We prove new explicit upper bounds on the leverage scores of Fourier sparse functions under both the Gaussian and Laplace measures. In particular, we study $s$-sparse functions of the form $f(x) = sum_{j=1}^s a_j e^{i lambda_j x}$ for coefficients $a_j in mathbb{C}$ and frequencies $lambda_j in mathbb{R}$. 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. It is the first oblivious sketching method with this property for any kernel besides the polynomial kernel, resolving an open question of [AKM+17,AKK+20b]. 2. Active Learning. They can be used as non-uniform sampling distributions for robust active learning when data follows a Gaussian or Laplace distribution. Using the framework of [AKM+19], we provide essentially optimal results for bandlimited and multiband interpolation, and Gaussian process regression. These results generalize existing work that only applies to uniformly distributed data.
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