无限维空间上的分辨率一致贪心神经近似
Resolution-Consistent Greedy Neural Approximation on Infinite-Dimensional Spaces
- Universidad Tecnológica Atlántico Mediterráneo – UTAMED（大西洋地中海科技大学（UTAMED））
- Universidad Cardenal Herrera-CEU, CEU Universities（红衣主教埃雷拉-CEU大学（CEU大学））
- CUNEF Universidad（库内夫大学）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
该研究针对含有限坐标观测的无限维输入浅层神经模型，构建了构造性近似与学习保证，通过完全修正贪心方法实现经验回归的统计复杂度与输入分辨率一致，并经合成实验验证了相关参数范围。
中文摘要 AI 辅助
我们针对通过有限个坐标观测到的具有无限维输入的浅层神经模型，构建了构造性近似与学习保证。该分析基于参数归一化神经字典及其关联的加权变差类，在此类中，近似误差可分解为依赖于分布的坐标截断项与贪心有限宽度项。对于经验回归，完全修正贪心过程可产生与保留输入分辨率统计复杂度一致的总体保证；该框架可扩展到希尔伯特值响应，且不明确依赖输出维度。该无维度表述是统计层面而非计算层面的：选择新神经元仍需求解非凸参数搜索问题。近期无限维通用近似结果所基于的拟波兰构造提供了一个启发性示例，合成实验验证了预测的分辨率、宽度及样本量范围。
英文摘要
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.
