We prove sharp upper and lower bounds for the approximation of Sobolev functions by sums of multivariate ridge functions, i.e., functions of the form $\mathbb{R}^d \ni x \mapsto \sum_{k=1}^n h_k(A_k x) \in \mathbb{R}$ with $h_k : \mathbb{R}^\ell \to \mathbb{R}$ and $A_k \in \mathbb{R}^{\ell \times d}$. We show that the order of approximation asymptotically behaves as $n^{-r/(d-\ell)}$, where $r$ is the regularity of the Sobolev functions to be approximated. Our lower bound even holds when approximating $L^\infty$-Sobolev functions of regularity $r$ with error measured in $L^1$, while our upper bound applies to the approximation of $L^p$-Sobolev functions in $L^p$ for any $1 \leq p \leq \infty$. These bounds generalize well-known results about the approximation properties of univariate ridge functions to the multivariate case. Moreover, we use these bounds to obtain sharp asymptotic bounds for the approximation of Sobolev functions using generalized translation networks and complex-valued neural networks.

翻译：我们证明了用多元脊函数之和逼近Sobolev函数时尖锐的上下界，其中脊函数形式为$\mathbb{R}^d \ni x \mapsto \sum_{k=1}^n h_k(A_k x) \in \mathbb{R}$，且满足$h_k : \mathbb{R}^\ell \to \mathbb{R}$和$A_k \in \mathbb{R}^{\ell \times d}$。我们证明逼近阶的渐近行为为$n^{-r/(d-\ell)}$，其中$r$为待逼近Sobolev函数的正则性。我们的下界结果甚至在以$L^1$误差度量逼近正则性为$r$的$L^\infty$-Sobolev函数时依然成立，而上界结果适用于任意$1 \leq p \leq \infty$时在$L^p$空间中逼近$L^p$-Sobolev函数。这些界限将关于单变量脊函数逼近性质的经典结果推广到多元情形。此外，我们利用这些界限获得了使用广义平移网络和复值神经网络逼近Sobolev函数时的尖锐渐近界。