SE-Sync: A certifiably correct algorithm for synchronization over the special Euclidean group

Abstract

Many important geometric estimation problems naturally take the form of synchronization over the special Euclidean group: estimate the values of a set of unknown group elements x1, . . . , xn ∈ SE(d) given noisy measurements of a subset of their pairwise relative transforms x −1 i xj. Examples of this class include the foundational problems of pose-graph simultaneous localization and mapping (SLAM) (in robotics), camera motion estimation (in computer vision), and sensor network localization (in distributed sensing), among others. This inference problem is typically formulated as a nonconvex maximum-likelihood estimation that is computationally hard to solve in general. Nevertheless, in this paper we present an algorithm that is able to efficiently recover certifiably globally optimal solutions of the special Euclidean synchronization problem in a non-adversarial noise regime. The crux of our approach is the development of a semidefinite relaxation of the maximum-likelihood estimation whose minimizer provides an exact MLE so long as the magnitude of the noise corrupting the available measurements falls below a certain critical threshold; furthermore, whenever exactness obtains, it is possible to verify this fact a posteriori, thereby certifying the optimality of the recovered estimate. We develop a specialized optimization scheme for solving large-scale instances of this semidefinite relaxation by exploiting its low-rank, geometric, and graph-theoretic structure to reduce it to an equivalent optimization problem defined on a low-dimensional Riemannian manifold, and then design a Riemannian truncated-Newton trust-region method to solve this reduction efficiently. Finally, we combine this fast optimization approach with a simple rounding procedure to produce our algorithm, SE-Sync. Experimental evaluation on a variety of simulated and real-world pose-graph SLAM datasets shows that SE-Sync is capable of recovering certifiably globally optimal solutions when the available measurements are corrupted by noise up to an order of magnitude greater than that typically encountered in robotics and computer vision applications, and does so more than an order of magnitude faster than the Gauss-Newton-based approach that forms the basis of current state-of-the-art techniques. * This report is an extended version of a paper presented at the 12 th International Workshop on the Algorithmic Foundations of Robotics [70].

Deterministic Guarantees for Burer‐Monteiro Factorizations of Smooth Semidefinite Programs

“…For general d , diagonal blocks of X of size d d are constrained to be identity matrices: this SDP is known as Orthogonal-Cut [6,9]. Among other uses, it appears as a relaxation of synchronization on Z 2 D f˙1g [1,5,21] and synchronization of rotations [14,28], with applications in stochastic block modeling (community detection) and SLAM (simultaneous localization and mapping for robotics).…”

Initialization of 3D pose graph optimization using Lagrangian duality

“…Luckily, very fast state-of-the-art solvers for the dual problem [D] exploiting the low-rank structure for our problem have appeared since the original submission of this work that unties the potential of the proposed procedures for application in virtually any PGO instance regardless of its size [28]. Future work will include further evaluation on large-scale problem exploring the use of these solvers.…”

Convex Global 3D Registration with Lagrangian Duality

“…This provides particularly good approximations for many problems that can be reformulated as a Quadratically Constrained Quadratic Program (QCQP), where the relaxed problem becomes a Semidefinite Program (SDP) [4,15]. Some problems involving rotations can be characterized as QCQPs, such as Pose Graph Optimization, for which recent literature applying the Lagrangian dual relaxation has shown impressive results finding globally optimal solutions based solely on convex relaxations [11,10,7,40,8].…”

“…Because of this, it has been customary in other problems involving rotations to relax the constraints by dropping the determinant constraint det(R) = +1 and keeping only the orthonormality constraints R R = I 3 , which amounts to performing the optimization in O(3) rather than in SO(3). This approach has provided tight relaxations for other problems [11,10,7,40]. For the registration problem however it works well only in a certain range of problems [32].…”