用几何与量子理论解析深度学习中的概率变换机制。
Quantum Geometry insights in Deep Learning
- 通过最优传输理论关联蒙日-安培方程与生成模型的概率变换。
- 发现协方差矩阵空间与量子几何中的CAH锥完全一致。
- 提出重整化群流视角，揭示深层特征学习的双重数学结构。
本文探讨蒙日-安培方程在深度学习中的基础作用，特别是在玻尔兹曼机和基于能量的模型中。我们回顾了玻尔兹曼学习的结构及其与自由能最小化的关联，建立最优传输理论与深度学习的联系，证明蒙日-安培方程支配生成模型中的概率变换。进一步引入量子几何视角，发现学习过程中出现的协方差矩阵空间恰好对应冯诺依曼代数理论中的康奈斯-阿拉基-豪格鲁普（CAH）锥。此外，我们提出一种基于重整化群（RG）流的替代方法，虽不同于最优传输视角，但揭示了蒙日-安培结构在学习动态中的另一种表现。这一双重视角深化了对层次化特征学习的数学理解，融合了统计力学、量子几何与深度学习理论。
原文摘要 · Abstract (English)
In this paper, we explore the fundamental role of the Monge-Ampère equation in deep learning, particularly in the context of Boltzmann machines and energy-based models. We first review the structure of Boltzmann learning and its relation to free energy minimization. We then establish a connection between optimal transport theory and deep learning, demonstrating how the Monge-Ampère equation governs probability transformations in generative models. Additionally, we provide insights from quantum geometry, showing that the space of covariance matrices arising in the learning process coincides with the Connes-Araki-Haagerup (CAH) cone in von Neumann algebra theory. Furthermore, we introduce an alternative approach based on renormalization group (RG) flow, which, while distinct from the optimal transport perspective, reveals another manifestation of the Monge-Ampère domain in learning dynamics. This dual perspective offers a deeper mathematical understanding of hierarchical feature learning, bridging concepts from statistical mechanics, quantum geometry, and deep learning theory.
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