在 $\widetilde{O}(mn \log \frac{\kappa}{\epsilon})$ 比特运算内求解线性系统

中文摘要 AI 辅助

我们给出一个确定性算法，用于求解非奇异线性系统 $Ax=b$，其中 $A\in\mathbb{R}^{n\times n}$ 有 $m$ 个非零元素，条件数为 $\kappa$，对任意相对残差容限 $0<\epsilon\le1/2$，在输入每个条目具有对数数量级比特数的情况下，使用 $\widetilde{O}(mn\log(\kappa/\epsilon))$ 比特运算。对于稀疏、多项式条件数且 $m=\widetilde{O}(n)$ 的系统，这给出了对于任意逆多项式精度的一个 $\widetilde{O}(n^2)$ 算法，改进了 Peng 和 Vempala 的算法（经 Nie 改进），在该情形下其运行时间约为 $O(n^{2.2707})$（使用当前最佳矩阵乘法指数），并在很大程度上缩小了精确算术中共轭梯度法的理想性能与有限精度计算可实现运行时间之间持续超过 70 年的差距。该算法出奇地简单。对于整数输入，我们将 Dixon 提升算法应用于扰动系统 $(A+I/R)x=b$，其中 $R$ 为合适的整数。缩放后，该系统的矩阵为 $RA+I\equiv I\pmod R$，因此其模逆是平凡的，每一步提升只需对 $O(\log R)$ 比特数进行稀疏矩阵-向量乘积（涉及 $A$）。快速有理重构随后恢复扰动系统的精确解，该解是原系统的一个 $\epsilon$-精确解。归一化和舍入将结果扩展到定点与浮点输入，浮点输出使用短整数有效数字和公共编码指数表示。一个可计算的证书消除了对 $\kappa$ 先验知识的需求。

英文摘要

We give a deterministic algorithm that solves a nonsingular linear system $Ax=b$, where $A\in\mathbb{R}^{n\times n}$ has $m$ nonzero entries and condition number $κ$, to any relative residual tolerance $0<ε\le1/2$ using $\widetilde{O(}mn\log(κ/ε))$ bit operations for inputs with logarithmically many bits per entry. For sparse, polynomially conditioned systems with $m=\widetilde{O}(n)$, this gives an $\widetilde{O}(n^2)$ algorithm for any inverse-polynomial accuracy, improving on the algorithm of Peng and Vempala, as sharpened by Nie, whose running time in this regime is approximately $O(n^{2.2707})$ with the best current matrix multiplication exponent, and largely closing a gap between the idealized performance of the conjugate gradient method in exact arithmetic and the running time achievable with finite-precision computation that has persisted for over 70 years. The algorithm is surprisingly simple. For integer inputs, we apply Dixon's lifting algorithm to the perturbed system $(A+I/R)x=b$ for a suitable integer $R$. After scaling, the matrix of this system is $RA+I\equiv I\pmod R$, so its modular inverse is trivial, and each lifting step needs only a sparse matrix-vector product with $A$ on $O(\log R)$-bit numbers. Fast rational reconstruction then recovers the exact solution of the perturbed system, which is an $ε$-accurate solution of the original one. Normalization and rounding extend the result to fixed-point and floating-point inputs, with floating-point outputs represented using short integer significands and a common encoded exponent. A computable certificate removes the need for prior knowledge of $κ$.

发表机构

• MIT（麻省理工学院）

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