Optimal Scalar Quantization for Matrix Multiplication: Closed-Form Density and Phase Transition

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.

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 $\mathrm{OPT}_{\mathrm{dyn}}+rν_{\max}^2/4$, where $\mathrm{OPT}_{\mathrm{dyn}}$ is the best admissible error and $ν_{\max}$ 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.

We study low-precision computation of C=AB with both factors quantized. We derive an exact finite-dimensional identity for the expected squared product error under independent, zero-mean entrywise errors with known variance fields; it holds exactly for non-overloading subtractive dither and for independent stochastic rounding, and we empirically assess deterministic round-to-nearest (RTN). Using the product-preserving equivalence AB=(AT)(T^{-1}B), we formulate contraction-gauge preconditioning: jointly choosing a factor representation and its sharing pattern before quantization. Preconditioning can reduce product error but may require extra transformed, quantized copies of the opposite operand: a shared transform needs one copy, a block-specific transform up to one per block. Within the bounded family of positive diagonal gauges (folds), a geometric program computes a globally optimal shared fold and a linear program decides whether the identity fold is already optimal. For other families we derive computable selection statistics -- tail index for scaling, profile spread for partitioning, coherence and weighted-Gram energy for rotations, slice-energy covariance for hierarchy depth -- with upper bounds for ranking heuristic candidates. Across twelve linear products from a trained three-block image classifier, median within-product rank correlations between dither-model predictions and deterministic-RTN errors are 0.937 at 8 bits and 0.918 at 4 bits. The GP fold cuts held-out product error over the identity fold by 18.0% (8-bit) and 20.5% (4-bit) in geometric mean, beats a SmoothQuant-style grid baseline at both precisions and on ten of twelve products, and lowers composed logit MSE by 15.4% and 26.4%. We thus provide exact stochastic product-error accounting, certified selection within the diagonal family, and a common objective for evaluating reusable transform candidates under RTN.