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While solving a research problem related to approximations in neural networks, I've faced the following problem which I have not been able to solve after trying different approaches for a while.

Let's say that we have a matrix $A \in \mathbb{R}^{n x n}$ whose entries are each random varianbles $a_{ij} = \sum_{k=1}^m x_{ik} x_{jk}$ where $x_{ab} \sim N(0, 1)$ and each of the random variables $x_{ab}$ are independent of the other ones (unless both indices of $x$ are the same, in which case it will be the same random variable).

The question is: can we bound the spectral norm of the matrix $A$? A simpler version of the question which has the same value for me, is if we can bound the spectral value of matrix $A$ while zero-ing out its diagonal values. I can see that the latter version might be simpler, as in the diagonal, the random variables $x_{ii}$ are not independent. But I'd definitely be happier to know about the general solution to this problem.

P.S.: I know that we can bound this value by $\infty$, but I would love to know if there is any tight bound already available on this. In particular, I would like to know about bounds that are dependent on the variable $m$, but any insight could be helpful for me and is much appreciated.

I know that the product of two independent standard normal random variables is a form of K-Distributions, as mentioned here, but as this distribution is not a sub-gaussian distribution I couldn't use the available resources that discuss spectral norm of random matrices whose entries are sub-gaussian. The other path that I've tried is bounding spectral norm by frobenius norm, but as the K-distribution is distributed around zero (and has mean zero), $x_{ab}^2$ does not have zero mean, and things got a bit messy as I couldn't come up with a concentration inequality to continue this.

Thanks a lot!

Amin
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  • Are the $x_{a,b}$ **jointly** independent? – PhoemueX Apr 02 '22 at 09:03
  • I'm not sure what you mean by $x_{a,b}$. In the question, $x_{ab}$ is one random variable identified by two indices. Could you please clarify? – Amin Apr 03 '22 at 23:06
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    It seems your matrix is a product of two (non-symmetric) Gaussian matrices. I would think a lot is known. – lcv Apr 03 '22 at 23:19
  • I actually couldn't write it as a product of two Gaussian matrices, because of the sum that is there. Do you have any idea on how it can be written as that kind of product? @lcv – Amin Apr 04 '22 at 03:26
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    Unless I'm mistaken, it seems that $A = XX^T$ with $X_{ij}=x_{ij}$. That should get you started – StratosFair Apr 04 '22 at 06:33

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