Approximability of p → q matrix norms: generalized krivine rounding and hypercontractive hardness

Abstract

We study the problem of computing the p → q operator norm of a matrix A in Rmxn, defined as ||A||p→q : = supx∈Rn\{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 is in [q, p], and the problem is hard to approximate within almost polynomial factors when 2 is not in [q,p]. For the case when 2 is in [q,p] we prove almost matching approximation and NP-hardness results.The regime when p 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