The universal approximation property of width-bounded networks has been studied as a dual of the classical universal approximation theorem for depth-bounded ones. There were several attempts to characterize the minimum width $w_{\min}$ enabling the universal approximation property; however, only a few of them found the exact values. In this work, we show that the minimum width for the universal approximation of $L^p$ functions from $[0,1]^{d_x}$ to $\mathbb R^{d_y}$ is exactly $\max\{d_x,d_y,2\}$ if an activation function is ReLU-Like (e.g., ReLU, GELU, Softplus). Compared to the known result $w_{\min}=\max\{d_x+1,d_y\}$ when the domain is ${\mathbb R^{d_x}}$, our result first shows that approximation on a compact domain requires smaller width than on ${\mathbb R^{d_x}}$. We next prove a lower bound on $w_{\min}$ for uniform approximation using general activation functions including ReLU: $w_{\min}\ge d_y+1$ if $d_x<d_y\le2d_x$. Together with our first result, this shows a dichotomy between $L^p$ and uniform approximations for general activation functions and input/output dimensions.

翻译：宽度受限网络的通用逼近性质，作为经典深度受限网络通用逼近定理的对偶问题被广泛研究。已有诸多工作尝试刻画实现通用逼近性质所需的最小宽度$w_{\min}$，但仅有少数研究精确确定了该数值。本文证明，当激活函数为类ReLU函数（如ReLU、GELU、Softplus）时，实现从$[0,1]^{d_x}$到$\mathbb R^{d_y}$的$L^p$函数通用逼近所需的最小宽度恰为$\max\{d_x,d_y,2\}$。相较于已知在定义域为${\mathbb R^{d_x}}$时$w_{\min}=\max\{d_x+1,d_y\}$的结果，本文首次证明紧致域上的逼近所需宽度小于${\mathbb R^{d_x}}$上的情形。进一步，针对包含ReLU在内的通用激活函数，我们证明了均匀逼近情形下$w_{\min}$的下界：当$d_x<d_y\le2d_x$时，$w_{\min}\ge d_y+1$。结合第一项结论，这揭示了对于通用激活函数及不同输入/输出维度，$L^p$逼近与均匀逼近之间存在根本性差异。