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 \[ \|A\|_{p\rightarrow q} ~:=~ \max_{x \,\in\,R^n \setminus \{0\}} \frac{\|Ax\|_q}{\|x\|_p} \] This problem generalizes thespectral norm of a matrix ($p=q=2$) and the Grothendieck problem ($p=\infty$,$q=1$), and has been widely studied in various regimes. When $p \geq q$, theproblem exhibits a dichotomy: constant factor approximation algorithms areknown if $2 \in [q,p]$, and the problem is hard to approximate within almostpolynomial factors when $2 \notin [q,p]$.

The regime when $p < q$, known as \emph{hypercontractive norms}, isparticularly significant for various applications but much less wellunderstood. 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 theproblem (which captures small-set expansion) and also proved hardness ofapproximation results based on the Exponential Time Hypothesis. However, noNP-hardness of approximation is known for these problems for any $p < q$.

We study the hardness of approximating matrix norms in both the above casesand 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 $NP \not\subseteqBPTIME(2^{\log^{O(1)}\!n})$. This suggests that, similar to the case of $p \geqq$, the hypercontractive setting may be qualitatively different when $2$ doesnot lie between $p$ and $q$.

- For all $p \geq q$ with $2 \in [q,p]$, we show $\|A\|_{p\rightarrow q}$ ishard to approximate within any factor than $1/(\gamma_{p^*} \cdot \gamma_q)$,where for any $r$, $\gamma_r$ denotes the $r^{th}$ norm of a gaussian, and$p^*$ is the dual norm of $p$.