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**1 - 3**of**3**### The non-convex Burer-Monteiro approach works on smooth semidefinite programs

"... Abstract Semidefinite programs (SDPs) can be solved in polynomial time by interior point methods, but scalability can be an issue. To address this shortcoming, over a decade ago, Burer and Monteiro proposed to solve SDPs with few equality constraints via rank-restricted, non-convex surrogates. Rema ..."

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Abstract Semidefinite programs (SDPs) can be solved in polynomial time by interior point methods, but scalability can be an issue. To address this shortcoming, over a decade ago, Burer and Monteiro proposed to solve SDPs with few equality constraints via rank-restricted, non-convex surrogates. Remarkably, for some applications, local optimization methods seem to converge to global optima of these non-convex surrogates reliably. Although some theory supports this empirical success, a complete explanation of it remains an open question. In this paper, we consider a class of SDPs which includes applications such as max-cut, community detection in the stochastic block model, robust PCA, phase retrieval and synchronization of rotations. We show that the low-rank Burer-Monteiro formulation of SDPs in that class almost never has any spurious local optima. This paper was corrected on April 9, 2018. Theorems 2 and 4 had the assumption that M (1) is a manifold. From this assumption it was stated that , which is not true in general. To ensure this identity, the theorems now make the stronger assumption that gradients of the constraints A(Y Y ) = b are linearly independent for all Y in M. All examples treated in the paper satisfy this assumption. Appendix D gives details.

### SYNC-RANK: ROBUST RANKING, CONSTRAINED RANKING AND RANK AGGREGATION VIA EIGENVECTOR AND SDP SYNCHRONIZATION

, 2015

"... Abstract. We consider the classic problem of establishing a statistical ranking of a set of n items given a set of inconsistent and incomplete pairwise comparisons between such items. Instantiations of this problem occur in numerous applications in data analysis (e.g., ranking teams in sports data), ..."

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Abstract. We consider the classic problem of establishing a statistical ranking of a set of n items given a set of inconsistent and incomplete pairwise comparisons between such items. Instantiations of this problem occur in numerous applications in data analysis (e.g., ranking teams in sports data), computer vision, and machine learning. We formulate the above problem of ranking with incomplete noisy information as an instance of the group synchronization problem over the group SO(2) of planar rotations, whose usefulness has been demonstrated in numerous applications in recent years in computer vision and graphics, sensor network localization and structural biology. Its least squares solution can be approximated by either a spectral or a semidefinite programming (SDP) relaxation, followed by a rounding procedure. We show extensive numerical simulations on both synthetic and real-world data sets (Premier League soccer games, a Halo 2 game tournament and NCAA College Basketball games), which show that our proposed method compares favorably to other ranking methods from the recent literature. Existing theoretical guarantees on the group synchronization problem imply lower bounds on the largest amount of noise permissible in the data while still achieving an approximate recovery of the ground truth ranking. We propose a similar synchronization-based algorithm for the rank-aggregation problem, which integrates in a globally consistent ranking many pairwise rank-offsets or partial rankings, given by different rating systems on the same set of items, an approach which yields significantly more accurate results than other aggregation methods, including Rank-Centrality, a recent state-of-the-art algorithm. Furthermore, we discuss the problem of semi-supervised ranking when there is available information on the ground truth rank of a subset of players, and propose an algorithm based on SDP

### Convex relaxations for certain inverse problems on graphs

, 2015

"... Many maximum likelihood estimation problems are known to be intractable in the worst case. A common approach is to consider convex relaxations of the max-imum likelihood estimator (MLE), and relaxations based on semidefinite program-ming (SDP) are among the most popular. This thesis focuses on a cer ..."

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Many maximum likelihood estimation problems are known to be intractable in the worst case. A common approach is to consider convex relaxations of the max-imum likelihood estimator (MLE), and relaxations based on semidefinite program-ming (SDP) are among the most popular. This thesis focuses on a certain class of graph-based inverse problems, referred to as synchronization-type problems. These are problems where the goal is to estimate a set of parameters from pairwise information between them. In this thesis, we investigate the performance of the SDP based approach for a range of problems of this type. While for many such problems, such as multi-reference alignment in signal processing, a precise explanation of their effectiveness remains a fascinating open problem, we rigorously establish a couple of remarkable phenomena. For example, in some instances (such as community detection under the stochas-tic block model) the solution to the SDP matches the ground truth parameters (i.e. achieves exact recovery) for information theoretically optimal regimes. This is estab-