Majorization Minimization Methods for Distributed Pose Graph Optimization
Taosha Fan, Todd D. Murphey
Abstract
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 ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathsf {MM\text{--}PGO}$</tex-math></inline-formula> ) that applies to a broad class of robust loss kernels. The <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathsf {MM\text{--}PGO}$</tex-math></inline-formula> method is guaranteed to converge to first-order critical points under mild conditions. Furthermore, noting that the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathsf {MM\text{--}PGO}$</tex-math></inline-formula> 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 ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathsf {AMM\text{--}PGO}^*$</tex-math></inline-formula> ) and without ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathsf {AMM\text{--}PGO}^{\#}$</tex-math></inline-formula> )—have faster convergence in contrast to the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathsf {MM\text{--}PGO}$</tex-math></inline-formula> method without sacrificing theoretical guarantees. In particular, the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathsf {AMM\text{--}PGO}^{\#}$</tex-math></inline-formula> 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathsf {AMM\text{--}PGO}^*$</tex-math></inline-formula> 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.