深度ReLU神经网络高维逼近的计算复杂性
Computation complexity of deep ReLU neural networks in high-dimensional approximation
摘要：
本文的目的是研究深度ReLU神经网络在单位立方体$\mathbb{I}^d:=[0,1]^d$上的混合光滑$H_\infty^α(\mathbb{I}^d)$的Hölder-Nikol'skii空间中逼近函数的计算复杂度。在这种情况下，对于任何函数$f\in H_\infty^α(\mathbb{I}^d)$，我们显式地构建了非自适应和自适应深度ReLU神经网络，其输出以规定的精度$\varepsilon$近似$f$，并证明了这种近似的计算复杂性的维度相关界限，其特征在于深度ReLU神经网络的大小和深度，显式地在$d$和$\varepsilon$中。我们的结果显示了深度ReLU神经网络的自适应近似方法优于非自适应方法。
英文摘要：
The purpose of the present paper is to study the computation complexity of deep ReLU neural networks to approximate functions in Hölder-Nikol'skii spaces of mixed smoothness $H_\infty^α(\mathbb{I}^d)$ on the unit cube $\mathbb{I}^d:=[0,1]^d$. In this context, for any function $f\in H_\infty^α(\mathbb{I}^d)$, we explicitly construct nonadaptive and adaptive deep ReLU neural networks having an output that approximates $f$ with a prescribed accuracy $\varepsilon$, and prove dimension-dependent bounds for the computation complexity of this approximation, characterized by the size and the depth of this deep ReLU neural network, explicitly in $d$ and $\varepsilon$. Our results show the advantage of the adaptive method of approximation by deep ReLU neural networks over nonadaptive one.
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