Variance reduction is a crucial idea for Monte Carlo simulation and the stochastic Lanczos quadrature method is a dedicated method to approximate the trace of a matrix function. Inspired by their advantages, we combine these two techniques to approximate the log-determinant of large-scale symmetric positive definite matrices. Key questions to be answered for such a method are how to construct or choose an appropriate projection subspace and derive guaranteed theoretical analysis. This paper applies some probabilistic approaches including the projection-cost-preserving sketch and matrix concentration inequalities to construct a suboptimal subspace. Furthermore, we provide some insights on choosing design parameters in the underlying algorithm by deriving corresponding approximation error and probabilistic error estimations. Numerical experiments demonstrate our method's effectiveness and illustrate the quality of the derived error bounds.

翻译：方差缩减是蒙特卡洛模拟中的关键思想，而随机Lanczos求积法是一种专门用于近似矩阵函数迹的方法。受其优势启发，我们将这两种技术相结合，以近似大规模对称正定矩阵的对数行列式。此类方法需要解决的关键问题是如何构造或选择合适的投影子空间，并推导出可靠的理论分析。本文采用包括投影成本保持草图与矩阵浓度不等式在内的概率方法来构造次优子空间。此外，通过推导相应的近似误差与概率误差估计，我们为底层算法中设计参数的选择提供了见解。数值实验证明了我们方法的有效性，并展示了所推导误差界线的质量。