An extension of the angular synchronization problem to the heterogeneous setting

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