揭示多变量岭函数逼近的最优误差规律，为神经网络设计提供理论支撑。
On best approximation by multivariate ridge functions with applications to generalized translation networks
- 基于多变量岭函数构建逼近模型，利用线性投影与非线性激活组合
- 逼近误差随参数数n增长呈n^{-r/(d-ℓ)}衰减，精度由函数光滑度决定
- 适用于广义平移网络和复值神经网络，理论指导深度学习架构设计
本文证明了用多元岭函数（形式为x↦∑ₖϱₖ(Aₖx)）逼近Sobolev函数时的严格上下界。当被逼近函数具有r阶可微性时，逼近误差渐近行为为n^{-r/(d-ℓ)}，其中d为输入维度，ℓ为投影后维度。下界在L¹误差下对L∞-Sobolev函数仍成立，上界适用于任意1≤p≤∞下的Lᵖ-Sobolev函数逼近。该结果推广了单变量岭函数的经典结论。进一步，将此结果应用于广义平移网络和复值神经网络，获得其逼近的精确渐近界。
原文摘要 · Abstract (English)
In this paper, we prove sharp upper and lower bounds for the approximation of Sobolev functions by sums of multivariate ridge functions, i.e., for approximation by functions of the form $\mathbb{R}^d \ni x \mapsto \sum_{k=1}^n \varrho_k(A_k x) \in \mathbb{R}$ with $\varrho_k : \mathbb{R}^\ell \to \mathbb{R}$ 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 (order of differentiability) 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 regarding the approximation properties of univariate ridge functions to the multivariate case. We use our results to obtain sharp asymptotic bounds for the approximation of Sobolev functions using generalized translation networks and complex-valued neural networks.
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