用微分几何解释softmax，揭示其背后的几何结构与动态机制。
Softmax as a Lagrangian-Legendrian Seam
- 将softmax视为概率单纯形上的接触流形界面，连接熵与对数求和的保守描述。
- 揭示偏置不变性对应李布流，KL散度提供到界面的距离度量。
- 为机器学习提供新视角，适合研究信息几何与模型可解释性的学者。
本文首次建立机器学习与现代微分几何之间的桥梁。我们证明，softmax中从逻辑值到概率的转换可建模为几何界面：在简单折叠辛邻域内的概率单纯形（接触屏）上，由负熵和对数求和-指数函数生成的两个保守描述通过一个Legendrian'接缝'相遇。偏置平移不变性表现为屏幕上的Reeb流，Fenchel-Young等式/KL差距提供了到该接缝的可计算距离。通过分析二分类与三分类情形，使图像具体化，并展望下一步方向：紧凑的逻辑值模型（射影或球面型）、全局不变量，以及与信息几何的联系——屏幕上动力学表现为复制者流。
原文摘要 · Abstract (English)
This note offers a first bridge from machine learning to modern differential geometry. We show that the logits-to-probabilities step implemented by softmax can be modeled as a geometric interface: two potential-generated, conservative descriptions (from negative entropy and log-sum-exp) meet along a Legendrian "seam" on a contact screen (the probability simplex) inside a simple folded symplectic collar. Bias-shift invariance appears as Reeb flow on the screen, and the Fenchel-Young equality/KL gap provides a computable distance to the seam. We work out the two- and three-class cases to make the picture concrete and outline next steps for ML: compact logit models (projective or spherical), global invariants, and connections to information geometry where on-screen dynamics manifest as replicator flows.
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