从微分流形视角揭示神经网络中连续吸引子的普适性
A Differential Manifold Perspective and Universality Analysis of Continuous Attractors in Artificial Neural Networks
- 基于微分流形理论构建分析框架
- 发现奇异值分层在常见模型中普遍存在
- 适合研究神经网络动力学与理论机制的学者
连续吸引子在生物和人工神经系统中对空间导航、记忆及深度学习优化至关重要。然而，现有研究缺乏统一框架来分析其在不同动力系统中的特性，限制了跨架构的通用性。本文从微分流形视角建立新分析框架，验证了与已有结论的兼容性，阐明了连续吸引子现象与局部雅可比矩阵特征值之间的关联，并证明了在常见分类模型与数据集中的奇异值分层具有普适性。这些发现表明连续吸引子可能广泛存在于一般神经网络中，凸显建立普遍理论的必要性，而所提框架因特征值与奇异值间的紧密数学联系，展现出良好的理论基础。
原文摘要 · Abstract (English)
Continuous attractors are critical for information processing in both biological and artificial neural systems, with implications for spatial navigation, memory, and deep learning optimization. However, existing research lacks a unified framework to analyze their properties across diverse dynamical systems, limiting cross-architectural generalizability. This study establishes a novel framework from the perspective of differential manifolds to investigate continuous attractors in artificial neural networks. It verifies compatibility with prior conclusions, elucidates links between continuous attractor phenomena and eigenvalues of the local Jacobian matrix, and demonstrates the universality of singular value stratification in common classification models and datasets. These findings suggest continuous attractors may be ubiquitous in general neural networks, highlighting the need for a general theory, with the proposed framework offering a promising foundation given the close mathematical connection between eigenvalues and singular values.
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