TITLE: On the induced matrix norm $\| \cdot \|_{2,\infty}$
QUESTION [5 upvotes]: The induced norm of the matrix $A$ as a map from $(\mathbb R^n , \| \cdot \|_p)$ to $(\mathbb R^m, \| \cdot \|_q)$ is given by
$$ \| A \|_{p,q} = \sup_{x\in\mathbb{R}^n\setminus \{0\}} \frac{\|Ax\|_q}{\|x\|_p}.$$
I would like to compute $\| \cdot \|_{2,\infty}$. In this paper: On the Calculation of the $l_2\to l_1$ Induced Matrix Norm, the authors presented the results for any $p,q\in\{1,2,\infty\}$, except for $p=2$ and $q=\infty$. 
So I would like to know if this is still an open problem. And if it is, then is there any result on finding a tight and easy-to-compute upper bound on the norm?
Thank you in advance.

REPLY [4 votes]: Computing such induced norms is a hard problem. For the case of $p=2$ and $q \ge 2$, have a look at this paper by Barak et al. to see how tricky the problem is.
Typically, for other than the nice cases of $1,2, \infty$ style, these norms are NP-hard to compute, with well-known results for the case $p \ge q$ (see also e.g.: "Matrix norms are NP-Hard to approximate"). The paper of Barak et al focuses on the hypercontractive case of $p < q$.
For the specific case of $p=2$ and $q=\infty$ mentioned above, I think the paper of Barak et al cited above mentions that the norm is just the largest 2-norm of any row of the matrix (thanks to N. Johnston for pointing this out).