统一分析随机特征与神经算子的泛化性能，揭示学习速率与所需神经元数量的关系。
Random Features for Operator-Valued Kernels: Bridging Kernel Methods and Neural Operators
- 提出覆盖多种正则化方法的统一理论框架
- 证明最优学习速率，并确定达到精度所需的最小神经元数
- 适用于目标不在希尔伯特空间的复杂场景，适合理论研究者
本文研究随机特征方法的泛化性质。分析将早期针对Tikhonov正则化的结果扩展到广泛的谱正则化技术，并进一步推广至算子值核设置。该统一框架使我们能够通过神经切线核（NTK）视角，对神经算子和神经网络进行严格理论分析。特别地，该框架可建立最优学习速率，并清晰阐明达到给定精度所需神经元数量。此外，我们在设定正确和设定错误两种情况下均建立了极小极大速率，完善并强化了以往特定核算法的研究结论。
原文摘要 · Abstract (English)
In this work, we investigate the generalization properties of random feature methods. Our analysis extends prior results for Tikhonov regularization to a broad class of spectral regularization techniques and further generalizes the setting to operator-valued kernels. This unified framework enables a rigorous theoretical analysis of neural operators and neural networks through the lens of the Neural Tangent Kernel (NTK). In particular, it allows us to establish optimal learning rates and provides a good understanding of how many neurons are required to achieve a given accuracy. Furthermore, we establish minimax rates in the well-specified case and also in the misspecified case, where the target is not contained in the reproducing kernel Hilbert space. These results sharpen and complete earlier findings for specific kernel algorithms.
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