The implicit trace estimation problem asks for an approximation of the trace of a square matrix, accessed via matrix-vector products (matvecs). This paper designs new randomized algorithms, XTrace and XNysTrace, for the trace estimation problem by exploiting both variance reduction and the exchangeability principle. For a fixed budget of matvecs, numerical experiments show that the new methods can achieve errors that are orders of magnitude smaller than existing algorithms, such as the Girard-Hutchinson estimator or the Hutch++ estimator. A theoretical analysis confirms the benefits by offering a precise description of the performance of these algorithms as a function of the spectrum of the input matrix. The paper also develops an exchangeable estimator, XDiag, for approximating the diagonal of a square matrix using matvecs.

翻译：摘要: 隐式迹估计问题要求通过矩阵-向量乘积（matvecs）访问一个方阵，并近似其迹。本文通过利用方差缩减和可交换性原理，设计了新的随机算法XTrace和XNysTrace用于迹估计问题。在固定matvecs预算下，数值实验表明，新方法能够实现比现有算法（如Girard-Hutchinson估计器或Hutch++估计器）小数个数量级的误差。理论分析通过精确描述这些算法性能随输入矩阵谱的变化，证实了其优势。本文还开发了一个可交换估计器XDiag，用于通过matvecs近似方阵的对角线。