正迹类算子的随机迹估计
Stochastic trace estimation for positive trace-class operators
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中文总结 AI 辅助
该研究为可分希尔伯特空间上的正迹类算子构建了无穷维Girard-Hutchinson与Hutch++迹估计器，证明其误差界与样本复杂度，提出实用截断实现并验证其在多类算子问题中精度可比现有算法且计算成本更低。
中文摘要 AI 辅助
隐式迹估计旨在近似只能通过矩阵-向量或算子-向量乘积访问的矩阵或线性算子的迹。在矩阵场景中，Girard-Hutchinson估计器通常需要$\u27e8\bigtriangleup\bigotimes\u27e9$次乘积以达到精度$\u27e8\bigtriangleup\bigotimes\u27e9$，而方差降低的Hutch++算法将半正定矩阵的样本复杂度降至$\u27e8\bigtriangleup\bigotimes\u27e9$。我们为可分希尔伯特空间上的正迹类算子开发了这些估计器的无穷维类似物。理想化估计器使用协方差由目标算子确定的高斯随机元，得到无偏的算子版本Girard-Hutchinson和Hutch++。我们证明了与有限维矩阵结果类似的高概率误差界；特别地，理想化infHutch++达到$\u27e8\bigtriangleup\bigotimes\u27e9$的样本复杂度。为实现实际计算，我们提出了将随机样本限制在有限维子空间的截断实现；对于固定样本预算，我们证明当截断维数趋于无穷时，截断infHutch++依分布收敛于其理想化对应版本。针对积分算子、态密度近似以及径向Dirac算子谱滤波的数值实验表明，这些截断估计器能达到与Zvonek、Horning和Townsend提出的ContHutch++算法相当的精度，同时在chebfun中使用更低次数的函数表示和更小的内部离散化。
英文摘要
Implicit trace estimation aims to approximate the trace of a matrix or linear operator accessible only through matrix-vector or operator-vector products. In the matrix setting, the Girard-Hutchinson estimator typically requires $\mathcal{O}(\varepsilon^{-2})$ products to achieve accuracy $\varepsilon$, while the variance-reduced Hutch++ algorithm reduces this sample complexity to $\mathcal{O}(\varepsilon^{-1})$ for positive semidefinite matrices. We develop infinite-dimensional analogues of these estimators for positive trace-class operators on separable Hilbert spaces. The idealized estimators use Gaussian random elements whose covariance is determined by the target operator, leading to unbiased operator versions of Girard-Hutchinson and Hutch++. We prove high-probability error bounds analogous to the finite-dimensional matrix results; in particular, idealized infHutch++ achieves $\mathcal{O}(\varepsilon^{-1})$ sample complexity. For practical computation, we introduce truncated implementations that restrict the random samples to finite-dimensional subspaces; for fixed sample budget, we show that truncated infHutch++ converges in distribution to its idealized counterpart as the truncation dimension tends to infinity. Numerical experiments with integral operators, density-of-states approximations, and spectral filtering for a radial Dirac operator show that these truncated estimators can achieve accuracy comparable to the ContHutch++ algorithm by Zvonek, Horning & Townsend while using lower-degree function representations and smaller internal discretizations in chebfun.
