论文标题
随机低级近似方法的一般错误分析
A general error analysis for randomized low-rank approximation methods
论文作者
论文摘要
我们提出了与光谱和Frobenius规范中给定实际基质的低级别近似相关的一般误差分析。首先，我们得出确定性误差界限，这些误差界限具有一些最小的假设。其次，我们在非标准高斯情况下的预期中得出了误差界限，假设对随机矩阵变量的非平凡均值和一般协方差矩阵。提出的分析概括并改善了Halko，Martinsson和Tropp提出的光谱和Frobenius规范的误差界限。第三，我们考虑随机的单数值分解，并在这种情况下在预期中专门提高我们的误差界限。在教学合成测试案例上进行的数值实验证明了新误差边界的紧密度。
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.
