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 general 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良好。对它们进行理论分析对于保证其行为至关重要，而在此意义上，随机低秩近似误差的随机分析发挥着核心作用。事实上，几种用于近似主导特征或奇异模态的随机方法都可以重写为低秩近似方法。然而，尽管算法种类繁多，现有用于分析的理论框架都依赖于协方差矩阵的特定结构，而这并不适用于所有算法。我们提出了一个通用框架，用于对中心化和非标准高斯矩阵的弗罗贝尼乌斯范数下的低秩近似误差进行随机分析。在协方差矩阵的最小假设下，我们推导了期望和概率意义上的精确界。我们的界限具有清晰的解释，使我们能够推导出性质并激发协方差矩阵的实际选择，从而产生高效的低秩近似算法。文献中最常用的界限已被证明是我们所提出的界限的一个特例，且我们提出的界限更为紧凑。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高性能，这与我们的界限所暗示的结果一致。