Towards Certifiable State Estimation under Observability Degeneracy

Certifiable methods have arisen as a means to guarantee global optimality of solutions to non-convex problems using convex semidefinite programming (SDP) relaxations. The most performant of these methods use a local solver to obtain the candidate solution, and then certify its optimality using efficient linear algebra techniques. However, for many problems of interest in robotics, this local-solve-then-certify approach is impeded by a form of degeneracy in the relaxation, leaving a costly optimization of the relaxation as the only recourse. In this paper, we introduce our Central-Path Certifier (CP-Cert), a certifiable method explicitly tailored to certify candidate optima to problems that exhibit this form of degeneracy. Using a candidate as a starting point, our approach seeks a nearby region of the feasible space -- known as the central path -- where a valid certificate can be readily obtained. The approach is kept efficient by exploiting indirect linear algebra techniques, problem sparsity, and parallelism. We apply CP-Cert to both matrix-weighted pose registration and pointcloud data association, whose novel SDP relaxation is of independent interest. On simulated examples, we explore the properties of this novel relaxation and show that CP-Cert is fast and scalable, achieving runtimes that are up to three orders of magnitude faster than state-of-the-art direct solvers. Finally, we combine these contributions into a certifiable, outlier-robust pose-estimation pipeline, which we apply to real-world data.

Robust and efficient state estimation is crucial for perception, navigation, and control in robotics. State estimation problems are conveniently modeled using the factor-graph framework as enabled by modern software packages such as GTSAM or g2o. However, the standard solvers included in such frameworks are local and may converge to poor local minima, posing significant safety concerns. Conversely, techniques based on convex relaxations have been shown to provide a means of globally solving or certifying many state estimation problems. However, these relaxations 1) often require substantial effort to formulate, and 2) may incur significantly higher cost compared to efficient local solvers, as they require solving a large semidefinite program (SDP). In this work, we address both shortcomings by 1) creating a new procedure within the GTSAM framework for automatically constructing convex SDP relaxations for any factor graphs with common factor and variable types, and by 2) exploiting the Bayes tree constructions native to GTSAM to decompose the SDP problem, leading to significant speedup in solver time for chordally sparse problems. We demonstrate the favorable scaling of this structure-exploiting global estimator compared to standard local solvers for two case studies: A 3D pose-graph SLAM problem with a ring factor graph and a 2D localization problem with a chain factor graph. The software framework is available at https://github.com/borglab/gtsam.

Parameter estimation in robotics and computer vision faces formidable challenges from both outlier contamination and nonconvex optimization landscapes. While M-estimation addresses the problem of outliers through robust loss functions, it creates severely nonconvex problems that are difficult to solve globally. Adaptive reweighting schemes provide one particularly appealing strategy for implementing M-estimation in practice: these methods solve a sequence of simpler weighted least squares (WLS) subproblems, enabling both the use of standard least squares solvers and the recovery of higher-quality estimates than simple local search. However, adaptive reweighting still crucially relies upon solving the inner WLS problems effectively, a task that remains challenging in many robotics applications due to the intrinsic nonconvexity of many common parameter spaces (e.g. rotations and poses). In this paper, we show how one can easily implement adaptively reweighted M-estimators with certifiably correct solvers for the inner WLS subproblems using only fast local optimization over smooth manifolds. Our approach exploits recent work on certifiable factor graph optimization to provide global optimality certificates for the inner WLS subproblems while seamlessly integrating into existing factor graph-based software libraries and workflows. Experimental evaluation on pose-graph optimization and landmark SLAM tasks demonstrates that our adaptively reweighted certifiable estimation approach provides higher-quality estimates than alternative local search-based methods, while scaling tractably to realistic problem sizes.