On the design of scalable, high-precision spherical-radial Fourier features
<|reference_start|>On the design of scalable, high-precision spherical-radial Fourier features: Approximation using Fourier features is a popular technique for scaling kernel methods to large-scale problems, with myriad applications in machine learning and statistics. This method replaces the integral representation of a shift-invariant kernel with a sum using a quadrature rule. The design of the latter is meant to reduce the number of features required for high-precision approximation. Specifically, for the squared exponential kernel, one must design a quadrature rule that approximates the Gaussian measure on $\mathbb{R}^d$. Previous efforts in this line of research have faced difficulties in higher dimensions. We introduce a new family of quadrature rules that accurately approximate the Gaussian measure in higher dimensions by exploiting its isotropy. These rules are constructed as a tensor product of a radial quadrature rule and a spherical quadrature rule. Compared to previous work, our approach leverages a thorough analysis of the approximation error, which suggests natural choices for both the radial and spherical components. We demonstrate that this family of Fourier features yields improved approximation bounds.<|reference_end|>
@article{belhadji2024on,
    title={On the design of scalable, high-precision spherical-radial Fourier
  features},
    author={Ayoub Belhadji, Qianyu Julie Zhu, Youssef Marzouk},
    journal={arXiv preprint arXiv:2408.13231},
    year={2024},
    archivePrefix={arXiv},
    eprint={2408.13231},
    primaryClass={stat.ML cs.LG}
}