Approximating the Little Grothendieck Problem over the Orthogonal Group

The little Grothendieck problem (a special case of Boolean quadratic optimization) consists of maximizing ∑ ij Cijxixj over binary variables xi ∈ {±1}, where C is a positive semidefinite matrix. In this paper we focus on a natural generalization of this problem, the little Grothendieck problem over the orthogonal group. Given C ∈ Rdn×dn a positive semidefinite matrix, the objective is to maximize ∑ ij tr ( C ijOiO T j ) restricting Oi to take values in the group of orthogonal matrices O(d), where Cij denotes the (ij)-th d × d block of C. We propose an approximation algorithm, to which we refer as Orthogonal-Cut, to solve the little Grothendieck problem over the group of orthogonal matrices O(d) and show a constant approximation ratio. Our method is based on semidefinite programming where the relaxation is inspired by the work of Goemans and Williamson in the context of the MaxCut problem. For a given d ≥ 1, we show a constant approximation ratio of α d, where αd is the expected average singular value of a d× d matrix with random Gaussian N ( 0, 1 d ) i.i.d. entries. For d = 1 we recover the known α 1 = 2/π approximation guarantee for the classical little Grothendieck problem. Orthogonal-Cut also serves as an approximation algorithm for several applications including the Procrustes problem where it improves over the best previously known approximation ratio of 1 2 √ 2 . The little Grothendieck problem falls under the larger class of problems approximated by an algorithm recently proposed in the context of the non-commutative Grothendieck inequality. Nonetheless, our approach is simpler and gives a better approximation ratio.