Certain classes of CUR algorithms, also referred to as cross or pseudoskeleton algorithms, are widely used for low-rank matrix approximation when direct access to all matrix entries is costly. Their key advantage lies in constructing a rank-r approximation by sampling only r columns and r rows of the target matrix. This property makes them particularly attractive for reduced-order modeling of nonlinear matrix differential equations, where nonlinear operations on low-rank matrices can otherwise produce high-rank or even full-rank intermediates that must subsequently be truncated to rank $r$. CUR cross algorithms bypass the intermediate step and directly form the rank-$r$ matrix. However, standard cross algorithms may suffer from loss of accuracy in some settings, limiting their robustness and broad applicability. In this work, we propose a cross oversampling algorithm that augments the intersection with additional sampled columns and rows. We provide an error analysis demonstrating that the proposed oversampling improves robustness. We also present an algorithm that adaptively selects the number of oversampling entries based on efficiently computable indicators. We demonstrate the performance of the proposed CUR algorithm for time integration of several nonlinear stochastic PDEs on low-rank matrix manifolds.

翻译：CUR算法（亦称交叉或伪骨架算法）的某些类别被广泛用于低秩矩阵逼近，当直接访问所有矩阵元素代价高昂时。其关键优势在于仅通过采样目标矩阵的r列和r行即可构建秩r逼近。这一特性使其在非线性矩阵微分方程的降阶建模中尤为引人注目，因为对低秩矩阵的非线性运算可能产生高秩甚至满秩的中间结果，随后必须截断至秩$r$。CUR交叉算法绕过了中间步骤，直接形成秩-$r$矩阵。然而，标准交叉算法在某些场景下可能面临精度损失问题，限制了其鲁棒性与广泛适用性。本研究提出一种交叉过采样算法，通过额外采样列与行来增强交集。我们提供的误差分析表明，所提出的过采样策略能提升算法鲁棒性。同时，我们提出一种基于高效可计算指标自适应选择过采样条目数量的算法。我们通过多个非线性随机偏微分方程在低秩矩阵流形上的时间积分案例，展示了所提出CUR算法的性能表现。