# A spectral mixture representation of isotropic kernels with application to random Fourier features

arXiv: https://arxiv.org/abs/2411.02770 · HTML: https://mlsanity.com/paper/2411.02770/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 79 references that are in the map.
- First posted on arXiv 5 Nov 2024.

## Builds on (most cited first, 7 of 7)

- [Generalization Properties of Learning with Random Features](https://mlsanity.com/paper/1602.04474/) (2016)
- [Quasi-Monte Carlo Feature Maps for Shift-Invariant Kernels](https://mlsanity.com/paper/1412.8293/) (2014)
- [Optimal Rates for Random Fourier Features](https://mlsanity.com/paper/1506.02155/) (2015)
- [Randomly Projected Additive Gaussian Processes for Regression](https://mlsanity.com/paper/1912.12834/) (2019)
- [Kernel Packet: An Exact and Scalable Algorithm for Gaussian Process Regression with Mat'ern Correlations](https://mlsanity.com/paper/2203.03116/) (2022)
- [On the Beta Prime Prior for Scale Parameters in High-Dimensional Bayesian Regression Models](https://mlsanity.com/paper/1807.06539/) (2018)
- [Beyond Mat'ern: On A Class of Interpretable Confluent Hypergeometric Covariance Functions](https://mlsanity.com/paper/1911.05865/) (2019)

## Built on by (most cited first, 5 of 5)

- [Scalable Random Wavelet Features: Efficient Non-Stationary Kernel Approximation with Convergence Guarantees](https://mlsanity.com/paper/2602.00987/) (2026)
- [Fast Gaussian process inference by exact Mat'ern kernel decomposition](https://mlsanity.com/paper/2508.01864/) (2025)
- [Bernstein-Schur Kernels: Random Features by Sketched Modulation and Radial Randomization](https://mlsanity.com/paper/2606.11255/) (2026)
- [Conditional Distribution Estimation for Functional Responses with Random Forests](https://mlsanity.com/paper/2608.08247/) (2026)
- [Signed random Fourier features for fast density estimation with indefinite kernels](https://mlsanity.com/paper/2608.29265/) (2026)

## Abstract (arXiv)

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 ℝd for all d≥1 can be decomposed as a scale mixture of α-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-invariant kernels, including exponential power kernels, and generalized Cauchy kernels, as well as newly introduced kernels such as the generalized Mat'ern, Tricomi, and Fox H kernels. In particular, we retrieve the fact that the spectral distributions of these kernels, which can only be explicited in terms of the Fox H special function, are scale mixtures of the multivariate Gaussian distribution, along with an explicit mixing distribution formula. This result has broad applications for support vector machines, kernel ridge regression, Gaussian processes, and other kernel-based machine learning techniques for which the random Fourier features technique is applicable.