This work considers the low-rank approximation of a matrix $A(t)$ depending on a parameter $t$ in a compact set $D \subset \mathbb{R}^d$. Application areas that give rise to such problems include computational statistics and dynamical systems. Randomized algorithms are an increasingly popular approach for performing low-rank approximation and they usually proceed by multiplying the matrix with random dimension reduction matrices (DRMs). Applying such algorithms directly to $A(t)$ would involve different, independent DRMs for every $t$, which is not only expensive but also leads to inherently non-smooth approximations. In this work, we propose to use constant DRMs, that is, $A(t)$ is multiplied with the same DRM for every $t$. The resulting parameter-dependent extensions of two popular randomized algorithms, the randomized singular value decomposition and the generalized Nystr\"{o}m method, are computationally attractive, especially when $A(t)$ admits an affine linear decomposition with respect to $t$. We perform a probabilistic analysis for both algorithms, deriving bounds on the expected value as well as failure probabilities for the $L^2$ approximation error when using Gaussian random DRMs. Both, the theoretical results and numerical experiments, show that the use of constant DRMs does not impair their effectiveness; our methods reliably return quasi-best low-rank approximations.

翻译：本文研究定义在紧致集 $D \subset \mathbb{R}^d$ 上关于参数 $t$ 的矩阵 $A(t)$ 的低秩近似问题。此类问题源于计算统计学和动力系统等应用领域。随机化算法因能高效实现低秩近似而日益流行，其通常通过将矩阵与随机降维矩阵相乘来实现。若直接对每个参数 $t$ 独立应用不同随机降维矩阵处理 $A(t)$，不仅计算成本高昂，还会导致近似结果的内在非光滑性。本文提出采用恒定随机降维矩阵策略，即对任意参数 $t$ 均使用同一随机降维矩阵与 $A(t)$ 相乘。由此衍生出的两种经典随机化算法——随机奇异值分解与广义Nyström方法——的参数扩展版本具有显著计算优势，尤其当 $A(t)$ 对参数 $t$ 具有仿射线性分解结构时。我们对两种算法进行概率分析，针对高斯随机降维矩阵情形，导出了期望误差界及$L^2$近似误差的失败概率上界。理论分析与数值实验均表明，采用恒定随机降维矩阵不影响算法有效性；所提方法可稳定提供准最优的低秩近似结果。