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 large
and comparable. In this regime, the random RFF Gram matrix no longer converges
to 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 our
analysis. This analysis also provides accurate estimates of training and test
regression errors for large $n,p,N$. Based on these estimates, a precise
characterization of two qualitatively different phases of learning, including
the phase transition between them, is provided; and the corresponding double
descent test error curve is derived from this phase transition behavior. These
results do not depend on strong assumptions on the data distribution, and they
perfectly match empirical results on real-world data sets.

本文讨论基于随机傅里叶核（RFF）的回归模型的精确渐进特征，研究表明在数据样本数、数据维度和特征空间维度等三个因素中为大且可比较的实际场景下，随机 RFF Gram 矩阵不再收敛于著名的极限高斯核矩阵而是有一个可处理的行为，双重下降测试误差曲线从这种相变行为中得出，该结果不依赖于数据分布的强假设。