تقرير
Synchronization on circles and spheres with nonlinear interactions
| العنوان: | Synchronization on circles and spheres with nonlinear interactions |
|---|---|
| المؤلفون: | Criscitiello, Christopher, Rebjock, Quentin, McRae, Andrew D., Boumal, Nicolas |
| سنة النشر: | 2024 |
| المجموعة: | Computer Science Mathematics |
| مصطلحات موضوعية: | Mathematics - Optimization and Control, Computer Science - Machine Learning, Mathematics - Dynamical Systems |
| الوصف: | We consider the dynamics of $n$ points on a sphere in $\mathbb{R}^d$ ($d \geq 2$) which attract each other according to a function $\varphi$ of their inner products. When $\varphi$ is linear ($\varphi(t) = t$), the points converge to a common value (i.e., synchronize) in various connectivity scenarios: this is part of classical work on Kuramoto oscillator networks. When $\varphi$ is exponential ($\varphi(t) = e^{\beta t}$), these dynamics correspond to a limit of how idealized transformers process data, as described by Geshkovski et al. (2024). Accordingly, they ask whether synchronization occurs for exponential $\varphi$. In the context of consensus for multi-agent control, Markdahl et al. (2018) show that for $d \geq 3$ (spheres), if the interaction graph is connected and $\varphi$ is increasing and convex, then the system synchronizes. What is the situation on circles ($d=2$)? First, we show that $\varphi$ being increasing and convex is no longer sufficient. Then we identify a new condition (that the Taylor coefficients of $\varphi'$ are decreasing) under which we do have synchronization on the circle. In so doing, we provide some answers to the open problems posed by Geshkovski et al. (2024). Comment: 28 pages, 1 figure |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2405.18273 |
| رقم الأكسشن: | edsarx.2405.18273 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |