Randomized Trace Estimation

Tyler Chen

Abstract: Suppose $A$ is a square matrix and $x$ is a random vector with mean zero and identity covariance. Then, $x^TAx$ forms an unbiased estimator for the trace of $A$. Since $x^TAx$ can be computed using only matrix-vector products with $A$, this simple observation allows us to estimate the trace of matrices for which an explicit representation is unknown or intractable to write down explicitly. Estimators of this form have proven to be a powerful algorithmic tool since their emergence in the late 1980s and early 1990s, and remain an important tool for domain problems in fields such as statistics, uncertainty quantification, machine learning, and computational quantum physics, etc. Recent years have seen a rapid growth in the development of randomized algorithms for trace estimation. Current state of the art algorithms are often based on the estimator described above, but offer provable theoretical improvements which are easily observed in practice. This talk provides an overview of recent developments in stochastic trace estimation techniques, with a particular focus on methods for estimating the trace of matrix functions such as the matrix exponential or matrix inverse.

Submission Type: Abstract