实对称矩阵对角线估计的蒙特卡罗方法
Monte Carlo Methods for Estimating the Diagonal of a Real Symmetric Matrix
摘要：
对于只能通过矩阵向量积获得的实对称矩阵，我们提出了计算对角元素的蒙特卡罗估计。我们的概率界normwise绝对和相对误差适用于蒙特卡罗估计的基础上随机Rademacher，稀疏Rademacher，规范化和非规范化高斯向量，并与有界的四阶矩向量。在我们的证明矩阵浓度不等式的新用途代表了一个系统的模型，为未来的分析。我们的界大多不依赖于矩阵的维数，目标不同的误差测量比现有的工作，并意味着估计的准确性增加与矩阵的对角优势。基于导数的全球敏感性指标的应用程序证实了这一点，合成测试矩阵的数值实验。我们建议不要在实践中使用稀疏Rademacher向量，这是许多随机草图和采样算法的基础，因为即使在大采样量下，它们也只能提供一位数的精度。
英文摘要：
For real symmetric matrices that are accessible only through matrix vector products, we present Monte Carlo estimators for computing the diagonal elements. Our probabilistic bounds for normwise absolute and relative errors apply to Monte Carlo estimators based on random Rademacher, sparse Rademacher, normalized and unnormalized Gaussian vectors, and to vectors with bounded fourth moments. The novel use of matrix concentration inequalities in our proofs represents a systematic model for future analyses. Our bounds mostly do not depend on the matrix dimension, target different error measures than existing work, and imply that the accuracy of the estimators increases with the diagonal dominance of the matrix. An application to derivative-based global sensitivity metrics corroborates this, as do numerical experiments on synthetic test matrices. We recommend against the use in practice of sparse Rademacher vectors, which are the basis for many randomized sketching and sampling algorithms, because they tend to deliver barely a digit of accuracy even under large sampling amounts.
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