An extension of the angular synchronization problem to the heterogeneous setting
Authors: Cucuringu Mihai , Tyagi Hemant 
Published: January 2022

<p style='text-indent:20px;'>Given an undirected measurement graph <inline-formula><tex-math id="M1">\begin{document}$ G = ([n], E) $\end{document}</tex-math></inline-formula>, the classical angular synchronization problem consists of recovering unknown angles <inline-formula><tex-math id="M2">\begin{document}$ \theta_1, \dots, \theta_n $\end{document}</tex-math></inline-formula> from a collection of noisy pairwise measurements of the form <inline-formula><tex-math id="M3">\begin{document}$ (\theta_i - \theta_j) \mod 2\pi $\end{document}</tex-math></inline-formula>, for each <inline-formula><tex-math id="M4">\begin{document}$ \{i, j\} \in E $\end{document}</tex-math></inline-formula>. This problem arises in a variety of applications, including computer vision, time synchronization of distributed networks, and ranking from preference relationships. In this paper, we consider a generalization to the setting where there exist <inline-formula><tex-math id="M5">\begin{document}$ k $\end{document}</tex-math></inline-formula> unknown groups of angles <inline-formula><tex-math id="M6">\begin{document}$ \theta_{l, 1}, \dots, \theta_{l, n} $\end{document}</tex-math></inline-formula>, for <inline-formula><tex-math id="M7">\begin{document}$ l = 1, \dots, k $\end{document}</tex-math></inline-formula>. For each <inline-formula><tex-math id="M8">\begin{document}$ {\left\{{{i, j}}\right\}} \in E $\end{document}</tex-math></inline-formula>, we are given noisy pairwise measurements of the form <inline-formula><tex-math id="M9">\begin{document}$ \theta_{\ell, i} - \theta_{\ell, j} $\end{document}</tex-math></inline-formula> for an <i>unknown</i> <inline-formula><tex-math id="M10">\begin{document}$ \ell \in \{1, 2, \ldots, k\} $\end{document}</tex-math></inline-formula>. This can be thought of as a natural extension of the angular synchronization problem to the heterogeneous setting of multiple groups of angles, where the measurement graph has an unknown edge-disjoint decomposition <inline-formula><tex-math id="M11">\begin{document}$ G = G_1 \cup G_2 \ldots \cup G_k $\end{document}</tex-math></inline-formula>, where the <inline-formula><tex-math id="M12">\begin{document}$ G_i $\end{document}</tex-math></inline-formula>'s denote the subgraphs of edges corresponding to each group. We propose a probabilistic generative model for this problem, along with a spectral algorithm for which we provide a detailed theoretical analysis in terms of robustness against both sampling sparsity and noise. The theoretical findings are complemented by a comprehensive set of numerical experiments, showcasing the efficacy of our algorithm under various parameter regimes. Finally, we consider an application of bi-synchronization to the graph realization problem, and provide along the way an iterative graph disentangling procedure that uncovers the subgraphs <inline-formula><tex-math id="M13">\begin{document}$ G_i $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M14">\begin{document}$ i = 1, \ldots, k $\end{document}</tex-math></inline-formula> which is of independent interest, as it is shown to improve the final recovery accuracy across all the experiments considered.</p>