Random-Feature Precision for Weak Diffusion Residuals

Random Fourier features can reduce the cost of kernel diagnostics, but their approximation error must be assessed together with the noise in the observations. This paper establishes sharp frequency budgets for a signed Gaussian-kernel U-statistic of weak diffusion residuals. The analysis considers a bounded class of scalar periodic velocity loops whose own endpoint law is Gaussian. For $n$ observations and unknown squared RKHS residual norm $q={\mathrm{WSD}}^2\ge b/n$, controlling additional mean-square error at level ${\epsilon }^2{q}^2$ uniformly over this class requires $\Theta (\sqrt{\log n}/{\epsilon }^2)$ frequencies. Preserving the exact-kernel estimator's risk within a factor of $1+{\epsilon }^2$ instead requires $\Theta (n/{\epsilon }^2)$. Both budgets are optimal over admissible iid importance-sampling laws shared across the signal class, and are attained by defensive Gaussian mixtures. The key analytical ingredient is a uniform data-risk profile that tracks signal-dependent variance and the interpolation-noise floor. Together with control of the data–frequency interaction, this profile yields a sharp Gaussian proposal-width threshold. Below it, intermediate signals determine the worst-case budget, which a boundary-only analysis misses. Population-risk calculations and noisy-statistic simulations examine the predicted budget gap and difficult signal scales on finite panels. At fixed tolerance, the two precision criteria lead to nearly linear and quadratic scalar evaluation costs, respectively.