A Tight Analysis of Hutchinson's Diagonal Estimator

08/05/2022

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by   Prathamesh Dharangutte, et al.

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Let 𝐀∈ℝ^n× n be a matrix with diagonal diag(𝐀) and let 𝐀̅ be 𝐀 with its diagonal set to all zeros. We show that Hutchinson's estimator run for m iterations returns a diagonal estimate d̃∈ℝ^n such that with probability (1-δ), d̃ - diag(𝐀)_2 ≤ c√(log(2/δ)/m)𝐀̅_F, where c is a fixed constant independent of all other parameters. This results improves on a recent result of [Baston and Nakatsukasa, 2022] by a log(n) factor, yielding a bound that is independent of the matrix dimension n.