Spectral Convergence of Random Feature Method in Multiple Dimensions
Abstract
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.
Related
- A hyperbolic nonlinear Schrödinger system in multiple spatial dimensions: structure, convergence, and numerical discretization
- Spectral Analysis of Gram Matrices with Missing at Random Observations: Convergence, Central Limit Theorems, and Applications in Statistical Inference
- Impact of Feature Scaling on the Convergence Speed of Linear SVM Solvers : An Empirical Benchmark
- Instance-Specific Sharp Convergence Rate of the Add-λ Estimator via Feature Space Decomposition
- Sharp convergence rate of the estimation error of LASSO via the Feature Space Decomposition method
BibTeX
@misc{ming2026spectral,
title = {{Spectral Convergence of Random Feature Method in Multiple Dimensions}},
author = {Pingbing Ming and Hao Yu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03401},
eprint = {2609.03401},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03401},
}
RIS
TY - UNPB
TI - Spectral Convergence of Random Feature Method in Multiple Dimensions
AU - Pingbing Ming
AU - Hao Yu
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.03401
UR - https://arxiv.org/abs/2609.03401
AB - 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.
ER -