The Error Probability of Random Fourier Features is Dimensionality Independent

We show that the error probability of reconstructing kernel matrices from Random Fourier Features for any shift-invariant kernel function is at most O(exp(−D)), where D is the number of random features. We also provide a matching informationtheoretic method-independent lower bound of Ω(exp(−D)) for standard Gaussian distributions. Compared to prior work, we are the first to show that the error probability for random Fourier features is independent of the dimensionality of data points as well as the size of their domain. As applications of our theory, we obtain dimension-independent bounds for kernel ridge regression and support vector machines.