标题： 随机迹和对角线估计器用于张量 显示英文标题

标题： Stochastic Trace and Diagonal Estimator for Tensors

摘要： 我们考虑在仅能通过张量-向量乘法访问张量的框架下，估计N阶张量（其中$N \geq 2$）的迹和对角线元素的问题。目标是通过最小化张量-向量查询次数来估计张量的对角线元素和迹。Hutchinson的开创性工作及其由Bekas等人扩展的版本分别使用矩阵-向量查询给出了给定矩阵的迹和对角线元素的无偏估计。然而，据我们所知，目前尚无关于使用张量-向量查询估计高阶张量的迹和对角线元素的类似结果。本文填补了这一空白，并在此模型下提出了张量的迹和对角线元素的无偏估计量。我们提出的方法可以看作是Hutchinson和Bekas等人估计量的推广，在N = 2时退化为他们的估计量。我们提供了对我们提议的严格理论分析，并通过支持性的模拟实验加以补充。 显示英文摘要

摘要： We consider the problem of estimating the trace and diagonal entries of an N-order tensor (where $N \geq 2$) under the framework where the tensor can only be accessed through tensor-vector multiplication. The aim is to estimate the tensor's diagonal entries and trace by minimizing the number of tensor-vector queries. The seminal work of Hutchinson and its extended version due to Bekas et al. give unbiased estimates of the trace and diagonal elements of a given matrix, respectively, using matrix-vector queries. However, to the best of our knowledge, no analogous results are known for estimating the trace and diagonal entries of higher-order tensors using tensor-vector queries. This paper addresses this gap and presents unbiased estimators for the trace and diagonal entries of a tensor under this model. Our proposed methods can be seen as generalizations of Hutchinson's and Bekas et al.'s estimators and reduce to their estimators when N = 2. We provide a rigorous theoretical analysis of our proposals and complement it with supporting simulations.