# On the Landscape of Synchronization Networks: A Perspective from Nonconvex Optimization

arXiv: https://arxiv.org/abs/1809.11083 · HTML: https://mlsanity.com/paper/1809.11083/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Sep 2018.

## Built on by (most cited first, 11 of 11)

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- [Rank-one matrix estimation: analytic time evolution of gradient descent dynamics](https://mlsanity.com/paper/2105.12257/) (2021)
- [Nonconvex Matrix Factorization is Geodesically Convex: Global Landscape Analysis for Fixed-rank Matrix Optimization From a Riemannian Perspective](https://mlsanity.com/paper/2209.15130/) (2022)
- [Finding Regularized Competitive Equilibria of Heterogeneous Agent Macroeconomic Models with Reinforcement Learning](https://mlsanity.com/paper/2303.04833/) (2023)
- [Cascaded Gaps: Towards Gap-Dependent Regret for Risk-Sensitive Reinforcement Learning](https://mlsanity.com/paper/2203.03110/) (2022)
- [A Complete Loss Landscape Analysis of Regularized Deep Matrix Factorization](https://mlsanity.com/paper/2506.20344/) (2025)
- [Taming Equilibrium Bias in Risk-Sensitive Multi-Agent Reinforcement Learning](https://mlsanity.com/paper/2405.02724/) (2024)
- [Benign landscape for Burer-Monteiro factorizations of MaxCut-type semidefinite programs](https://mlsanity.com/paper/2411.03103/) (2024)

## Abstract (arXiv)

Studying the landscape of nonconvex cost function is key towards a better understanding of optimization algorithms widely used in signal processing, statistics, and machine learning. Meanwhile, the famous Kuramoto model has been an important mathematical model to study the synchronization phenomena of coupled oscillators over various network topologies. In this paper, we bring together these two seemingly unrelated objects by investigating the optimization landscape of a nonlinear function E(boldsymbolθ) = frac12sum1≤ i,j≤ n aij(1-cos(θi - θj)) associated to an underlying network and exploring the relationship between the existence of local minima and network topology. This function arises naturally in Burer-Monteiro method applied to mathbbZ2 synchronization as well as matrix completion on the torus. Moreover, it corresponds to the energy function of the homogeneous Kuramoto model on complex networks for coupled oscillators. We prove the minimizer of the energy function is unique up to a global translation under deterministic dense graphs and ErdHos-R'enyi random graphs with tools from optimization and random matrix theory. Consequently, the stable equilibrium of the corresponding homogeneous Kuramoto model is unique and the basin of attraction for the synchronous state of these coupled oscillators is the whole phase space minus a set of measure zero. In addition, our results address when the Burer-Monteiro method recovers the ground truth exactly from highly incomplete observations in mathbbZ2 synchronization and shed light on the robustness of nonconvex optimization algorithms against certain types of so-called monotone adversaries. Numerical simulations are performed to illustrate our results.