Estimation of matrix trace using machine learning

TL;DR

This paper introduces a machine learning-based method for estimating the trace of a matrix using fewer probing vectors, achieving accuracy comparable to traditional stochastic estimators with many more random vectors.

Contribution

The paper proposes a novel trace estimator that replaces random noise vectors with machine learning-determined probing vectors, reducing computational cost significantly.

Findings

Achieves similar accuracy with only about 10 probing vectors as traditional methods do with 10,000 random vectors.

Provides an unbiased estimator for the expectation of functions of traces.

Discusses bias correction and estimator quality evaluation.

Abstract

We present a new trace estimator of the matrix whose explicit form is not given but its matrix multiplication to a vector is available. The form of the estimator is similar to the Hutchison stochastic trace estimator, but instead of the random noise vectors in Hutchison estimator, we use small number of probing vectors determined by machine learning. Evaluation of the quality of estimates and bias correction are discussed. An unbiased estimator is proposed for the calculation of the expectation value of a function of traces. In the numerical experiments with random matrices, it is shown that the precision of trace estimates with $\mathcal{O}(10)$ probing vectors determined by the machine learning is similar to that with $\mathcal{O}(10000)$ random noise vectors.