Inapproximability of Matrix p→q Norms

This problem generalizes the spectral norm of a matrix (p = q = 2) and the Grothendieck problem (p = ∞, q = 1), and has been widely studied in various regimes. When p ≥ q, the problem exhibits a dichotomy: constant factor approximation algorithms are known if 2 ∈ [q, p], and the problem is hard to approximate within almost polynomial factors when 2 / ∈ [q, p]. The regime when p < q, known as 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 NPhardness of approximation is known for these problems for any p < q. We prove the first NP-hardness result for approximating hypercontractive norms. We show that for any 1 < p < q < ∞ with 2 / ∈ [p, q], ‖A‖p→q is hard to approximate within 2O((log n) 1−ε) assuming NP 6⊆ BPTIME ( 2(log n) O(1) ) .