Solving SDPs for synchronization and MaxCut problems via the Grothendieck inequality
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Proceedings of the 2017 Conference on Learning Theory, PMLR 65:14761515, 2017.
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 highdimensional problems. In order to cope with this problem, Burer and Monteiro proposed a nonconvex rankconstrained formulation, which has good performance in practice but is still poorly understood theoretically. In this paper we study the rankconstrained version of SDPs arising in MaxCut and in $\mathbb Z_2$ and $\rm SO(d)$ synchronization problems. We establish a Grothendiecktype 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 trustregion method to this nonconvex problem, while constraining the rank to be of order one. For the MaxCut problem, our inequality implies that any local maximizer of the rankconstrained SDP provides a $(1  1/(k1)) \times 0.878$ 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 twogroups 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 informationtheoretically optimal methods.
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