Random Fourier Features for the RBF Kernel

Builds an n x D feature map \(\Phi\) such that \(\Phi \Phi^\top \approx K\) for the RBF kernel \(K(x, y) = \exp(-\|x-y\|^2 / (2\sigma^2))\), via Rahimi & Recht (2007): draw \(\omega_k \sim N(0, \sigma^{-2} I_d)\) and \(b_k \sim U[0, 2\pi]\), then \(\phi_k(x) = \sqrt{2/D} \cos(\omega_k^\top x + b_k)\).

Usage

rff_features(x, kernel = kernel_spec("rbf"), D = 200L, seed = NULL)

Arguments

x: Numeric matrix n x d (or vector).
kernel: A kernel_spec() with type = "rbf". Bandwidth may be "median" (resolved against x) or a positive numeric.
D: Integer. Number of random features. Larger D -> better approximation but higher memory / compute. Default 200L.
seed: Integer or NULL. Random seed.

Value

An object of class "kernel_factor" with components:

F: The n x D random-feature matrix.
method: "rff".
m: Equal to D.
kernel: Resolved kernel spec.
n: Number of rows in the input.
omega, b: Random draws (kept for reproducible re-encoding).

Details

Currently RBF-only; other shift-invariant kernels (Matern with specific nu) require their own Fourier spectra and are not yet implemented.

References

Rahimi, A., & Recht, B. (2007). Random features for large-scale kernel machines. NeurIPS, 20.

Examples

set.seed(1)
x <- matrix(stats::rnorm(2000L), ncol = 2L)
f <- rff_features(x, D = 150L, seed = 1L)
dim(f$F)
#> [1] 1000  150