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This paper establishes spectral convergence of the random feature method (RFM) for multidimensional targets across various function classes, including Sobolev and bandlimited functions. The authors derive high-probability approximation estimates and demonstrate that a single random space can approximate every target within a specified source ball, achieving spectral accuracy across all admissible error norms. Additionally, they provide error estimates for RFM discretizations in multidimensional elliptic boundary value problems, revealing a trade-off between high accuracy and severe ill-conditioning due to spectral approximation.
A single random feature space can achieve high spectral accuracy for multidimensional targets while exposing a critical trade-off with ill-conditioning.
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.
