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I'm working on a paper that requires bounding

$$\Pr\left[|\vec x^\top Q \vec y| >= t\right]$$ where $Q$ is a matrix (happens to be symmetric) and $\vec x,\vec y$ are iid real mean-zero subgaussian random vectors. As I understand it, the Hanson-Wright inequality bounds this as $$2\exp\left(-c \cdot\min\left(\frac{t^2}{k^4 ||Q||_F^2}, \frac{t}{k^2||Q||_2}\right)\right)$$ where $k$ is the subgaussian parameter and $c$ is an absolute constant.

But I'm trying to derive an actual numerical bound. Is there a paper which gives an actual value of $c$, preferably a tight one? Possibly it would be different for the two sides of the min(). Thanks.

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    $\begingroup$ For $Q$ sparse, lemma 3.1 in [1] shows $c=\frac{1}{4e}$ works for the diagonal terms. The off diagonal terms are a bit messy but there are some constants appearing later in the same paper. [1]: www-personal.umich.edu/~shuhengz/research/sparseHW.pdf $\endgroup$
    – Josiah Park
    Commented Nov 29, 2018 at 1:23
  • $\begingroup$ @JosiahPark the link above doesn't work anymore. Here is one that does: arxiv.org/pdf/1510.05517.pdf If in the future this one doesn't work, for reference the title is "Sparse Hanson-Wright inequalities for subgaussian quadratic forms" by Shuheng Zhou. $\endgroup$
    – LSK21
    Commented Jul 26, 2023 at 9:23

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