变因子范围和网格步长而不改变 $C$。具体而言，我们在可逆内基变换和正交外旋转下寻找最小的前导期望平方误差。在无界格上的独立、零均值减性抖动噪声下，我们证明了仅输出的界 $E_{\rm lead} \ge (c_A+c_B)/K \Vert AB\Vert_*^2$，其中 $K$ 是内维度，$c_A$ 和 $c_B$ 是归一化噪声方差，$\Vert AB\Vert_*$ 是核范数。该界是紧的：当阶为 $K$ 的 Hadamard 矩阵存在时（包括所有 2 的幂），SVD 对齐的 Hadamard 构造达到下确界，而 SVD 对齐的 DCT 构造对每个 $K$ 在因子 2 内。没有外旋转时，Gram 矩阵平衡最小化分解能量，有限集展平在 $C\log(K(m+n))$ 内达到该界。对于 2 的幂 $K$，条件期望在 $O((m+n)K^2)$ 精确实数运算中确定性选择 Hadamard 符号。合成实验验证了两种构造，并说明了正则化与条件数之间的权衡。这些结果刻画了全规范最优，并量化了保留行和列索引的代价。
英文摘要
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.
Comments17 pages, 2 figures. Code: https://github.com/piyush314/gauge-floors
