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 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：本文研究定义在紧集$D \subset \mathbb{R}^d$上、依赖于参数$t$的矩阵$A(t)$的低秩近似问题。此类问题在计算统计学与动力系统等领域具有重要应用。随机算法因其在低秩近似中的高效性而日益普及，其通常通过将矩阵与随机降维矩阵（DRM）相乘实现。若直接将该方法应用于$A(t)$，需对每个参数$t$使用独立不同的DRM，这不仅导致计算成本高昂，且会产生本质上的非光滑近似。本文提出采用恒定DRM策略，即对所有参数$t$使用同一DRM乘$A(t)$。由此衍生出两种经典随机算法（随机奇异值分解与广义Nyström方法）的参数依赖扩展，在$A(t)$关于$t$具有仿射线性分解时展现显著计算优势。我们针对两种算法进行概率分析，在使用高斯随机DRM的条件下，推导了近似误差的期望界与失效概率界。理论结果与数值实验均表明：采用恒定DRM不会降低算法效能，所提方法能可靠地恢复拟最优低秩近似。