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.

翻译：我们显示,从随机 Fourier 特性中重建高山内核功能的内核矩阵的误差概率最高为$\ mathcal{O}(R ⁇ 2/3}\ frem(-D)$),其中$D是随机特性的数量,$R$是数据域的直径。我们还提供了一个信息-理论方法(独立)下限$Omega((1-\ exp(-R ⁇ 2)))\ exp(-D))$。与先前的工作相比,我们是第一个显示随机Fourier特性的误差概率独立于数据点的维度。作为我们理论的应用,我们获得了内核脊回归和支持矢量机器的维独立边框。