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I have two independent random gaussian matrix $A$ and $B \in R^{d\times n}$, and i want to compute an upper bound of the probability that

$$Pr(\left\| (A-B)(A+B)^T\right\| \leq a)$$

One method might be considering that since the two matrices are gaussian, we shall have that $(A-B)$ and $(A+B)$ are independent gaussian matrices, denoted as $C$ and $D$.

We would then consider to compute the probability

$$Pr(\left\| CD^T\right\| \leq a)$$

we shall note that $CD^T$ is nearly a gaussian matrix with coefficients related with $d$ and $n$. We can thus bound this value by using the concentration bound of two-norm of gaussian matrix. However, this method is can only give a bound of $exp(-cd)$. Is there any better result?

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  • $\begingroup$ In what regime? I.e., how (if anyhow) should $a$ depend on $n$ and $d$ and what bound do you expect or will be happy with? $\endgroup$
    – fedja
    Commented Jul 26, 2023 at 12:43
  • $\begingroup$ Thank you so much for that. Let the errror parameter being $\epsilon$. I would expect that $a = \epsilon n$ and get a bound of $\exp (-c poly(d,\frac{1}{\epsilon^2}) )$. $\endgroup$
    – Shouitch Chuu
    Commented Jul 26, 2023 at 15:06

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