Title: Solving SDPs for synchronization and MaxCut problems via the Grothendieck inequality,
Abstract: A number of statistical estimation problems can be addressed by semidefinite
programs (SDP). While SDPs are solvable in polynomial time using interior point
methods, in practice generic SDP solvers do not scale well to high-dimensional
problems. In order to cope with this problem, Burer and Monteiro proposed a
non-convex rank-constrained formulation, which has good performance in practice
but is still poorly understood theoretically.
In this paper we study the rank-constrained version of SDPs arising in MaxCut
and in synchronization problems. We establish a Grothendieck-type inequality
that proves that all the local maxima and dangerous saddle points are within a
small multiplicative gap from the global maximum. We use this structural
information to prove that SDPs can be solved within a known accuracy, by
applying the Riemannian trust-region method to this non-convex problem, while
constraining the rank to be of order one. For the MaxCut problem, our
inequality implies that any local maximizer of the rank-constrained SDP
provides a $ (1 - 1/(k-1)) \times 0.878$ approximation of the MaxCut, when the
rank is fixed to $k$.
We then apply our results to data matrices generated according to the
Gaussian ${\mathbb Z}_2$ synchronization problem, and the two-groups stochastic
block model with large bounded degree. We prove that the error achieved by
local maximizers undergoes a phase transition at the same threshold as for
information-theoretically optimal methods.