# Solving SDPs for synchronization and MaxCut problems via the Grothendieck inequality

arXiv: https://arxiv.org/abs/1703.08729 · HTML: https://mlsanity.com/paper/1703.08729/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Mar 2017.

## Built on by (most cited first, 8 of 8)

- [Gradient-Free Methods for Deterministic and Stochastic Nonsmooth Nonconvex Optimization](https://mlsanity.com/paper/2209.05045/) (2022)
- [Escaping Saddle Points for Nonsmooth Weakly Convex Functions via Perturbed Proximal Algorithms](https://mlsanity.com/paper/2102.02837/) (2021)
- [Dynamic Ranking and Translation Synchronization](https://mlsanity.com/paper/2207.01455/) (2022)
- [Nonconvex Matrix Factorization is Geodesically Convex: Global Landscape Analysis for Fixed-rank Matrix Optimization From a Riemannian Perspective](https://mlsanity.com/paper/2209.15130/) (2022)
- [Momentum-inspired Low-Rank Coordinate Descent for Diagonally Constrained SDPs](https://mlsanity.com/paper/2106.08775/) (2021)
- [Efficient semidefinite bounds for multi-label discrete graphical models](https://mlsanity.com/paper/2111.12491/) (2021)
- [Inference and Optimization for Engineering and Physical Systems](https://mlsanity.com/paper/2208.13880/) (2022)
- [Benign landscape for Burer-Monteiro factorizations of MaxCut-type semidefinite programs](https://mlsanity.com/paper/2411.03103/) (2024)

## Abstract (arXiv)

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)) × 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 Z2 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.