用几何公理化框架揭示神经网络的运行机制。
Axiomatizing Neural Networks via Pursuit of Subspaces
- 提出子空间追寻假说，用几何公理解释神经网络行为。
- 统一解释了表征结构、计算机制与泛化性能等核心问题。
- 为深度学习理论提供可推理的数学基础，适合研究者参考。
尽管深度神经网络在多个领域取得了显著成果，但其内在机制仍不清晰，常被视为黑箱。这一经验性能与理论理解之间的鸿沟，类似于古典几何学尚未建立公理体系的早期阶段。本文提出子空间追寻（PoS）假说，构建了一套基于几何公理的神经网络行为框架。这些公理及其推论为浅层与深层架构中的表征、计算与泛化提供了统一视角。该框架能对深度学习中的基本问题给出几何解释，包括表征结构、架构机制与泛化行为，为建立连贯的理论基础迈出关键一步。
原文摘要 · Abstract (English)
While deep neural networks have achieved remarkable success across a wide range of domains, their underlying mechanisms remain poorly understood, and they are often regarded as black boxes. This gap between empirical performance and theoretical understanding poses a challenge analogous to the pre-axiomatic stage of classical geometry. In this work, we introduce the Pursuit of Subspaces (PoS) hypothesis, an axiomatic framework that formulates neural network behavior through a set of geometric postulates. These axioms, together with their derived consequences, provide a unified perspective on representation, computation, and generalization in both shallow and deep architectures. We show that this framework yields geometric explanations for fundamental questions in deep learning, including representation structure, architectural mechanisms, and generalization behavior, offering a principled step toward a coherent theoretical foundation.
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