关于不定矩阵的随机迹估计及其在行列式中的应用
On randomized trace estimates for indefinite matrices with an application to determinants
摘要：
随机迹估计是一种流行且研究良好的技术，其通过计算随机向量$X$的许多样本的$x^T Bx$的平均值来近似大规模矩阵$B$的迹。通常，$B$是对称正定的（SPD），但许多应用会产生不确定的$B$。最值得注意的是，这是对数行列式估计的情况，这是统计学习中突出的任务，例如高斯过程回归的极大似然估计。随机跟踪估计的分析，包括尾部边界，主要集中在SPD的情况下。在这项工作中，我们推导出新的随机跟踪估计适用于不定$B$与Rademacher或高斯随机向量的尾部界限。这些界限显着改善现有的结果不确定$B$，减少所需的样本数的一个因素$n$或甚至更多，其中$n$是$B$的大小。即使对于SPD矩阵，我们的工作也改进了Roosta-Khorasani和Ascher关于Rademacher向量的现有结果。这项工作还分析了随机跟踪估计与Lanczos方法的组合，用于逼近$f(A)$的轨迹。特别注意的是矩阵对数，这是需要对数行列式估计。我们改进和扩展了已有的结果，不仅涵盖Rademacher，而且高斯随机向量。
英文摘要：
Randomized trace estimation is a popular and well studied technique that approximates the trace of a large-scale matrix $B$ by computing the average of $x^T Bx$ for many samples of a random vector $X$. Often, $B$ is symmetric positive definite (SPD) but a number of applications give rise to indefinite $B$. Most notably, this is the case for log-determinant estimation, a task that features prominently in statistical learning, for instance in maximum likelihood estimation for Gaussian process regression. The analysis of randomized trace estimates, including tail bounds, has mostly focused on the SPD case. In this work, we derive new tail bounds for randomized trace estimates applied to indefinite $B$ with Rademacher or Gaussian random vectors. These bounds significantly improve existing results for indefinite $B$, reducing the the number of required samples by a factor $n$ or even more, where $n$ is the size of $B$. Even for an SPD matrix, our work improves an existing result by Roosta-Khorasani and Ascher for Rademacher vectors. This work also analyzes the combination of randomized trace estimates with the Lanczos method for approximating the trace of $f(A)$. Particular attention is paid to the matrix logarithm, which is needed for log-determinant estimation. We improve and extend an existing result, to not only cover Rademacher but also Gaussian random vectors.
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