An extension of the angular synchronization problem to the heterogeneous setting

Given an undirected measurement graph $G = ([n], E)$, the classical angular synchronization problem consists of recovering unknown angles $\theta1,\dots,\thetan$ from a collection of noisy pairwise measurements of the form $(\thetai - \thetaj) \mod 2\pi$, for each ${i,j} \in E$. This problem arises in a variety of applications, including computer vision, time synchronization of distributed networks, and ranking from preference relationships. In this paper, we consider a generalization to the setting where there exist $k$ unknown groups of angles $\theta{l,1}, \dots,\theta{l,n}$, for $l=1,\dots,k$. For each $ {i,j} \in E$, we are given noisy pairwise measurements of the form $\theta{\ell,i} - \theta{\ell,j}$ for an unknown $\ell \in {1,2,\ldots,k}$. This can be thought of as a natural extension of the angular synchronization problem to the heterogeneous setting of multiple groups of angles, where the measurement graph has an unknown edge-disjoint decomposition $G = G1 \cup G2 \ldots \cup Gk$, where the $Gi$'s denote the subgraphs of edges corresponding to each group. We propose a probabilistic generative model for this problem, along with a spectral algorithm for which we provide a detailed theoretical analysis in terms of robustness against both sampling sparsity and noise. The theoretical findings are complemented by a comprehensive set of numerical experiments, showcasing the efficacy of our algorithm under various parameter regimes. Finally, we consider an application of bi-synchronization to the graph realization problem, and provide along the way an iterative graph disentangling procedure that uncovers the subgraphs $G_i$, $i=1,\ldots,k$ which is of independent interest, as it is shown to improve the final recovery accuracy across all the experiments considered.