用代数几何解析神经网络，揭示模型复杂度与训练特性的深层联系。
Algebra Unveils Deep Learning -- An Invitation to Neuroalgebraic Geometry
- 通过多项式激活的神经网络构建半代数簇函数空间。
- 将维度、奇点等代数几何特征与泛化能力、表达力关联。
- 为深度学习提供新理论视角，适合理论研究者参考。
本文倡导从代数几何视角研究由机器学习模型参数化的函数空间。聚焦于具有多项式激活函数的代数模型，其对应的函数空间构成半代数簇。我们建立了一个映射表，将这些簇的代数几何不变量（如维度、次数、奇点）与机器学习的核心特性（如样本复杂度、表达能力、训练动态、隐式偏差）相对应。同时综述相关文献，并探讨超越代数领域的思想。本工作奠定了连接代数几何与深度学习的新研究方向——神经代数几何的基础。
原文摘要 · Abstract (English)
In this position paper, we promote the study of function spaces parameterized by machine learning models through the lens of algebraic geometry. To this end, we focus on algebraic models, such as neural networks with polynomial activations, whose associated function spaces are semi-algebraic varieties. We outline a dictionary between algebro-geometric invariants of these varieties, such as dimension, degree, and singularities, and fundamental aspects of machine learning, such as sample complexity, expressivity, training dynamics, and implicit bias. Along the way, we review the literature and discuss ideas beyond the algebraic domain. This work lays the foundations of a research direction bridging algebraic geometry and deep learning, that we refer to as neuroalgebraic geometry.
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