Approximating Matrix p-norms

Abstract

We consider the problem of computing the q->p norm of a matrix A, which is defined for p,q \ge 1, as |A|_{q->p} = max_{x !=0 } |Ax|_p / |x|_q. This is in general a non-convex optimization problem, and is a natural generalization of the well-studied question of computing singular values (this corresponds to p=q=2). Different settings of parameters give rise to a variety of known interesting problems (such as the Grothendieck problem when p=1 and q=\infty). However, very little is understood about the approximability of the problem for different values of p,q. Our first result is an efficient algorithm for computing the q->p norm of matrices with non-negative entries, when q \ge p \ge 1. The algorithm we analyze is based on a natural fixed point iteration, which can be seen as an analog of power iteration for computing eigenvalues. We then present an application of our techniques to the problem of constructing a scheme for oblivious routing in the l_p norm. This makes constructive a recent existential result of Englert and R\"acke [ER] on O(log n)-competitive oblivious routing schemes (which they make constructive only for p=2). On the other hand, when we do not have any restrictions on the entries (such as non-negativity), we prove that the problem is NP-hard to approximate to any constant factor, for 2 < p \le q, and p \le q < 2 (these are precisely the ranges of p,q with p\le q, where constant factor approximations are not known). In this range, our techniques also show that if NP does not have quasi-polynomial time algorithms, the q->p cannot be approximated to a factor 2^{(log n)^{1-eps}}, for any \eps>0.