光滑组合函数在低内在维度集上的最优神经网络近似
Optimal neural network approximation of smooth compositional functions on sets with low intrinsic dimension
AI总结：
本文提出了一种在低内在维度集上近似光滑组合函数的深度神经网络方法，通过结构假设和组合模型统一了目标函数和协变量的约束，实现了更高效的近似速率和收敛性分析。
摘要：
我们研究了在结构假设下深度ReLU网络的近似和统计学习性质，以缓解维度灾难。我们证明了在低Minkowski维度集上定义的s-霍尔德光滑函数的最小最大均匀近似速率，使用具有灵活宽度和深度的全连接网络，即使在经典全维度设置中，也通过logarithmic因子改进了现有结果。关键的技术成分是深度ReLU网络的新记忆结果，它使密集架构能够高效地进行点拟合。我们进一步引入了一类组合模型，其中每个组件函数都是光滑的，并且作用于低内在维度的域。该框架将统计学习文献中两种常见的假设——目标函数的结构约束和协变量的低维性——统一到一个模型中。我们证明深度网络可以以由组合中最难函数决定的速率近似这些函数。作为应用，我们推导了非参数回归经验风险最小化改进的收敛速率，能够适应光滑性、组合结构和内在维度性。
英文摘要：
We study approximation and statistical learning properties of deep ReLU networks under structural assumptions that mitigate the curse of dimensionality. We prove minimax-optimal uniform approximation rates for $s$-Hölder smooth functions defined on sets with low Minkowski dimension using fully connected networks with flexible width and depth, improving existing results by logarithmic factors even in classical full-dimensional settings. A key technical ingredient is a new memorization result for deep ReLU networks that enables efficient point fitting with dense architectures. We further introduce a class of compositional models in which each component function is smooth and acts on a domain of low intrinsic dimension. This framework unifies two common assumptions in the statistical learning literature, structural constraints on the target function and low dimensionality of the covariates, within a single model. We show that deep networks can approximate such functions at rates determined by the most difficult function in the composition. As an application, we derive improved convergence rates for empirical risk minimization in nonparametric regression that adapt to smoothness, compositional structure, and intrinsic dimensionality.
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