Inapproximability of Matrix p→ q Norms

We study the problem of computing the p→ q norm of a matrix A ∈ R^m × n, defined as A_p→ q := max_x ∈ R^n ∖{0}Ax_q/x_p 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 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 < ∞ with 2 ∉ [p,q], A_p→ q is hard to approximate within 2^O(log^1-ϵn) assuming NP ⊈BPTIME(2^log^O(1)n). This suggests that, similar to the case of p ≥ q, the hypercontractive setting may be qualitatively different when 2 does not lie between p and q. - For all p ≥ q with 2 ∈ [q,p], we show A_p→ q is hard to approximate within any factor than 1/(γ_p^*·γ_q), where for any r, γ_r denotes the r^th norm of a gaussian, and p^* is the dual norm of p.