We consider the problem of distributed pose graph optimization (PGO) that has important applications in multirobot simultaneous localization and mapping (SLAM). We propose the majorization minimization (MM) method for distributed PGO ($\mathsf {MM\text{--}PGO}$) that applies to a broad class of robust loss kernels. The$\mathsf {MM\text{--}PGO}$method is guaranteed to converge to first-order critical points under mild conditions. Furthermore, noting that the$\mathsf {MM\text{--}PGO}$method is reminiscent of proximal methods, we leverage Nesterov's method and adopt adaptive restarts to accelerate convergence. The resulting accelerated MM methods for distributed PGO—both with a master node in the network ($\mathsf {AMM\text{--}PGO}^*$) and without ($\mathsf {AMM\text{--}PGO}^{\#}$)—have faster convergence in contrast to the$\mathsf {MM\text{--}PGO}$method without sacrificing theoretical guarantees. In particular, the$\mathsf {AMM\text{--}PGO}^{\#}$method, which needs no master node and is fully decentralized, features a novel adaptive restart scheme and has a rate of convergence comparable to that of the$\mathsf {AMM\text{--}PGO}^*$method using a master node to aggregate information from all the nodes. The efficacy of this work is validated through extensive applications to 2-D and 3-D SLAM benchmark datasets and comprehensive comparisons against existing state-of-the-art methods, indicating that our MM methods converge faster and result in better solutions to distributed PGO.

We present Theseus, an efficient application-agnostic open source library for differentiable nonlinear least squares (DNLS) optimization built on PyTorch, providing a common framework for end-to-end structured learning in robotics and vision. Existing DNLS implementations are application specific and do not always incorporate many ingredients important for efficiency. Theseus is application-agnostic, as we illustrate with several example applications that are built using the same underlying differentiable components, such as second-order optimizers, standard costs functions, and Lie groups. For efficiency, Theseus incorporates support for sparse solvers, automatic vectorization, batching, GPU acceleration, and gradient computation with implicit differentiation and direct loss minimization. We do extensive performance evaluation in a set of applications, demonstrating significant efficiency gains and better scalability when these features are incorporated. Project page: https://sites.google.com/view/theseus-ai/

We propose a novel sparse constrained formulation and from it derive a real-time optimization method for 3D human pose and shape estimation. Our optimization method, SCOPE (Sparse Constrained Optimization for 3D human Pose and shapE estimation), is orders of magnitude faster (avg. 4ms convergence) than existing optimization methods, while being mathematically equivalent to their dense unconstrained formulation under mild assumptions. We achieve this by exploiting the underlying sparsity and constraints of our formulation to efficiently compute the Gauss-Newton direction. We show that this computation scales linearly with the number of joints and measurements of a complex 3D human model, in contrast to prior work where it scales cubically due to their dense unconstrained formulation. Based on our optimization method, we present a real-time motion capture framework that estimates 3D human poses and shapes from a single image at over 30 FPS. In benchmarks against state-of-the-art methods on multiple public datasets, our framework outperforms other optimization methods and achieves competitive accuracy against regression methods. Project page with code and videos: https://sites.google.com/view/scope-human/.

In this paper, we consider the problem of distributed pose graph optimization (PGO) that has extensive applications in multi-robot simultaneous localization and mapping (SLAM). We propose majorization minimization methods for distributed PGO and show that our methods are guaranteed to converge to first-order critical points under mild conditions. Furthermore, since our methods rely a proximal operator of distributed PGO, the convergence rate can be significantly accelerated with Nesterov's method, and more importantly, the acceleration induces no compromise of convergence guarantees. In addition, we also present accelerated majorization minimization methods for the distributed chordal initialization that have a quadratic convergence, which can be used to compute an initial guess for distributed PGO. The efficacy of this work is validated through applications on a number of 2D and 3D SLAM datasets and comparisons with existing state-of-the- art methods, which indicates that our methods have faster convergence and result in better solutions to distributed PGO.

In this article, we consider the problem of planar graph-based simultaneous localization and mapping (SLAM) that involves both poses of the autonomous agent and positions of observed landmarks. We present complex (CPL)-SLAM, an efficient and certifiably correct algorithm to solve planar graph-based SLAM using the complex number representation. We formulate and simplify planar graph-based SLAM as the maximum likelihood estimation on the product of unit complex numbers, and relax this nonconvex quadratic complex optimization problem to convex complex semidefinite programming (SDP). Furthermore, we simplify the corresponding complex SDP to Riemannian staircase optimization (RSO) on the complex oblique manifold that can be solved with the Riemannian trust region method. In addition, we prove that the SDP relaxation and RSO simplification are tight as long as the noise magnitude is below a certain threshold. The efficacy of this work is validated through applications of CPL-SLAM and comparisons with existing state-of-the-art methods on planar graph-based SLAM, which indicates that our proposed algorithm is capable of solving planar graph-based SLAM certifiably, and is more efficient in numerical computation and more robust to measurement noise than existing state-of-the-art methods. The C++ code for CPL-SLAM is available at https://github.com/MurpheyLab/CPL-SLAM.

In this paper, we present CPL-Sync, a certifiably correct algorithm to solve planar pose graph optimization (PGO) using the complex number representation. We formulate planar PGO as the maximum likelihood estimation (MLE) on the product of unit complex numbers, and relax this nonconvex quadratic complex optimization problem to complex semidefinite programming (SDP). Furthermore, we simplify the corresponding semidefinite programming to Riemannian staircase optimization (RSO) on complex oblique manifolds that can be solved with the Riemannian trust region (RTR) method. In addition, we prove that the SDP relaxation and RSO simplification are tight as long as the noise magnitude is below a certain threshold. The efficacy of this work is validated through comparisons with existing methods as well as applications on planar PGO in simultaneous localization and mapping (SLAM), which indicates that the proposed algorithm is capable of solving planar PGO certifiably, and is more efficient in numerical computation and more robust to measurement noises than existing state-of-the-art methods. The C++ code for CPL-Sync is available at https://github.com/fantaosha/CPL-Sync.