In this paper, we investigate diagonal estimation for large or implicit matrices, aiming to develop a novel and efficient stochastic algorithm that incorporates adaptive parameter selection. We explore the influence of different eigenvalue distributions on diagonal estimation and analyze the necessity of introducing the projection method and adaptive parameter optimization into the stochastic diagonal estimator. Based on this analysis, we derive a lower bound on the number of random query vectors needed to satisfy a given probabilistic error bound, which forms the foundation of our adaptive stochastic diagonal estimation algorithm. Finally, numerical experiments demonstrate the effectiveness of the proposed estimator for various matrix types, showcasing its efficiency and stability compared to other existing stochastic diagonal estimation methods.

翻译：本文研究大规模或隐式矩阵的对角估计问题，旨在提出一种融合自适应参数选择的新型高效随机算法。我们探究了不同特征值分布对对角估计的影响，分析了在随机对角估计器中引入投影方法与自适应参数优化的必要性。基于此分析，我们推导出满足给定概率误差界所需随机查询向量数量的下界，这构成了自适应随机对角估计算法的理论基础。最后，数值实验证明了所提估计器对多种矩阵类型的有效性，相较于现有随机对角估计方法，其展现出更优的效率和稳定性。