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Life sciences · Preprint
arXiv · September 3, 2026
Raises a question worth testing. It does not answer one.
This is a theoretical mathematics preprint that proves spectral convergence bounds for random feature methods applied to multidimensional function approximation across several regularity classes. The work establishes that a single random feature space can simultaneously approximate all targets in a ball with rates ranging from super-exponential to algebraic, and identifies a trade-off between approximation accuracy and numerical conditioning. No empirical validation, clinical application, or peer-reviewed publication is reported.
Preprint.
Spectral convergence of RFM proved for Sobolev, Gevrey, ultra-analytic, and bandlimited function classes in multiple dimensions Approximation rates range from super-exponential to algebraic depending on target regularity Random feature matrices exhibit super-exponential singular-value decay with Fourier features and exponential decay with tanh features
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This is a theoretical mathematics preprint proving spectral convergence properties of random feature methods; it is not peer reviewed, contains no empirical validation, and raises mechanistic questions rather than answering applied or clinical ones.
Graded across the dimensions that decide whether you should act, each from what the source actually supports. There is no single score, and where a dimension was not assessed it says so.
What is missing. This record has no reported figures. That is a gap in the analysis, not a judgement about the study.
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.
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