提出粒子系统同步的通用判据，解决Transformer简化模型同步性难题。
Synchronization of mean-field models on the circle
- 基于交互函数三阶导数的L1范数，建立全局同步判据。
- 证明当β≥-0.16时系统全局同步，优于此前0≤β≤1的结论。
- 揭示β<-2/3时无法同步，为模型设计提供理论边界。
本文研究定义在单位圆上的n个相互作用粒子的平均场模型，是经典Kuramoto模型的推广。若从几乎任意初始状态出发，所有粒子最终汇聚于圆上同一点，则称发生全局同步。本文提出以粒子相互作用函数三阶导数的L1范数为指标的通用同步准则。作为应用，解决了关于自注意力动力学（Transformer的简化模型）的一个猜想：证明当β≥-0.16时系统实现全局同步，显著扩展了此前Criscitiello, Rebjock, McRae, Boumal（2024）给出的0≤β≤1范围。同时证明当β< -2/3时全局同步不成立。
原文摘要 · Abstract (English)
This paper considers a mean-field model of $n$ interacting particles whose state space is the unit circle, a generalization of the classical Kuramoto model. Global synchronization is said to occur if after starting from almost any initial state, all particles coalesce to a common point on the circle. We propose a general synchronization criterion in terms of $L_1$-norm of the third derivative of the particle interaction function. As an application we resolve a conjecture for the so-called self-attention dynamics (stylized model of transformers), by showing synchronization for all $β\ge -0.16$, which significantly extends the previous bound of $0\le β\le 1$ from Criscitiello, Rebjock, McRae, and Boumal (2024). We also show that global synchronization does not occur when $β< -2/3$.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。