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 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 (Found Comput Math, 15(5):1187–1212, 2015) for Rademacher vectors. This work also analyzes the combination of randomized trace estimates with the Lanczos method for approximating the trace of f(B). 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.

随机轨迹估计是一种流行且经过充分研究的技术，它通过计算随机向量X 的许多样本的\(x^T Bx\)的平均值来近似大规模矩阵B的轨迹。通常情况下，乙是对称正定（SPD），但许多应用产生不确定乙。最值得注意的是，对数行列式估计就是这种情况，这是一项在统计学习中具有突出特点的任务，例如在高斯过程回归的最大似然估计中。对包括尾部边界在内的随机迹估计的分析主要集中在 SPD 案例上。在这项工作中，我们为应用于不定B 的随机迹估计得出了新的尾部边界与 Rademacher 或高斯随机向量。这些界限显着改善了不定B 的现有结果，将所需样本的数量减少了一个因子n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（Found Comput Math, 15(5):1187–1212, 2015）对 Rademacher 向量的现有结果。这项工作还分析了随机迹估计与 Lanczos 方法的组合，用于逼近f ( B）。特别注意矩阵对数，这是对数行列式估计所需的。我们改进和扩展了现有结果，不仅涵盖了 Rademacher，还涵盖了高斯随机向量。