  We show that the error probability of reconstructing kernel matrices from
Random Fourier Features for the Gaussian kernel function is at most
$\mathcal{O}(R^{2/3} \exp(-D))$, where $D$ is the number of random features and
$R$ is the diameter of the data domain. We also provide an
information-theoretic method-independent lower bound of $\Omega((1-\exp(-R^2))
\exp(-D))$. 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 applications of our theory, we obtain dimension-independent
bounds for kernel ridge regression and support vector machines.
