A new network with super-approximation power is introduced. This network is built with Floor ($\lfloor x\rfloor$) or ReLU ($max\{0,x\}$) activation function in each neuron; hence, we call such networks Floor-ReLU networks. For any hyperparameters $N\in {\mathbb{N}}^{+}$ and $L\in {\mathbb{N}}^{+}$, we show that Floor-ReLU networks with width $max\{d,5N+13\}$ and depth $64dL+3$ can uniformly approximate a Hölder function $f$ on ${[0,1]}^d$ with an approximation error $3\lambda d^{\alpha /2}{N}^{-\alpha \sqrt{L}}$, where $\alpha \in (0,1]$ and $\lambda$ are the Hölder order and constant, respectively. More generally for an arbitrary continuous function $f$ on ${[0,1]}^d$ with a modulus of continuity ${\omega }_f(·)$, the constructive approximation rate is ${\omega }_f\left(\sqrt{d}{N}^{-\sqrt{L}}\right)+2{\omega }_f\left(\sqrt{d}\right)N^{-\sqrt{L}}$. As a consequence, this new class of networks overcomes the curse of dimensionality in approximation power when the variation of ${\omega }_f\left(r\right)$ as $r\to 0$ is moderate (e.g., ${\omega }_f\left(r\right)\lesssim r^{\alpha }$ for Hölder continuous functions), since the major term to be considered in our approximation rate is essentially $\sqrt{d}$ times a function of $N$ and $L$ independent of $d$ within the modulus of continuity.

介绍了一种具有超逼近能力的新网络。这个网络是用 Floor ($\lfloor X\rfloor$) 或 ReLU ($最大限度\{0,X\}$) 每个神经元的激活函数；因此，我们称这种网络为 Floor-ReLU 网络。对于任何超参数$N\in {\mathbb{N}}^{+}$ 和 $升\in {\mathbb{N}}^{+}$，我们展示了具有宽度的 Floor-ReLU 网络 $最大限度\{d,5N+13\}$ 和深度 $64d升+3$ 可以一致地逼近一个 Hölder 函数 $F$ 在 ${[0,1]}^d$ 有近似误差 $3\lambda d^{\alpha /2}{N}^{——\alpha \sqrt{升}}$， 在哪里 $\alpha \in (0,1]$ 和 $\lambda$分别是 Hölder 阶和常数。更一般地用于任意连续函数$F$ 在 ${[0,1]}^d$ 具有连续性模数 ${\omega }_F(·)$，建设性逼近率为 ${\omega }_F\left(\sqrt{d}{N}^{——\sqrt{升}}\right)+2{\omega }_F\left(\sqrt{d}\right)N^{——\sqrt{升}}$. 因此，当${\omega }_F\left(r\right)$ 作为 $r\to 0$ 是中等的（例如， ${\omega }_F\left(r\right)\lesssim r^{\alpha }$ 对于 Hölder 连续函数），因为在我们的近似率中要考虑的主要项本质上是 $\sqrt{d}$ 次函数 $N$ 和 $升$ 独立于 $d$ 在连续性模数内。