Inapproximability of Matrix \(\boldsymbol{p \rightarrow q}\) Norms

We study the problem of computing the \(p\rightarrow q\) norm of a matrix \(A \in{\mathbb{R}}^{m \times n}\) , defined as \(\|A\|_{p\rightarrow q} \:= \max _{x \in{\mathbb{R}}^n \setminus \{0\}} \frac{\|Ax\|_{q}}{\|x\|_{p}}\) . 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 \(p \lt q\) , known as hypercontractive norms, is particularly significant for various applications but much less well understood. The case with \(p=2\) and \(q \gt 2\) was studied by Barak et al. [Proceedings of the 44th Annual ACM Symposium on Theory of Computing, 2012, pp. 307–326], who gave subexponential 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 \(p \lt q\) . We prove the first NP-hardness result (under randomized reductions) for approximating hypercontractive norms. We show that for any \(1\lt p \lt q \lt \infty\) with \(2 \notin{[p,q]}\) , \(\|A\|_{p\rightarrow q}\) is hard to approximate within \(2^{O((\log n)^{1-\epsilon })}\) assuming \(\textrm{NP} \not \subseteq \textrm{BPTIME}(2^{(\log n)^{O(1)}})\) . En route to the above result, we also prove almost tight results for the case when \(p \geq q\) with \(2 \in{[q,p]}\) .