We propose a general error analysis related to the low-rank approximation of a given real matrix in both the spectral and Frobenius norms. First, we derive deterministic error bounds that hold with some minimal assumptions. Second, we derive error bounds in expectation in the non-standard Gaussian case, assuming a non-trivial mean and a general covariance matrix for the random matrix variable. The proposed analysis generalizes and improves the error bounds for spectral and Frobenius norms proposed by Halko, Martinsson and Tropp. Third, we consider the Randomized Singular Value Decomposition and specialize our error bounds in expectation in this setting. Numerical experiments on an instructional synthetic test case demonstrate the tightness of the new error bounds.

翻译：我们建议对光谱和弗罗贝尼乌斯规范中某个真实矩阵的低位近似值进行一般性错误分析。 首先,我们得出带有某些最低假设的确定性错误界限。 其次,我们得出非标准高斯案例的预期误差界限,假设随机矩阵变量的非三重平均值和一般共变矩阵。 拟议的分析对Halko、Martinsson和Tropp提出的光谱和弗罗贝尼乌斯规范的误差界限进行了归纳和改进。 第三,我们考虑了随机 Singular值分解,并专门确定了我们在此背景下的预期误差界限。 在一个指示性合成测试案例上的数值实验显示了新误差界限的严密性。