On Sketching the $q$ to $p$ norms

Abstract: 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 $$|A|<em>{q \to p} = \sup</em>{x \in \mathbb{R}^d \backslash \vec{0}} \frac{|Ax|<em>p}{|x|_q}$$, 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 $$L \in \mathbb{R}^{nd} \rightarrow \mathbb{R}<sup>k$$ is a $(k,\alpha)$-sketching family if from $L(A)$, one can approximate $|A|</em>{q \to p}$ 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.