用热带几何设计可解释的双曲正切多层感知机初始化，让决策边界从一开始就匹配预设形状。
From Universal Approximation Theorem to Tropical Geometry of Multi-Layer Perceptrons
- 基于热带几何构造平面上的纯双曲正切网络，满足泛化逼近定理的有限求和形式
- 初始化时决策边界即贴合指定形状，训练后可进一步优化
- 无需使用ReLU，实现形状驱动的可解释初始化，适合需要可控决策边界的场景
我们通过神经网络的热带几何视角重新审视泛化逼近定理（UAT），为双曲正切多层感知机（MLP）提出一种几何感知的可构造初始化方法。热带几何表明，修正线性单元（ReLU）网络的决策函数具有组合结构，常被描述为热带有理函数——即两个热带多项式的差。聚焦于二维平面二分类问题，我们设计了完全由双曲正切构成的MLP，其形式符合UAT的有限求和结构：有限个平移与缩放后的仿射函数双曲正切之线性组合。这些模型在初始化阶段即能生成与预设形状一致的决策边界，若需进一步优化，可通过标准训练实现。该方法在平面上实现了热带视角与光滑网络之间的实用桥梁，无需采用ReLU架构即可实现可解释、形状驱动的初始化。本文重点在二维空间中进行构造与实证演示；高维理论分析与扩展留待未来工作。
原文摘要 · Abstract (English)
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.
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