Fourier Sparse Leverage Scores and Approximate Kernel Learning

We prove new explicit upper bounds on the leverage scores of Fourier sparse functions under both the Gaussian and Laplace measures. Bounding Fourier sparse leverage scores under various measures is of pure mathematical interest in approximation theory, and our work extends existing results for the uniform measure [Erd17,CP19a]. Practically, our bounds are motivated by two important applications in machine learning: 1. Kernel Approximation. They yield a new random Fourier features algorithm for approximating Gaussian and Cauchy (rational quadratic) kernel matrices. For low-dimensional data, our method uses a near optimal number of features, and its runtime is polynomial in the *statistical dimension* of the approximated kernel matrix.