Published on Tue Dec 29 2020

### An extension of the angular synchronization problem to the heterogeneous setting

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In this paper, we consider a generalization to the setting where there exist unknown groups of angles. We propose a probabilistic generative model for this problem, along with a spectral algorithm for which we provide a detailed theoretical analysis. The theoretical findings are complemented by a comprehensive set of numerical experiments.

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###### Abstract

Given an undirected measurement graph , the classical angular synchronization problem consists of recovering unknown angles from a collection of noisy pairwise measurements of the form , for each . 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 unknown groups of angles , for . For each $\{i,j\} \in E$, we are given noisy pairwise measurements of the form for an unknown . 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 , where the '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 , which is of independent interest, as it is shown to improve the final recovery accuracy across all the experiments considered.