Robust Angular Synchronization via Directed Graph Neural Networks

TL;DR

GNNSync tackles angular synchronization and its $k$-synchronization extension by reframing the problem as directed-graph learning and training end-to-end with robust, cycle-consistency–aware losses. A DIMPA-based digraph GNN provides directional embeddings, followed by learned initial angle estimates and a few projected gradient steps using a Hermitian similarity matrix to produce final angles. The authors introduce the upset and cycle-consistency losses, with a confidence-based reweighting mechanism to downweight noisy edges, and demonstrate strong robustness to high noise across synthetic and real-world SNL-like datasets, outperforming classical spectral/SDP and compelling baselines. The work contributes a generalizable, supervision-free framework that leverages directed graph structure and angular geometry and shows promise for extensions to broader groups, including Euc(2), as well as potential supervised or patch-based graph realization tasks.

Abstract

The angular synchronization problem aims to accurately estimate (up to a constant additive phase) a set of unknown angles $θ_1, \dots, θ_n\in[0, 2π)$ from $m$ noisy measurements of their offsets $θ_i-θ_j \;\mbox{mod} \; 2π.$ Applications include, for example, sensor network localization, phase retrieval, and distributed clock synchronization. An extension of the problem to the heterogeneous setting (dubbed $k$-synchronization) is to estimate $k$ groups of angles simultaneously, given noisy observations (with unknown group assignment) from each group. Existing methods for angular synchronization usually perform poorly in high-noise regimes, which are common in applications. In this paper, we leverage neural networks for the angular synchronization problem, and its heterogeneous extension, by proposing GNNSync, a theoretically-grounded end-to-end trainable framework using directed graph neural networks. In addition, new loss functions are devised to encode synchronization objectives. Experimental results on extensive data sets demonstrate that GNNSync attains competitive, and often superior, performance against a comprehensive set of baselines for the angular synchronization problem and its extension, validating the robustness of GNNSync even at high noise levels.

Figures (37)

• Figure 1: Low noise.
• Figure 2: High noise.
• Figure 4: GNNSync overview: starting from an adjacency matrix $\mathbf{A}$ encoding (noisy) pairwise offsets and an input feature matrix $\mathbf{X}$, GNNSync first applies a directed GNN to learn node embeddings $\mathbf{Z}$. It then calculates the inner product with a learnable vector (or $k$ learnable vectors for $k>1$) to produce the initial estimated angles $r_{i,l}^{(0)} \in [0, 2\pi)$ for $l\in\{1,\dots,k\}$, after rescaling. It then applies several projected gradient steps to the initial angle estimates to obtain the final angle estimates, $r_{i,l} \in [0, 2\pi)$. Let the ground-truth angle matrix be $\mathbf{R}\in\mathbb{R}^{n\times k}.$ The loss function is applied to the output angle matrix $\mathbf{r}$, given $\mathbf{A},$ while the final evaluation is based on $\mathbf{R}$ and $\mathbf{r}.$ Orange frames indicate trainable vectors/matrices, green squares fixed inputs, the red square the final estimated angles (outputs), and the yellow circles the loss function and evaluation.
• Figure 5: MSE performance on angular synchronization ($k=1$). Error bars indicate one standard deviation. Dashed lines highlight GNNSync variants.
• Figure 6: MSE performance on $k$-synchronization for $k\in\{2,3,4\}$. $p$ is the network density and $\eta$ is the noise level. Error bars indicate one standard deviation. Dashed lines highlight GNNSync variants.

Theorems & Definitions (6)

• Proposition 1
• Proposition 2
• Proposition 3
• proof
• Proposition 4
• proof