Probabilistic Permutation Synchronization using the Riemannian Structure of the Birkhoff Polytope [article]

Tolga Birdal, Umut Şimşekli
2019 arXiv   pre-print
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
more » ... rst 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.
arXiv:1904.05814v1 fatcat:ngqtrdaakzfujbqm3scvkptajq