提出无限维输入下神经网络的统一近似理论，实现分辨率无关的误差控制。
Resolution-Consistent Greedy Neural Approximation on Infinite-Dimensional Spaces
- 基于参数归一化字典与加权变差类，构建可解释的近似框架
- 在有限样本下，误差随宽度增长呈可控下降，且不依赖输入分辨率
- 适用于高维输出场景，适合研究泛化性与可扩展性的研究人员
我们为具有无限维输入但仅观测有限坐标的浅层神经网络建立了构造性近似与学习保证。分析基于参数归一化的神经字典及其对应的加权变差类。在此类中，近似误差分解为依赖分布的坐标截断项和贪婪有限宽度项。对于经验回归，一种全修正贪婪算法在总体上提供保证，其统计复杂度在保留的输入分辨率上保持一致。该框架可推广至希尔伯特空间值响应，且不显式依赖输出维度。这些无维度陈述是统计性质的，而非计算性质：选择新神经元仍需解决非凸参数搜索问题。最近的无限维通用逼近结果所依赖的准波兰构造提供了动机，合成实验验证了预测的分辨率、宽度与样本量范围。
原文摘要 · Abstract (English)
We develop constructive approximation and learning guarantees for shallow neural models with infinite-dimensional inputs observed through finitely many coordinates. The analysis is based on a parameter-normalized neural dictionary and its associated weighted variation class. Within this class, the approximation error separates into a distribution-dependent coordinate-truncation term and a greedy finite-width term. For empirical regression, a fully-corrective greedy procedure yields population guarantees whose statistical complexity is uniform in the retained input resolution. The same framework extends to Hilbert-valued responses without an explicit dependence on the output dimension. The dimension-free statements are statistical, not computational: selecting a new neuron still requires solving a nonconvex parameter-search problem. The quasi-Polish construction underlying recent infinite-dimensional universal approximation results provides a motivating example, and synthetic experiments illustrate the predicted resolution, width, and sample-size regimes.
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