# Generalized Spectral Kernels

arXiv: https://arxiv.org/abs/1506.02236 · HTML: https://mlsanity.com/paper/1506.02236/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Jun 2015.

## Built on by (most cited first, 17 of 17)

- [On Learning the Transformer Kernel](https://mlsanity.com/paper/2110.08323/) (2021)
- [Scalable Variational Gaussian Processes via Harmonic Kernel Decomposition](https://mlsanity.com/paper/2106.05992/) (2021)
- [Nonstationary multi-output Gaussian processes via harmonizable spectral mixtures](https://mlsanity.com/paper/2202.09233/) (2022)
- [Characteristic kernels on Hilbert spaces, Banach spaces, and on sets of measures](https://mlsanity.com/paper/2206.07588/) (2022)
- [Adversarial Robustness Guarantees for Gaussian Processes](https://mlsanity.com/paper/2104.03180/) (2021)
- [Hida-Mat'ern Kernel](https://mlsanity.com/paper/2107.07098/) (2021)
- [Mixed Variable Bayesian Optimization with Frequency Modulated Kernels](https://mlsanity.com/paper/2102.12792/) (2021)
- [Marginalising over Stationary Kernels with Bayesian Quadrature](https://mlsanity.com/paper/2106.07452/) (2021)
- [Online Estimation of the Koopman Operator Using Fourier Features](https://mlsanity.com/paper/2212.01503/) (2022)
- [Learning High-Dimensional Distributions with Latent Neural Fokker-Planck Kernels](https://mlsanity.com/paper/2105.04538/) (2021)
- [Non-Gaussian Process Regression](https://mlsanity.com/paper/2209.03117/) (2022)
- [On the Identifiability and Interpretability of Gaussian Process Models](https://mlsanity.com/paper/2310.17023/) (2023)
- [Nonstationary Sparse Spectral Permanental Process](https://mlsanity.com/paper/2410.03581/) (2024)
- [Multi-View Oriented GPLVM: Expressiveness and Efficiency](https://mlsanity.com/paper/2502.08253/) (2025)
- [PolyMicros: Bootstrapping a Foundation Model for Polycrystalline Material Structure](https://mlsanity.com/paper/2506.11055/) (2025)
- [Scalable Deep Basis Kernel Gaussian Processes](https://mlsanity.com/paper/2505.18526/) (2025)
- [Regular Fourier Features for Nonstationary Gaussian Processes](https://mlsanity.com/paper/2602.23006/) (2026)

## Abstract (arXiv)

In this paper we propose a family of tractable kernels that is dense in the family of bounded positive semi-definite functions (i.e. can approximate any bounded kernel with arbitrary precision). We start by discussing the case of stationary kernels, and propose a family of spectral kernels that extends existing approaches such as spectral mixture kernels and sparse spectrum kernels. Our extension has two primary advantages. Firstly, unlike existing spectral approaches that yield infinite differentiability, the kernels we introduce allow learning the degree of differentiability of the latent function in Gaussian process (GP) models and functions in the reproducing kernel Hilbert space (RKHS) in other kernel methods. Secondly, we show that some of the kernels we propose require fewer parameters than existing spectral kernels for the same accuracy, thereby leading to faster and more robust inference. Finally, we generalize our approach and propose a flexible and tractable family of spectral kernels that we prove can approximate any continuous bounded nonstationary kernel.