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：本文考虑在紧集 $D \subset \mathbb{R}^d$ 中依赖于参数 $t$ 的矩阵 $A(t)$ 的低秩逼近问题。此类问题的应用领域包括计算统计学和动力系统。随机化算法是执行低秩逼近日益流行的策略，通常通过将矩阵与随机降维矩阵（DRMs）相乘来实现。若将此类算法直接应用于 $A(t)$，需对每个 $t$ 采用不同且独立的DRM，这不仅计算昂贵，还会导致本质上的非光滑逼近。本研究提出使用恒定DRM，即对所有 $t$ 均采用相同DRM与 $A(t)$ 相乘。两种流行随机化算法——随机奇异值分解和广义Nyström方法——的参数依赖扩展版本因此具有计算优势，尤其当 $A(t)$ 关于 $t$ 满足仿射线性分解时。我们对两种算法进行概率分析，在采用高斯随机DRM的条件下，推导近似误差的期望界及失败概率。理论结果与数值实验均表明，使用恒定DRM不会损害算法有效性；我们的方法能够可靠地返回准最优的低秩逼近。