# Non-Stationary Spectral Kernels

arXiv: https://arxiv.org/abs/1705.08736 · HTML: https://mlsanity.com/paper/1705.08736/

## Key facts

- Cited by 38 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 May 2017.

## Built on by (most cited first, 30 of 38)

- [Uncertainty Quantification in Machine Learning for Engineering Design and Health Prognostics: A Tutorial](https://mlsanity.com/paper/2305.04933/) (2023)
- [On Learning the Transformer Kernel](https://mlsanity.com/paper/2110.08323/) (2021)
- [Kernel Neural Operators (KNOs) for Scalable, Memory-efficient, Geometrically-flexible Operator Learning](https://mlsanity.com/paper/2407.00809/) (2024)
- [Neural Spectral Marked Point Processes](https://mlsanity.com/paper/2106.10773/) (2021)
- [Exact Gaussian Processes for Massive Datasets via Non-Stationary Sparsity-Discovering Kernels](https://mlsanity.com/paper/2205.09070/) (2022)
- [Adaptive Inducing Points Selection For Gaussian Processes](https://mlsanity.com/paper/2107.10066/) (2021)
- [Nonstationary multi-output Gaussian processes via harmonizable spectral mixtures](https://mlsanity.com/paper/2202.09233/) (2022)
- [A Robust Asymmetric Kernel Function for Bayesian Optimization, with Application to Image Defect Detection in Manufacturing Systems](https://mlsanity.com/paper/2109.10898/) (2021)
- [Recent Advances in Data-Driven Wireless Communication Using Gaussian Processes: A Comprehensive Survey](https://mlsanity.com/paper/2103.10134/) (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)
- [Hierarchical-Hyperplane Kernels for Actively Learning Gaussian Process Models of Nonstationary Systems](https://mlsanity.com/paper/2303.10022/) (2023)
- [The Minecraft Kernel: Modelling correlated Gaussian Processes in the Fourier domain](https://mlsanity.com/paper/2103.06950/) (2021)
- [A Temporal Kernel Approach for Deep Learning with Continuous-time Information](https://mlsanity.com/paper/2103.15213/) (2021)
- [Learning High-Dimensional Distributions with Latent Neural Fokker-Planck Kernels](https://mlsanity.com/paper/2105.04538/) (2021)
- [Accurate Remaining Useful Life Prediction with Uncertainty Quantification: a Deep Learning and Nonstationary Gaussian Process Approach](https://mlsanity.com/paper/2109.12111/) (2021)
- [Bayesian Complementary Kernelized Learning for Multidimensional Spatiotemporal Data](https://mlsanity.com/paper/2208.09978/) (2022)
- [Uncertainty Disentanglement with Non-stationary Heteroscedastic Gaussian Processes for Active Learning](https://mlsanity.com/paper/2210.10964/) (2022)
- [Bayesian Convolutional Deep Sets with Task-Dependent Stationary Prior](https://mlsanity.com/paper/2210.12363/) (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)
- [Scalable Random Wavelet Features: Efficient Non-Stationary Kernel Approximation with Convergence Guarantees](https://mlsanity.com/paper/2602.00987/) (2026)
- [Multi-View Oriented GPLVM: Expressiveness and Efficiency](https://mlsanity.com/paper/2502.08253/) (2025)
- [Probabilistic Wind Power Modelling via Heteroscedastic Non-Stationary Gaussian Processes](https://mlsanity.com/paper/2505.09026/) (2025)
- [STACI: Spatio-Temporal Aleatoric Conformal Inference](https://mlsanity.com/paper/2505.21658/) (2025)
- [Bayesian Optimization over Bounded Domains with the Beta Product Kernel](https://mlsanity.com/paper/2506.16316/) (2025)
- [Stochastic Collapsed Variational Inference for Structured Gaussian Process Regression Network](https://mlsanity.com/paper/2106.00719/) (2021)
- [Wasserstein-Splitting Gaussian Process Regression for Heterogeneous Online Bayesian Inference](https://mlsanity.com/paper/2107.12797/) (2021)
- [Remaining useful life prediction with uncertainty quantification: development of a highly accurate model for rotating machinery](https://mlsanity.com/paper/2109.11579/) (2021)

## Abstract (arXiv)

We propose non-stationary spectral kernels for Gaussian process regression. We propose to model the spectral density of a non-stationary kernel function as a mixture of input-dependent Gaussian process frequency density surfaces. We solve the generalised Fourier transform with such a model, and present a family of non-stationary and non-monotonic kernels that can learn input-dependent and potentially long-range, non-monotonic covariances between inputs. We derive efficient inference using model whitening and marginalized posterior, and show with case studies that these kernels are necessary when modelling even rather simple time series, image or geospatial data with non-stationary characteristics.