随机Lanczos求积用于线性代数中的计算不确定性

中文摘要 AI 辅助

将矩阵函数应用于向量是大型科学计算中的常见操作，例如贝叶斯推断。当矩阵对称正定且函数为其逆时，Lanczos算法仅通过矩阵-向量乘积即可给出无矩阵近似。在固定步数后停止，会使近似保持在已探索的Krylov子空间上，但对其正交补空间不赋予任何内容，而该补空间包含最小的特征值，因此对逆的贡献最大。这种截断通常被视为静默的数值误差，而非可测量的不确定性。我们证明，单次Lanczos运行既能产生逆的无矩阵近似，又能产生关于精确结果的高斯分布，其协方差衡量了截断所丢弃的内容，并随着步数增加而集中。该协方差以两部分恢复补空间：截断边界处的秩一耦合，由Lanczos残差精确给出；以及剩余主体上的各向同性项，其大小通过投影随机Lanczos求积估计，在所有探针间重用单个Krylov基。我们证明了一致性，偏差随求积深度指数衰减，方差达到蒙特卡洛速率。

英文摘要

Applying a matrix function to a vector is a common operation in large-scale scientific computing, such as Bayesian inference. When the matrix is symmetric positive definite and the function is its inverse, the Lanczos algorithm gives a matrix-free approximation from matrix-vector products alone. Stopping after a fixed number of steps keeps the approximation on the explored Krylov subspace but assigns nothing to its orthogonal complement, which holds the smallest eigenvalues and therefore the largest contributions to the inverse. This truncation is usually treated as a silent numerical error rather than as measurable uncertainty. We show that a single Lanczos run yields both a matrix-free approximation of the inverse and a Gaussian distribution over the exact result, whose covariance measures what the truncation discarded and concentrates as the number of steps grows. The covariance restores the complement in two parts: a rank-one coupling at the truncation boundary, given exactly by the Lanczos residual, and an isotropic term on the remaining bulk, whose size we estimate by Projected Stochastic Lanczos Quadrature, reusing one Krylov basis across all probes. We prove consistency, with bias decaying exponentially in the quadrature depth and variance at the Monte Carlo rate

发表机构

• University of Dodoma（多多马大学）
• African Institute for Mathematical Sciences Research and Innovation Centre（非洲数学科学研究与创新中心）
• University of Tübingen（蒂宾根大学）
• Tübingen AI Center（蒂宾根人工智能中心）

机构由 AI 辅助整理，请以论文原文为准。