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An extension of the angular synchronization problem to the heterogeneous setting

Mihai Cucuringu 1 Hemant Tyagi 2
2 MODAL - MOdel for Data Analysis and Learning
Inria Lille - Nord Europe, LPP - Laboratoire Paul Painlevé - UMR 8524, METRICS - Evaluation des technologies de santé et des pratiques médicales - ULR 2694, Polytech Lille - École polytechnique universitaire de Lille, Université de Lille, Sciences et Technologies
Abstract : 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 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 ,i − θ ,j for an unknown ∈ {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. .. ∪ G k , 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.
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Submitted on : Thursday, January 7, 2021 - 11:28:47 AM
Last modification on : Monday, January 10, 2022 - 1:41:15 PM
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  • HAL Id : hal-03101682, version 1

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Mihai Cucuringu, Hemant Tyagi. An extension of the angular synchronization problem to the heterogeneous setting. Foundations of Data Science, American Institute of Mathematical Sciences, In press. ⟨hal-03101682⟩

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