A low-rank approximation of a parameter-dependent matrix $A(t)$ is an important task in the computational sciences appearing for example in dynamical systems and compression of a series of images. In this work, we introduce AdaCUR, an efficient algorithm for computing a low-rank approximation of parameter-dependent matrices via CUR decomposition. The key idea for this algorithm is that for nearby parameter values, the column and row indices for the CUR decomposition can often be reused. AdaCUR is rank-adaptive, certifiable and has complexity that compares favourably against existing methods. A faster algorithm which we call FastAdaCUR that prioritizes speed over accuracy is also given, which is rank-adaptive and has complexity which is at most linear in the number of rows or columns, but without certification.

翻译：参数依赖矩阵$A(t)$的低秩逼近是计算科学中的一项重要任务，例如在动力系统和图像序列压缩中均有应用。本文提出AdaCUR算法，一种通过CUR分解计算参数依赖矩阵低秩逼近的高效方法。该算法的核心思想在于：对于相邻参数值，CUR分解的列索引和行索引通常可重复使用。AdaCUR算法具有秩自适应性、可验证性，且计算复杂度优于现有方法。本文同时提出更快速的FastAdaCUR算法，该算法以速度为优先目标而适当降低精度要求，具有秩自适应性且计算复杂度至多与矩阵行数或列数呈线性关系，但不具备可验证性。