# Spectral Convergence of Random Feature Method in Multiple Dimensions

arXiv: https://arxiv.org/abs/2609.03401 · HTML: https://mlsanity.com/paper/2609.03401/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 81 references that are in the map.
- First posted on arXiv 3 Sep 2026.

## Builds on (most cited first, 20 of 20)

- [Generalization Properties of Learning with Random Features](https://mlsanity.com/paper/1602.04474/) (2016)
- [On the Equivalence between Kernel Quadrature Rules and Random Feature Expansions](https://mlsanity.com/paper/1502.06800/) (2015)
- [Towards a Mathematical Understanding of Neural Network-Based Machine Learning: what we know and what we don't](https://mlsanity.com/paper/2009.10713/) (2020)
- [The Random Feature Model for Input-Output Maps between Banach Spaces](https://mlsanity.com/paper/2005.10224/) (2020)
- [Local Extreme Learning Machines and Domain Decomposition for Solving Linear and Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2012.02895/) (2020)
- [On the approximation of functions by tanh neural networks](https://mlsanity.com/paper/2104.08938/) (2021)
- [Approximation Bounds for Random Neural Networks and Reservoir Systems](https://mlsanity.com/paper/2002.05933/) (2020)
- [On Fast Leverage Score Sampling and Optimal Learning](https://mlsanity.com/paper/1810.13258/) (2018)
- [Exponential Convergence of the Deep Neural Network Approximation for Analytic Functions](https://mlsanity.com/paper/1807.00297/) (2018)
- [Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality](https://mlsanity.com/paper/2106.08900/) (2021)
- [Deep Randomized Neural Networks](https://mlsanity.com/paper/2002.12287/) (2020)
- [Operator Learning Using Random Features: A Tool for Scientific Computing](https://mlsanity.com/paper/2408.06526/) (2024)
- [Fast Statistical Leverage Score Approximation in Kernel Ridge Regression](https://mlsanity.com/paper/2103.05238/) (2021)
- [Full error analysis of the random deep splitting method for nonlinear parabolic PDEs and PIDEs](https://mlsanity.com/paper/2405.05192/) (2024)
- [Random Projection Neural Networks of Best Approximation: Convergence theory and practical applications](https://mlsanity.com/paper/2402.11397/) (2024)
- [Concentration of Random Feature Matrices in High-Dimensions](https://mlsanity.com/paper/2204.06935/) (2022)
- [Random Fourier Features via Fast Surrogate Leverage Weighted Sampling](https://mlsanity.com/paper/1911.09158/) (2019)
- [Optimal Rates and Saturation for Noiseless Kernel Ridge Regression](https://mlsanity.com/paper/2402.15718/) (2024)
- [Trace Regularity PINNs: Enforcing H^frac12(partial Omega) for Boundary Data](https://mlsanity.com/paper/2510.16817/) (2025)
- [A Structure-Adaptive Random Feature Method for High-Dimensional Elliptic PDEs](https://mlsanity.com/paper/2607.19786/) (2026)

## Abstract (arXiv)

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.