用哈密顿系统构建神经网络，提升金融欺诈与网络安全检测能力
A Hamiltonian driven Geometric Construction of Neural Networks via the Lognormal family, Application to Financial Fraud Detection and to Network Security
- 基于对数正态分布的统计流形构造网络，由哈密顿动力学驱动
- 权重矩阵旋转由SU(1,1)群作用决定，激活函数源自系统的辛结构
- 适用于高维流形数据，特别适合金融与网络安全场景
本文提出一种在统计流形上构建神经网络的新方法，基于对数正态分布。通过将哈密顿系统等价于该流形上的梯度流，定义网络输入值为哈密顿动力学的坐标系，自然嵌入庞加莱圆盘。核心贡献在于从几何原理推导网络组件：突触权重矩阵的旋转由SU(1,1)群在圆盘上的李群作用决定，激活函数则源于系统的辛结构。由此获得完整的权重矩阵（含平移向量）及输出值。
原文摘要 · Abstract (English)
We presents a method for constructing neural networks intrinsically on statistical manifolds via the lognormal distribution. We demonstrate this approach by formulating a neural network architecture directly on statistical manifold. The construction is driven by the Hamiltonian system that is equivalent to the gradient flow on this manifold. We define the network's input values using the coordinate system of this Hamiltonian dynamics, naturally embedded in the Poincar$\acute{e}$ disk. The core of our contribution lies in the derivation of the network's components from geometric principles: the rotation component of the synaptic weight matrix is determined by the Lie group action of $SU(1,1)$ on the disk, while the activation function emerges from the symplectic structure of the system. We subsequently obtain the complete weight matrix, including its translation vector, and the resulting output values.
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