Probabilistic Permutation Synchronization using the Riemannian Structure of the Birkhoff Polytope
This addresses the challenge of robustly aligning correspondences in computer vision or related fields, representing a novel method for a known bottleneck.
The paper tackles the problem of synchronizing correspondences across multiple sets of objects or images by introducing a new geometric and probabilistic approach, achieving high-quality multi-graph matching results with faster convergence and reliable confidence estimates on synthetic and real datasets.
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.