This paper examines the approximation of log-determinant for large-scale symmetric positive definite matrices. Inspired by the variance reduction technique, we split the approximation of $\log\det(A)$ into two parts. The first to compute is the trace of the projection of $\log(A)$ onto a suboptimal subspace, while the second is the trace of the projection on the corresponding orthogonal complementary space. For these two approximations, the stochastic Lanczos quadrature method is used. Furthermore, in the construction of the suboptimal subspace, we utilize a projection-cost-preserving sketch to bound the size of the Gaussian random matrix and the dimension of the suboptimal subspace. We provide a rigorous error analysis for our proposed method and explicit lower bounds for its design parameters, offering guidance for practitioners. We conduct numerical experiments to demonstrate our method's effectiveness and illustrate the quality of the derived bounds.

翻译：本文研究大规模对称正定矩阵的对数行列式逼近问题。受方差缩减技术启发，我们将对数行列式 $\log\det(A)$ 的逼近分解为两部分：首先计算 $\log(A)$ 在次优子空间上投影的迹，其次计算其在相应正交补空间上投影的迹。对这两个逼近采用随机Lanczos求积方法。此外，在构造次优子空间时，我们利用保投影代价草图来控制高斯随机矩阵的规模及次优子空间的维度。我们为所提方法提供了严格的误差分析，并给出了设计参数的显式下界，为实践者提供指导。通过数值实验验证了该方法的有效性，并展示了所推演界的质量。