Suppose $A\in\mathbb R^{n\times m}$ is a random matrix with $n < m$, and each entry $A_{ij}$ follows i.i.d. Gaussian distribution $N(0,1/n)$. I want to know whether we can upper bound the spectral norm $\|A\|$ and $\|A^+\|$ with high probability ($A^+$ is the Moore-Penrose pseudo-inverse of $A$).
It would also be good to upper bound the expectations $E[\|A\|^p]$ for positive integer $p$. But I believe the expectations could be easily bounded once we have "with high probability" bounds.
Finally got the answer. For bounding $\|A\|^p$ one can use the rectangular matrix Bernstein inequality. For bounding $\|A^+\|$ here is a wonderful paper that provides upper bounds for all eigenvalues of a sum of random matrices: [GT11] Alex Gittens and Joel Tropp. Tail bounds for all eigenvalues of a sum of random matrices. arXiv:1104.4513, 2011. Basically you need $O(r\log r)$ more instances to bound the $r$-th eigenvalue.