We revisit the Universal Approximation Theorem(UAT) through the lens of the tropical geometry of neural networks and introduce a constructive, geometry-aware initialization for sigmoidal multi-layer perceptrons (MLPs). Tropical geometry shows that Rectified Linear Unit (ReLU) networks admit decision functions with a combinatorial structure often described as a tropical rational, namely a difference of tropical polynomials. Focusing on planar binary classification, we design purely sigmoidal MLPs that adhere to the finite-sum format of UAT: a finite linear combination of shifted and scaled sigmoids of affine functions. The resulting models yield decision boundaries that already align with prescribed shapes at initialization and can be refined by standard training if desired. This provides a practical bridge between the tropical perspective and smooth MLPs, enabling interpretable, shape-driven initialization without resorting to ReLU architectures. We focus on the construction and empirical demonstrations in two dimensions; theoretical analysis and higher-dimensional extensions are left for future work.

翻译：我们通过神经网络的热带几何视角重新审视万能逼近定理（UAT），并提出一种针对S型多层感知机（MLP）的构造性、几何感知初始化方法。热带几何表明，修正线性单元（ReLU）网络所对应的决策函数具有一种组合结构，常被描述为热带有理式，即两个热带多项式之差。聚焦于平面二分类问题，我们设计了纯S型多层感知机，使其严格遵循UAT的有限和形式：即仿射函数的平移缩放S型函数的有限线性组合。所得模型在初始化阶段即可产生与预设形状对齐的决策边界，并可通过常规训练进一步优化。这为热带几何视角与平滑多层感知机之间搭建了实用桥梁，使得无需依赖ReLU架构即可实现可解释、形状驱动的初始化。本文重点阐述二维场景下的构造方法与实证演示，理论分析及高维扩展将留待未来工作探讨。