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