The Error Probability of Random Fourier Features is Dimensionality Independent

TL;DR

This paper proves that the error probability in reconstructing Gaussian kernel matrices using Random Fourier Features does not depend on data dimensionality, providing new bounds and implications for kernel methods.

Contribution

It introduces the first dimension-independent error probability bounds for Random Fourier Features with Gaussian kernels, advancing theoretical understanding.

Findings

Error probability is at most (R^{2/3} \, ext{exp}(-D))

Provides a lower bound of ((1- ext{exp}(-R^2)) \, ext{exp}(-D))

Dimension-independent bounds for kernel ridge regression and SVMs

Abstract

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.