发表机构
Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences; School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院数学与系统科学研究院计算数学与科学工程计算研究所; 中国科学院大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究证明了高维随机特征方法的谱收敛性，推导了其离散化误差估计与奇异值衰减特性，揭示了谱逼近精度与病态性的关联机制。
AI 中文摘要
我们首先证明了随机特征方法（RFM）对索伯列夫、格雷夫里、超解析和带限类中的高维目标函数具有谱收敛性。该分析建立了由核积分算子生成的插值尺度下的一般高概率逼近估计。在仅由采样特征决定的单个事件上，一个随机空间可逼近规定源球中的所有目标；此外，对每个目标，单个系数向量定义了一个逼近式，其在所有容许误差范数中同时达到谱精度。对于适配正则性的频率分布和增长频率窗口上的均匀分布，所得收敛速率从超指数到代数不等，具体取决于目标的正则性。其次，我们为强形式和弱形式的RFM离散化建立了抽象误差估计，从而将前述逼近界转化为高维二阶椭圆边值问题和特征值问题的收敛估计。最后，对于随机特征矩阵（RFMtxs），我们证明其在采用傅里叶特征时具有超指数奇异值衰减，在采用tanh特征时具有指数衰减，同时给出了相应的条件数下界。该分析揭示了一个共同机制：产生高精度的相同谱逼近也会导致严重的病态性。
英文摘要
We first prove spectral convergence of the random feature method (RFM) for multidimensional targets in Sobolev, Gevrey, ultra-analytic, and bandlimited classes. The analysis establishes general high-probability approximation estimates in the interpolation scale generated by a kernel integral operator. On a single event determined only by the sampled features, one random space approximates every target in a prescribed source ball; moreover, for each target, a single coefficient vector defines an approximant that attains spectral accuracy simultaneously in all admissible error norms. For both regularity-adapted frequency distributions and uniform distributions on growing frequency windows, the resulting rates range from super-exponential to algebraic, depending on the regularity of the target. Second, we establish abstract error estimates for strong- and weak-form RFM discretizations, thereby converting the preceding approximation bounds into convergence estimates for multidimensional second-order elliptic boundary value and eigenvalue problems. Finally, for random feature matrices (RFMtxs), we prove super-exponential singular-value decay with Fourier features and exponential decay with $\tanh$ features, together with corresponding condition-number lower bounds. The analysis identifies a common mechanism: the same spectral approximation that yields high accuracy also drives severe ill-conditioning.
Comments48 pages, 1 figure, 2 tables
