  We initiate the study of data dimensionality reduction, or sketching, for the
$q\to p$ norms. Given an $n \times d$ matrix $A$, the $q\to p$ norm, denoted
$\|A\|_{q \to p} = \sup_{x \in \mathbb{R}^d \backslash \vec{0}}
\frac{\|Ax\|_p}{\|x\|_q}$, is a natural generalization of several matrix and
vector norms studied in the data stream and sketching models, with applications
to datamining, hardness of approximation, and oblivious routing. We say a
distribution $S$ on random matrices $L \in \mathbb{R}^{nd} \rightarrow
\mathbb{R}^k$ is a $(k,\alpha)$-sketching family if from $L(A)$, one can
approximate $\|A\|_{q \to p}$ up to a factor $\alpha$ with constant
probability. We provide upper and lower bounds on the sketching dimension $k$
for every $p, q \in [1, \infty]$, and in a number of cases our bounds are
tight. While we mostly focus on constant $\alpha$, we also consider large
approximation factors $\alpha$, as well as other variants of the problem such
as when $A$ has low rank.
