Randomized algorithms have proven to perform well on a large class of numerical linear algebra problems. Their theoretical analysis is critical to provide guarantees on their behaviour, and in this sense, the stochastic analysis of the randomized low-rank approximation error plays a central role. Indeed, several randomized methods for the approximation of dominant eigen- or singular modes can be rewritten as low-rank approximation methods. However, despite the large variety of algorithms, the existing theoretical frameworks for their analysis rely on a specific structure for the covariance matrix that is not adapted to all the algorithms. We propose a unified framework for the stochastic analysis of the low-rank approximation error in Frobenius norm for centered and non-standard Gaussian matrices. Under minimal assumptions on the covariance matrix, we derive accurate bounds both in expectation and probability. Our bounds have clear interpretations that enable us to derive properties and motivate practical choices for the covariance matrix resulting in efficient low-rank approximation algorithms. The most commonly used bounds in the literature have been demonstrated as a specific instance of the bounds proposed here, with the additional contribution of being tighter. Numerical experiments related to data assimilation further illustrate that exploiting the problem structure to select the covariance matrix improves the performance as suggested by our bounds.

翻译：随机算法已被证明在大量数值线性代数问题上表现优异。其理论分析对于提供算法行为保证至关重要，就此而言，随机低秩近似误差的随机分析起着核心作用。事实上，多种用于近似主特征模态或奇异模态的随机方法均可重写为低秩近似方法。然而，尽管算法种类繁多，现有理论分析框架所依赖的协方差矩阵特定结构并不适用于所有算法。我们提出了一个统一框架，用于分析中心化和非标准高斯矩阵在Frobenius范数下的低秩近似误差的随机性。在协方差矩阵的最小假设条件下，我们推导出了精确的期望界和概率界。所得界限具有清晰的物理解释，使我们能够推导出协方差矩阵的性质并启发实际选择，从而设计出高效的低秩近似算法。文献中最常用的误差界已被证明是本文所提界限的特例，且本文界限具有更紧致的附加优势。与数据同化相关的数值实验进一步表明，利用问题结构选择协方差矩阵可提升算法性能，这与我们推导的误差界所揭示的规律一致。