SIAM Journal on Matrix Analysis and Applications, Volume 42, Issue 1, Page 202-223, January 2021.
A few matrix-vector multiplications with random vectors are often sufficient to obtain reasonably good estimates for the norm of a general matrix or the trace of a symmetric positive semi-definite matrix. Several such probabilistic estimators have been proposed and analyzed for standard Gaussian and Rademacher random vectors. In this work, we consider the use of rank-one random vectors, that is, Kronecker products of (smaller) Gaussian or Rademacher vectors. It is not only cheaper to sample such vectors but it can sometimes also be much cheaper to multiply a matrix with a rank-one vector instead of a general vector. In this work, theoretical and numerical evidence is given that the use of rank-one instead of unstructured random vectors still leads to good estimates. In particular, it is shown that our rank-one estimators multiplied with a modest constant constitute, with high probability, upper bounds of the quantity of interest. Partial results are provided for the case of lower bounds. The application of our techniques to condition number estimation for matrix functions is illustrated.

中文翻译：

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随机秩一向量的范数和迹估计

SIAM Journal on Matrix Analysis and Applications，第 42 卷，第 1 期，第 202-223 页，2021 年 1 月。
一些矩阵向量与随机向量的乘法通常足以获得对一般矩阵的范数或对称半正定矩阵的迹的相当好的估计。已经针对标准高斯和 Rademacher 随机向量提出并分析了几个这样的概率估计器。在这项工作中，我们考虑使用秩一随机向量，即（较小的）高斯或 Rademacher 向量的克罗内克积。对此类向量进行采样不仅成本更低，而且有时将矩阵与秩为一的向量相乘而不是一般向量也更便宜。在这项工作中，给出了理论和数值证据，表明使用秩一而不是非结构化随机向量仍然可以得到良好的估计。特别是，结果表明，我们的一级估计量乘以一个适度的常数，以高概率构成感兴趣数量的上限。为下限的情况提供了部分结果。说明了我们的技术在矩阵函数条件数估计中的应用。