ReLU神经网络对光滑泛函算子的逼近：维度衰减与误差分析

中文摘要 AI 辅助

我们研究了在无限维可分希尔伯特空间上，通过深度ReLU神经网络对光滑标量值泛函的一致逼近问题。将泛函输入写为$X(t)=\sum_{d\geq1}\xi_d\nu_d(t)$，我们通过$w_ds_d$量化坐标$d$的重要性，其中$s_d$界定相应基得分的量级，$w_d$控制目标泛函的方向Fréchet敏感性。我们的构造性分析结合了坐标截断、各向异性划分、局部泰勒逼近和ReLU网络实现，同时允许保留坐标之间的无限制交互。我们建立了均匀逼近误差的一般非渐近上界，以及基于伪维数的互补的最坏情况逼近误差下界。在广义指数坐标衰减$w_ds_d\asymp\exp(-cd^\rho)$（其中$\rho>0$）下，上界和下界在前导阶上匹配，从而产生近乎最优的逼近速率，该速率在网络预算的对数上是拉伸指数的。这是第一项通过坐标幅度和方向敏感性的联合维度衰减来显式刻画无限维泛函输入的神经网络逼近误差的工作。

英文摘要

We study the uniform approximation of smooth scalar-valued functionals on an infinite-dimensional separable Hilbert space by ReLU neural networks. A key feature in deep learning for functional data is the varying importance of different coordinates/dimensions. Representing the functional input in a basis expansion, we quantify the importance of each coordinate through both the magnitude of its corresponding basis score and the directional sensitivity of the target functional. Our analysis combines coordinate truncation, anisotropic partitioning, local Taylor approximation, and ReLU network realization, while allowing unrestricted interactions among the retained coordinates. We establish a general nonasymptotic upper bound for the uniform approximation error and a complementary pseudo-dimension-based lower bound for the worst-case approximation error. Under generalized exponential coordinate decay $w_ds_d\asymp\exp(-cd^ρ)$, with $ρ>0$, the upper and lower bounds match at the leading order, which is stretched-exponential in the logarithm of the network size budget, and thus yield the nearly optimal approximation rate. This is the first work to characterize neural network approximation error for infinite-dimensional functional inputs explicitly through the joint dimensional decay of coordinate magnitudes and directional sensitivities.

发表机构

• City University of Hong Kong（香港城市大学）

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