# Scalable L'evy Process Priors for Spectral Kernel Learning

arXiv: https://arxiv.org/abs/1802.00530 · HTML: https://mlsanity.com/paper/1802.00530/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Feb 2018.

## Built on by (most cited first, 7 of 7)

- [Deep neural networks with dependent weights: Gaussian Process mixture limit, heavy tails, sparsity and compressibility](https://mlsanity.com/paper/2205.08187/) (2022)
- [Recent Advances in Data-Driven Wireless Communication Using Gaussian Processes: A Comprehensive Survey](https://mlsanity.com/paper/2103.10134/) (2021)
- [Marginalising over Stationary Kernels with Bayesian Quadrature](https://mlsanity.com/paper/2106.07452/) (2021)
- [Modelling Non-Smooth Signals with Complex Spectral Structure](https://mlsanity.com/paper/2203.06997/) (2022)
- [Non-Gaussian Process Regression](https://mlsanity.com/paper/2209.03117/) (2022)
- [Shallow and Deep Nonparametric Convolutions for Gaussian Processes](https://mlsanity.com/paper/2206.08972/) (2022)
- [ALAS: Additive Learnable Alpha-Stable Kernels for Flexible Bayesian Optimization](https://mlsanity.com/paper/2607.18282/) (2026)

## Abstract (arXiv)

Gaussian processes are rich distributions over functions, with generalization properties determined by a kernel function. When used for long-range extrapolation, predictions are particularly sensitive to the choice of kernel parameters. It is therefore critical to account for kernel uncertainty in our predictive distributions. We propose a distribution over kernels formed by modelling a spectral mixture density with a L'evy process. The resulting distribution has support for all stationary covariances--including the popular RBF, periodic, and Mat'ern kernels--combined with inductive biases which enable automatic and data efficient learning, long-range extrapolation, and state of the art predictive performance. The proposed model also presents an approach to spectral regularization, as the L'evy process introduces a sparsity-inducing prior over mixture components, allowing automatic selection over model order and pruning of extraneous components. We exploit the algebraic structure of the proposed process for O(n) training and O(1) predictions. We perform extrapolations having reasonable uncertainty estimates on several benchmarks, show that the proposed model can recover flexible ground truth covariances and that it is robust to errors in initialization.