An extension of the angular synchronization problem to the heterogeneous setting
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Given an undirected measurement graph G = ([n], E), the classical angular synchronization problem consists of recovering unknown angles θ_1,…,θ_n from a collection of noisy pairwise measurements of the form (θ_i - θ_j) 2π, for each {i,j}∈ 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 θ_l,1, …,θ_l,n, for l=1,…,k. For each {i,j}∈ E, we are given noisy pairwise measurements of the form θ_ℓ,i - θ_ℓ,j for an unknown ℓ∈{1,2,…,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 = G_1 ∪ G_2 …∪ G_k, where the G_i'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,…,k which is of independent interest, as it is shown to improve the final recovery accuracy across all the experiments considered.
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