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arXiv · 2208.03268

A Tight Analysis of Hutchinson's Diagonal Estimator

Prathamesh Dharangutte·Christopher Musco
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Abstract

Let $\mathbf{A}\in \mathbb{R}^{n\times n}$ be a matrix with diagonal $\text{diag}(\mathbf{A})$ and let $\bar{\mathbf{A}}$ be $\mathbf{A}$ with its diagonal set to all zeros. We show that Hutchinson's estimator run for $m$ iterations returns a diagonal estimate $\tilde{d}\in \mathbb{R}^n$ such that with probability $(1-\delta)$, $$\|\tilde{d} - \text{diag}(\mathbf{A})\|_2 \leq c\sqrt{\frac{\log(2/\delta)}{m}}\|\bar{\mathbf{A}}\|_F,$$ 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 $n$.

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Prathamesh Dharangutte, Christopher Musco. 2022-08-05. A Tight Analysis of Hutchinson's Diagonal Estimator. https://arxiv.org/abs/2208.03268

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