输入直连的深层神经网络可逼近任意连续函数，仅需非线性激活函数。
Universal Approximation Theorem for Input-Connected Multilayer Perceptrons
- 每层隐藏单元直接连接原始输入，增强信息传递路径。
- 证明了只要激活函数非线性，深度结构就能逼近任意连续函数。
- 适用于一维和多维输入，理论严谨且具通用性。
我们提出输入直连多层感知机（IC-MLP），一种前馈神经网络架构，其中每个隐藏单元除接收前一层输出外，还通过仿射变换直接连接原始输入。首先在单变量情形下研究该架构，给出任意有限层数的IC-MLP的显式系统描述，包括网络函数的迭代公式。在此设定下，证明了通用逼近定理：若激活函数为非线性，则深度IC-MLP可逼近实数轴闭区间上的任意连续函数。随后将分析拓展至向量值输入，建立了在ℝⁿ紧子集上连续函数的相应通用逼近定理。
原文摘要 · Abstract (English)
We present the Input-Connected Multilayer Perceptron (IC-MLP), a feedforward neural network architecture in which each hidden neuron receives, in addition to the outputs of the preceding layer, a direct affine connection from the raw input. We first study this architecture in the univariate setting and give an explicit and systematic description of IC-MLPs with an arbitrary finite number of hidden layers, including iterated formulas for the network functions. In this setting, we prove a universal approximation theorem showing that deep IC-MLPs can approximate any continuous function on a closed interval of the real line if and only if the activation function is nonlinear. We then extend the analysis to vector-valued inputs and establish a corresponding universal approximation theorem for continuous functions on compact subsets of $\mathbb{R}^n$.
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