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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.19559 |
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| _version_ | 1866918398972133376 |
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| author | Ang, Calvin Kim, Sungyoon Pilanci, Mert |
| author_facet | Ang, Calvin Kim, Sungyoon Pilanci, Mert |
| contents | We study entrywise scalar quantization of two matrices prior to multiplication. Given $A\in R^{m\times k}$ and $B\in R^{k\times n}$, we quantize entries of $A$ and $B$ independently using scalar quantizers with $K_X$ and $K_Y$ levels per entry, and form $\widehat C=\widehat A\,\widehat B$. The objective is to minimize the matrix multiplication mean-squared error (MSE) $E[\|{AB-\widehat A\widehat B}\|_F^2]$ under a pair-i.i.d.\ inner-product model. In the high-resolution regime $K_X,K_Y\to\infty$, we derive a sharp $K^{-2}$ asymptotic expansion for $\mathcal{E}$, identify the exact optimal leading constants, and characterize asymptotically optimal quantization center densities in terms of conditional second moments. We then specialize to correlated Gaussian multiplicative pairs, obtaining a closed-form optimal point density \[ λ^\star(u)\ \propto\ \exp\!\left(-\frac{u^2}{6}\right)\bigl((1-ρ^2)+ρ^2u^2\bigr)^{1/3}, \qquad u=\frac{x}{σ_X}, \] with the same form for $y/σ_Y$, and prove a correlation-driven phase transition: the density is unimodal at the origin for $|ρ|\leq 1/\sqrt{3}$ and becomes bimodal for $|ρ|>1/\sqrt{3}$ with peaks at $u_{\mathrm{peak}}=\pm\sqrt{3-1/ρ^2}$. We show our method's applicability in synthetic experiments such as matrix multiplication quantization and least squares optimization, as well as quantization of large language model key and query activations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_19559 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Optimal Scalar Quantization for Matrix Multiplication: Closed-Form Density and Phase Transition Ang, Calvin Kim, Sungyoon Pilanci, Mert Information Theory Artificial Intelligence We study entrywise scalar quantization of two matrices prior to multiplication. Given $A\in R^{m\times k}$ and $B\in R^{k\times n}$, we quantize entries of $A$ and $B$ independently using scalar quantizers with $K_X$ and $K_Y$ levels per entry, and form $\widehat C=\widehat A\,\widehat B$. The objective is to minimize the matrix multiplication mean-squared error (MSE) $E[\|{AB-\widehat A\widehat B}\|_F^2]$ under a pair-i.i.d.\ inner-product model. In the high-resolution regime $K_X,K_Y\to\infty$, we derive a sharp $K^{-2}$ asymptotic expansion for $\mathcal{E}$, identify the exact optimal leading constants, and characterize asymptotically optimal quantization center densities in terms of conditional second moments. We then specialize to correlated Gaussian multiplicative pairs, obtaining a closed-form optimal point density \[ λ^\star(u)\ \propto\ \exp\!\left(-\frac{u^2}{6}\right)\bigl((1-ρ^2)+ρ^2u^2\bigr)^{1/3}, \qquad u=\frac{x}{σ_X}, \] with the same form for $y/σ_Y$, and prove a correlation-driven phase transition: the density is unimodal at the origin for $|ρ|\leq 1/\sqrt{3}$ and becomes bimodal for $|ρ|>1/\sqrt{3}$ with peaks at $u_{\mathrm{peak}}=\pm\sqrt{3-1/ρ^2}$. We show our method's applicability in synthetic experiments such as matrix multiplication quantization and least squares optimization, as well as quantization of large language model key and query activations. |
| title | Optimal Scalar Quantization for Matrix Multiplication: Closed-Form Density and Phase Transition |
| topic | Information Theory Artificial Intelligence |
| url | https://arxiv.org/abs/2603.19559 |