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[2411.02770] A spectral mixture representation of isotropic kernels with application to random Fourier features
Summary
This paper presents a spectral mixture representation of isotropic kernels, enhancing the application of Random Fourier Features (RFF) in machine learning. It introduces a decomposition method applicable to various multivariate kernels, expanding the usability of RFF beyond th...
Why It Matters
The findings provide a significant advancement in machine learning by offering a broader framework for applying Random Fourier Features to various kernels. This can improve the efficiency and effectiveness of kernel-based methods, which are crucial in many machine learning applications such as support vector machines and Gaussian processes.
Key Takeaways
- Introduces a new spectral mixture representation for isotropic kernels.
- Expands the application of Random Fourier Features beyond Gaussian kernels.
- Provides a simple spectral sampling formula for various multivariate kernels.
- Identifies mixing distributions as a function of the kernel, enhancing understanding.
- Implications for improving kernel-based machine learning techniques.
Computer Science > Machine Learning arXiv:2411.02770 (cs) [Submitted on 5 Nov 2024 (v1), last revised 22 Feb 2026 (this version, v4)] Title:A spectral mixture representation of isotropic kernels with application to random Fourier features Authors:Nicolas Langrené, Xavier Warin, Pierre Gruet View a PDF of the paper titled A spectral mixture representation of isotropic kernels with application to random Fourier features, by Nicolas Langren\'e and 2 other authors View PDF Abstract:Rahimi and Recht (2007) introduced the idea of decomposing positive definite shift-invariant kernels by randomly sampling from their spectral distribution for machine learning applications. This famous technique, known as Random Fourier Features (RFF), is in principle applicable to any such kernel whose spectral distribution can be identified and simulated. In practice, however, it is usually applied to the Gaussian kernel because of its simplicity, since its spectral distribution is also Gaussian. Clearly, simple spectral sampling formulas would be desirable for broader classes of kernels. In this paper, we show that the spectral distribution of positive definite isotropic kernels in $\mathbb{R}^{d}$ for all $d\geq1$ can be decomposed as a scale mixture of $\alpha$-stable random vectors, and we identify the mixing distribution as a function of the kernel. This constructive decomposition provides a simple and ready-to-use spectral sampling formula for many multivariate positive definite shift-invari...
