  We present an entirely new geometric and probabilistic approach to
synchronization of correspondences across multiple sets of objects or images.
In particular, we present two algorithms: (1) Birkhoff-Riemannian L-BFGS for
optimizing the relaxed version of the combinatorially intractable cycle
consistency loss in a principled manner, (2) Birkhoff-Riemannian Langevin Monte
Carlo for generating samples on the Birkhoff Polytope and estimating the
confidence of the found solutions. To this end, we first introduce the very
recently developed Riemannian geometry of the Birkhoff Polytope. Next, we
introduce a new probabilistic synchronization model in the form of a Markov
Random Field (MRF). Finally, based on the first order retraction operators, we
formulate our problem as simulating a stochastic differential equation and
devise new integrators. We show on both synthetic and real datasets that we
achieve high quality multi-graph matching results with faster convergence and
reliable confidence/uncertainty estimates.
