Inapproximability of Matrix $p\rightarrow q$ Norms

We study the problem of computing the $p\rightarrow q$ norm of a matrix $A \in R^{m \times n}$, defined as [ |A|{p\rightarrow q} ~:=~ \max{x \,\in\, Rⁿ \setminus {0}} \frac{|Ax|q}{|x|p} ] This problem generalizes the spectral norm of a matrix ($p=q=2$) and the Grothendieck problem ($p=\infty$, $q=1$), and has been widely studied in various regimes. When $p \geq q$, the problem exhibits a dichotomy: constant factor approximation algorithms are known if $2 \in [q,p]$, and the problem is hard to approximate within almost polynomial factors when $2 \notin [q,p]$. The regime when $p < q$, known as \emph{hypercontractive norms}, is particularly significant for various applications but much less well understood. The case with $p = 2$ and $q > 2$ was studied by [Barak et al, STOC'12] who gave sub-exponential algorithms for a promise version of the problem (which captures small-set expansion) and also proved hardness of approximation results based on the Exponential Time Hypothesis. However, no NP-hardness of approximation is known for these problems for any $p < q$. We study the hardness of approximating matrix norms in both the above cases and prove the following results: - We show that for any $1< p < q < \infty$ with $2 \notin [p,q]$, $|A|{p\rightarrow q}$ is hard to approximate within $2^{{O(\log{1-\epsilon}!n)}$} assuming $NP \not\subseteq BPTIME(2^{{\log{O(1)}!n})$.} This suggests that, similar to the case of $p \geq q$, the hypercontractive setting may be qualitatively different when $2$ does not lie between $p$ and $q$. - For all $p \geq q$ with $2 \in [q,p]$, we show $|A|{p\rightarrow q}$ is hard to approximate within any factor than $1/(\gamma{p^*} \cdot \gammaq)$, where for any $r$, $\gamma_r$ denotes the $r^{th}$ norm of a gaussian, and $p^*$ is the dual norm of $p$.