发表机构
UCLA; Michigan State; Univ. British Columbia（加州大学洛杉矶分校; 密歇根州立大学; 不列颠哥伦比亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过外代数视角重新诠释自适应交叉逼近的残差公式，分离影响性能的几何因素，并据此提出加权质量主元规则，在实验中优于贪婪主元选择。
AI 中文摘要
自适应交叉逼近（ACA）通过从矩阵中选取的行和列构造低秩CUR逼近，这使得它在单个元素查询成本低廉但形成或反复乘以完整矩阵不可行时具有吸引力。然而，其实际性能无法由现有的最坏情况界很好地解释，这些界可能呈指数级大，且常常无法预测贪婪最大元素主元选择何时表现不佳。通过将矩阵视为交叉格兰姆矩阵，我们借助外代数的视角重新诠释了CUR残差的经典行列式公式。在该表示中，每个残差项是两个$(k+1)$-刃之间的加权内积除以主元块的归一化体积。该公式分离了三种效应：奇异向量行的加权幅度、它们的角度几何以及所选主元的条件数。由此，我们恢复了一个经典的$\sigma_{k+1}$型估计，刻画了单个主元何时消除或近乎消除额外的行和列，获得了半正定矩阵的拟最优性估计，并通过一个基于角度的显式公式界定了秩一更新。我们利用该框架解释了贪婪ACA在非对称螺旋核示例上的失败，并为径向基伽辽金刚度矩阵提出了一种几何感知的加权质量主元选择规则。在报告的实验中，加权质量主元选择相对于贪婪主元选择和随机主元Cholesky，通常能降低Frobenius残差，同时在并行硬件上保持相当的运行时间。
英文摘要
Adaptive cross approximation (ACA) constructs low-rank CUR approximations from selected rows and columns of a matrix, making it attractive when individual entries are inexpensive to query but forming or repeatedly multiplying by the full matrix is not. Its practical performance, however, is not well explained by existing worst-case bounds, which can be exponentially large and often fail to predict when greedy largest-entry pivoting performs poorly. By viewing the matrix as a cross-gram matrix, we reinterpret the classical determinantal formula for the CUR residual through the lens of exterior algebra. In this representation, each residual entry is a weighted inner product between two $(k+1)$-blades divided by the normalized volume of the pivot block. The formula separates three effects: the weighted magnitudes of singular-vector rows, their angular geometry, and the conditioning of the selected pivots. From this we recover a classical $σ_{k+1}$-type estimate, characterize when one pivot annihilates or nearly annihilates additional rows and columns, obtain a quasi-optimality estimate for positive-semidefinite matrices, and bound the rank-one update through an explicit angle-based formula. We use this framework to explain the failure of greedy ACA on an asymmetric spiral-kernel example and to motivate a geometry-aware weighted-mass pivoting rule for radial-basis Galerkin stiffness matrices. In the reported experiments, weighted-mass pivoting often reduces the Frobenius residual relative to greedy pivoting and randomly pivoted Cholesky while retaining comparable runtime on parallel hardware.
Comments28 pages, 15 figures
