We introduce Tempered Geodesic Markov Chain Monte Carlo (TG-MCMC) algorithm for initializing pose graph optimization problems, arising in various scenarios such as SFM (structure from motion) or SLAM (simultaneous localization and mapping). TG-MCMC is first of its kind as it unites global non-convex optimization on the spherical manifold of quaternions with posterior sampling, in order to provide both reliable initial poses and uncertainty estimates that are informative about the quality of solutions. We devise theoretical convergence guarantees and extensively evaluate our method on synthetic and real benchmarks. Besides its elegance in formulation and theory, we show that our method is robust to missing data, noise and the estimated uncertainties capture intuitive properties of the data.

This paper presents a stochastic gradient Monte Carlo approach defined on a Cartesian product of SE(3), a domain commonly used to characterize problems in structure-from-motion (SFM) among other areas. The algorithm is parameterized by an inverse temperature such that when the value goes to inifinity, the algorithm is implicitly operating on a delta function with it's peak at the maximum of the base distribution. The proposed algorithm is formulated as a SDE and a splitting scheme is proposed to integrate it. A theoretical analysis on the SDE and its discretization is explored, showing that 1) the resulting Markov process has the appropriate invariant distribution and 2) the sampler will draw samples close to the maximum of the posterior (in terms of expectation of the unnormalized log posterior). Along with the algorithm, a model is defined using the Bingham distribution to characterize typical SFM posteriors which is then used to perform experiments with the algorithm.

We introduce Tempered Geodesic Markov Chain Monte Carlo (TG-MCMC) algorithm for initializing pose graph optimization problems, arising in various scenarios such as SFM (structure from motion) or SLAM (simultaneous localization and mapping). TG-MCMC is first of its kind as it unites global non-convex optimization on the spherical manifold of quaternions with posterior sampling, in order to provide both reliable initial poses and uncertainty estimates that are informative about the quality of solutions. We devise theoretical convergence guarantees and extensively evaluate our method on synthetic and real benchmarks. Besides its elegance in formulation and theory, we show that our method is robust to missing data, noise and the estimated uncertainties capture intuitive properties of the data. Papers published at the Neural Information Processing Systems Conference.