Stochastic Trace and Diagonal Estimator for Tensors

TL;DR

This paper introduces unbiased estimators for the trace and diagonal entries of N-order tensors using tensor-vector queries, extending matrix-based methods to higher dimensions with theoretical analysis and simulations.

Contribution

It generalizes Hutchinson's and Bekas et al.'s estimators to tensors, providing the first unbiased methods for trace and diagonal estimation in this setting.

Findings

Proposed estimators are unbiased for tensor trace and diagonal entries.

Theoretical analysis confirms estimator efficiency and accuracy.

Simulations demonstrate practical effectiveness of the methods.

Abstract

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.