Synchronization on circles and spheres with nonlinear interactions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912850105073664 |
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| author | Criscitiello, Christopher Rebjock, Quentin McRae, Andrew D. Boumal, Nicolas |
| author_facet | Criscitiello, Christopher Rebjock, Quentin McRae, Andrew D. Boumal, Nicolas |
| contents | 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 $φ$ of their inner products. When $φ$ is linear ($φ(t) = t$), 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 $φ$ is exponential ($φ(t) = e^{βt}$), these dynamics correspond to a limit of how idealized transformers process data, as described by Geshkovski et al. (2025). Accordingly, they ask whether synchronization occurs for exponential $φ$.
The answer depends on the dimension $d$. 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 $φ$ is increasing and convex, then the system synchronizes. We give a separate proof of this result.
What is the situation on circles ($d=2$)? First, we show that $φ$ being increasing and convex is no longer sufficient (even for complete graphs). Then we identify a new condition under which we do have synchronization on the circle (namely, if the Taylor coefficients of $φ'$ are decreasing). As a corollary, this provide synchronization for exponential $φ$ with $β\in (0, 1]$. The proofs are based on nonconvex landscape analysis. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_18273 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Synchronization on circles and spheres with nonlinear interactions Criscitiello, Christopher Rebjock, Quentin McRae, Andrew D. Boumal, Nicolas Optimization and Control Machine Learning Dynamical Systems 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 $φ$ of their inner products. When $φ$ is linear ($φ(t) = t$), 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 $φ$ is exponential ($φ(t) = e^{βt}$), these dynamics correspond to a limit of how idealized transformers process data, as described by Geshkovski et al. (2025). Accordingly, they ask whether synchronization occurs for exponential $φ$. The answer depends on the dimension $d$. 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 $φ$ is increasing and convex, then the system synchronizes. We give a separate proof of this result. What is the situation on circles ($d=2$)? First, we show that $φ$ being increasing and convex is no longer sufficient (even for complete graphs). Then we identify a new condition under which we do have synchronization on the circle (namely, if the Taylor coefficients of $φ'$ are decreasing). As a corollary, this provide synchronization for exponential $φ$ with $β\in (0, 1]$. The proofs are based on nonconvex landscape analysis. |
| title | Synchronization on circles and spheres with nonlinear interactions |
| topic | Optimization and Control Machine Learning Dynamical Systems |
| url | https://arxiv.org/abs/2405.18273 |