提出连续极限框架，解决分布式表示学习中的发散难题。
Continuous Limits of Coupled Flows in Representation Learning
- 将分布式学习建模为黎曼流形上的耦合慢-快动力系统。
- 证明权重始终收敛至空间度量主特征空间，避免参数爆炸。
- 揭示线性可分、正交解耦特征在稳态下自发涌现。
现代表示学习严重依赖全局误差信号，而基于局部交互的去中心化算法提供了根本性的分布式替代方案。然而，这些离散动力学在连续数据流形上的宏观收敛性质尚未得到理论阐明，常面临参数爆炸问题。本文通过在黎曼流形上将去中心化学习形式化为耦合的慢-快动力系统，填补了这一空白。首先，利用测度论极限，证明离散空间转移统一收敛至过阻尼朗之万随机微分方程。其次，通过伊藤-泊松再生核与拉萨尔不变性原理的随机扩展，确立表示权重无条件避免发散，并严格对齐空间度量的主特征空间。最后，构建全耦合空间-参数流的联合李雅普诺夫泛函，证明全局耗散性，并展示正交解耦、线性可分特征在稳态下自发产生。本框架连接离散算法与连续随机分析，为去中心化表示学习提供正式理论基准。
原文摘要 · Abstract (English)
While modern representation learning relies heavily on global error signals, decentralized algorithms driven by local interactions offer a fundamental distributed alternative. However, the macroscopic convergence properties of these discrete dynamics on continuous data manifolds remain theoretically unresolved, notoriously suffering from parameter explosion. We bridge this gap by formalizing decentralized learning as a coupled slow-fast dynamical system on Riemannian manifolds. First, using measure-theoretic limits, we prove that the discrete spatial transitions converge uniformly to an overdamped Langevin stochastic differential equation. Second, via the Itô-Poisson resolvent and a stochastic extension of LaSalle's Invariance Principle, we establish that the representation weights unconditionally avoid divergence and align strictly with the principal eigenspace of the spatial measure. Finally, we construct a joint Lyapunov functional for the fully coupled spatial-parametric flow. This proves global dissipativity and demonstrates that orthogonally disentangled, linearly separable features emerge spontaneously at the stationary limit. Our framework bridges discrete algorithms with continuous stochastic analysis, providing a formal theoretical baseline for decentralized representation learning.
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