用于对数行列式逼近的次优子空间构造
Suboptimal subspace construction for log-determinant approximation
AI总结：
针对大规模对称正定矩阵的对数行列式逼近，本文结合方差缩减与随机 Lanczos 求积，利用保投影代价草图和矩阵集中不等式构造次优子空间，并给出误差估计与实验验证。
摘要：
方差缩减是 Monte Carlo 模拟中的一个关键思想，而随机 Lanczos 求积法是逼近矩阵函数迹的专用方法。受二者优势启发，我们将这两种技术结合起来，以逼近大规模对称正定矩阵的对数行列式。这种方法需要回答的关键问题是，如何构造或选择合适的投影子空间，并得出具有保证的理论分析。本文应用包括保投影代价草图和矩阵集中不等式在内的若干概率方法来构造一个次优子空间。此外，我们通过推导相应的逼近误差和概率误差估计，对底层算法中设计参数的选择提供了一些见解。数值实验证明了我们方法的有效性，并说明了所推导误差界的质量。
英文摘要：
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.
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