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arXiv · 2405.17501

Geometry of Critical Sets and Existence of Saddle Branches for Two-layer Neural Networks

Leyang Zhang·Yaoyu Zhang·Tao Luo
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Abstract

This paper presents a comprehensive analysis of critical point sets in two-layer neural networks. To study such complex entities, we introduce the critical embedding operator and critical reduction operator as our tools. Given a critical point, we use these operators to uncover the whole underlying critical set representing the same output function, which exhibits a hierarchical structure. Furthermore, we prove existence of saddle branches for any critical set whose output function can be represented by a narrower network. Our results provide a solid foundation to the further study of optimization and training behavior of neural networks.

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Leyang Zhang, Yaoyu Zhang, Tao Luo. 2024-05-26. Geometry of Critical Sets and Existence of Saddle Branches for Two-layer Neural Networks. https://arxiv.org/abs/2405.17501

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