Bayesian Pose Graph Optimization via Bingham Distributions and Tempered Geodesic MCMC
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.  The algorithm is validated on synthetic experiments along with some real SFM datasets.  The problem that is considered (sampling and optimization over pose graphs on SE(3)^n) is an important one.  The proposed algorithm seems suitable and has at least a small amount of novelty.  However, there are some problems with both the theoretical and experimental analysis which are detailed below.  The paper has potential, but the issues need to be addressed before it is suitable for publication.  - The theorem as stated in the paper has omitted a key condition which somewhat limits its applicability to the scenario indicated in the paper.  Specifically, according to Lemma S1, the theorem only applies for values of \beta \leq D for some finite constant D.  This is not discussed at all in the main paper and only shows up in the statement of the lemma in the supplementary.  Importantly, it undermines a major line of argument for the paper, namely that the method provides a mechanism for either MCMC sampling from the posterior *and* a way to find the maximum of the posterior by setting \beta = \infty.  As far as I've understood the results, the proof of the theorem is not valid as \beta approaches \infty.  - The paper makes claims which are unsubstantiated by the theoretical results.  In particular, lines 248-250 claims that the theorem shows that the algorithm will sample close to the global optimizer x^*.  This is not what is actually said by the theorem.  Rather the theorem states that expected value of U(x) = -\log(p(x)) will be close to U(x^*). Note that this is significant in the case where the posterior non-convex and may be multimodal which line 248 specifically mentions.  - A discussion of the tuning of several key hyperparameters is missing in the paper, limiting reproducibility.  In particular, how are N (number of integration steps) and h (step size) set experimentally?  Are they tuned per experiment or fixed?  How sensitive are the results to these (and other) hyperparameters. Similarly, how exactly is the synthetic data generated?  Is source code going to be released?  This will address some (though not all) of these concerns.  There are also several other less significant issues.  - Figure 2 (in particular 2(e)) is not well explained, 2e in particular is never even referenced and the caption doesn't explain it clearly. - Fig 2e implies and Lines 284-5 discuss combining error in rotation with error in translation.  How is this done?  === UPDATE: I have read and considered the authors feedback and would not argue against acceptance of this paper.  However, I remain concerned about the looseness of the arguments around the applicability of the theoretical results and, in particular, the disconnect between how they're presented in the paper vs how they're actually defined in the appendix.