Approximating the little Grothendieck problem over the orthogonal and unitary groups

The little Grothendieck problem consists of maximizing ∑ _{i j}C_{i j}x_ix_j for a positive semidefinite matrix C, over binary variables x_i∈ { ± 1 }. In this paper we focus on a natural generalization of this problem, the little Grothendieck problem over the orthogonal group. Given C∈ R^{d n × d n} a positive semidefinite matrix, the objective is to maximize ∑ijtr(CijTOiOjT) restricting O_i to take values in the group of orthogonal matrices O_d, where C_{i j} denotes the (ij)-th d× d block of C. We propose an approximation algorithm, which we refer to 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. For a given d≥ 1 , we show a constant approximation ratio of α_R(d) ², where α_R(d) is the expected average singular value of a d× d matrix with random Gaussian N(0,1d) i.i.d. entries. For d= 1 we recover the known α_R(1) ²= 2 / π approximation guarantee for the classical little Grothendieck problem. Our algorithm and analysis naturally extends to the complex valued case also providing a constant approximation ratio for the analogous little Grothendieck problem over the Unitary Group U_d. 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 122. The little Grothendieck problem falls under the larger class of problems approximated by a recent algorithm proposed in the context of the non-commutative Grothendieck inequality. Nonetheless, our approach is simpler and provides better approximation with matching integrality gaps. Finally, we also provide an improved approximation algorithm for the more general little Grothendieck problem over the orthogonal (or unitary) group with rank constraints, recovering, when d= 1 , the sharp, known ratios.