An extension of the angular synchronization problem to the heterogeneous setting - Cucuringu, M; Tyagi, H (American Institute of Mathematical Sciences, 2022)
Given an undirected measurement graph G = ([n],E), the classical angular synchronization problem consists of recovering unknown angles θ1, . . . , θn from a collection of noisy pairwise measurements of the form (θi−θj ) mod 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 θl,1, . . . , θl,n, for l = 1, . . . , k. For each {i, j} ∈ E, we are given noisy pairwise measurements of the form θℓ,i − θℓ,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 = G1 ∪G2 . . . ∪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 Gi, i = 1, . . . , k which is of independent interest, as it is shown to improve the final recovery accuracy across all the experiments considered.