The little Grothendieck problem consists of maximizing
∑ijCijxixj 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 a dn x dn positive semidefinite matrix, the objective
is to maximize ∑ijTr(CijTOiOjT) restricting Oi to take
values in the group of orthogonal matrices, where Cij denotes the (ij)-th
d x d block of C. We propose an approximation algorithm, which we refer to as
Orthogonal-Cut, to solve this problem 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 αR(d)2, where αR(d) is
the expected average singular value of a d x d matrix with random Gaussian
N(0,1/d) i.i.d. entries. For d=1 we recover the known αR(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 problem over the
Unitary Group.
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~221. The little
Grothendieck problem falls under the class of problems approximated by a recent
algorithm proposed in the context of the non-commutative Grothendieck
inequality. Nonetheless, our approach is simpler and it provides a more
efficient algorithm with better approximation ratios and 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.Comment: Updates in version 2: extension to the complex valued (unitary group)
case, sharper lower bounds on the approximation ratios, matching integrality
gap, and a generalized rank constrained version of the problem. Updates in
version 3: Improvement on the expositio