从几何角度解析神经网络，提升模型可解释性与扩展性
Geometric Properties and Graph-Based Optimization of Neural Networks: Addressing Non-Linearity, Dimensionality, and Scalability
- 用几何度量和图结构分析网络内部机制
- 突破线性可分限制，优化复杂度与维度平衡
- 适合关注模型可解释性与高效架构的研究者
深度学习模型因复杂的层级变换常被视为黑箱。识别合适架构对在有限数据下最大化预测性能至关重要。理解神经网络的几何特性涉及对其结构、激活函数及高维空间中变换行为的分析，这些特性影响学习、表征与决策过程。本研究通过几何度量与图结构探索神经网络，基于 arXiv:2007.06559 的基础工作，解决对神经网络所操作数据流形的几何结构理解不足的问题，该结构影响分类、优化与表征。我们识别出三大挑战：(1) 克服线性可分性局限，(2) 管理维度-复杂度权衡，(3) 通过图表示提升可扩展性。为此，提出利用非线性激活函数，通过剪枝与迁移学习优化网络复杂度，并构建高效的图基模型。研究成果深化了对神经网络几何的理解，助力开发更鲁棒、可扩展且可解释的模型。
原文摘要 · Abstract (English)
Deep learning models are often considered black boxes due to their complex hierarchical transformations. Identifying suitable architectures is crucial for maximizing predictive performance with limited data. Understanding the geometric properties of neural networks involves analyzing their structure, activation functions, and the transformations they perform in high-dimensional space. These properties influence learning, representation, and decision-making. This research explores neural networks through geometric metrics and graph structures, building upon foundational work in arXiv:2007.06559. It addresses the limited understanding of geometric structures governing neural networks, particularly the data manifolds they operate on, which impact classification, optimization, and representation. We identify three key challenges: (1) overcoming linear separability limitations, (2) managing the dimensionality-complexity trade-off, and (3) improving scalability through graph representations. To address these, we propose leveraging non-linear activation functions, optimizing network complexity via pruning and transfer learning, and developing efficient graph-based models. Our findings contribute to a deeper understanding of neural network geometry, supporting the development of more robust, scalable, and interpretable models.
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