Random Multitype Spanning Forests for Synchronization on Sparse Graphs

Random diffusions are a popular tool in Monte-Carlo estimations, with well established algorithms such as Walk-on-Spheres (WoS) going back several decades. In this work, we introduce diffusion estimators for the problems of angular synchronization and smoothing on graphs, in the presence of a rotation associated to each edge. Unlike classical WoS algorithms that are point-wise estimators, our diffusion estimators allow for global estimations by propagating along the branches of random spanning subgraphs called multi-type spanning forests. Building upon efficient samplers based on variants of Wilson's algorithm, we show that our estimators outperform standard numerical-linear-algebra solvers in challenging instances, depending on the topology and density of the graph.