In this paper, we explore the fundamental role of the Monge-Amp\`ere 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-Amp\`ere 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-Amp\`ere 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.

翻译：本文探讨了蒙日-安培方程在深度学习中的基础性作用，特别是在玻尔兹曼机与基于能量的模型中的表现。我们首先回顾了玻尔兹曼学习的基本结构及其与自由能最小化的关系。随后，我们建立了最优传输理论与深度学习之间的联系，论证了蒙日-安培方程如何支配生成模型中的概率分布变换。此外，我们从量子几何的角度提出新见解，指出学习过程中产生的协方差矩阵空间与冯·诺依曼代数理论中的Connes-Araki-Haagerup（CAH）锥相吻合。进一步地，我们引入了基于重整化群（RG）流的替代研究路径，该方法虽与最优传输视角不同，却揭示了学习动力学中蒙日-安培作用域的另一种表现形式。这种双重研究视角为层级特征学习提供了更深刻的数学理解，从而搭建起统计力学、量子几何与深度学习理论之间的概念桥梁。