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This article characterizes the exact asymptotics of random Fourier feature(RFF) regression, in the realistic setting where the number of data samples$n$, their dimension $p$, and the dimension of feature space $N$ are all largeand comparable. In this regime, the random RFF Gram matrix no longer convergesto the well-known limiting Gaussian kernel matrix (as it does when $N \to\infty$ alone), but it still has a tractable behavior that is captured by ouranalysis. This analysis also provides accurate estimates of training and testregression errors for large $n,p,N$. Based on these estimates, a precisecharacterization of two qualitatively different phases of learning, includingthe phase transition between them, is provided; and the corresponding doubledescent test error curve is derived from this phase transition behavior. Theseresults do not depend on strong assumptions on the data distribution, and theyperfectly match empirical results on real-world data sets.

A Random Matrix Analysis of Random Fourier Features: Beyond the Gaussian Kernel, a Precise Phase Transition, and the Corresponding Double Descent