Fourier sparse leverage scores and approximate kernel learning

Tamás Erdélyi, Cameron Musco, Christopher Musco

    Research output: Contribution to journalConference articlepeer-review


    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) = Psj=1 ajeiλjx for coefficients aj ∈ C and frequencies λj ∈ 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.

    Original languageEnglish (US)
    JournalAdvances in Neural Information Processing Systems
    StatePublished - 2020
    Event34th Conference on Neural Information Processing Systems, NeurIPS 2020 - Virtual, Online
    Duration: Dec 6 2020Dec 12 2020

    ASJC Scopus subject areas

    • Computer Networks and Communications
    • Information Systems
    • Signal Processing


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