# Nonstationary multi-output Gaussian processes via harmonizable spectral mixtures

arXiv: https://arxiv.org/abs/2202.09233 · HTML: https://mlsanity.com/paper/2202.09233/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 33 references that are in the map.
- First posted on arXiv 18 Feb 2022.

## Builds on (most cited first, 8 of 8)

- [Gaussian Process Kernels for Pattern Discovery and Extrapolation](https://mlsanity.com/paper/1302.4245/) (2013)
- [Non-Stationary Spectral Kernels](https://mlsanity.com/paper/1705.08736/) (2017)
- [Spectral Mixture Kernels for Multi-Output Gaussian Processes](https://mlsanity.com/paper/1709.01298/) (2017)
- [Generalized Spectral Kernels](https://mlsanity.com/paper/1506.02236/) (2015)
- [Harmonizable mixture kernels with variational Fourier features](https://mlsanity.com/paper/1810.04416/) (2018)
- [MOGPTK: The Multi-Output Gaussian Process Toolkit](https://mlsanity.com/paper/2002.03471/) (2020)
- [Bayesian Nonparametric Spectral Estimation](https://mlsanity.com/paper/1809.02196/) (2018)
- [Gaussian process imputation of multiple financial series](https://mlsanity.com/paper/2002.05789/) (2020)

## Built on by (most cited first, 8 of 8)

- [Robust and Conjugate Gaussian Process Regression](https://mlsanity.com/paper/2311.00463/) (2023)
- [Scalable Multi-Output Gaussian Processes with Stochastic Variational Inference](https://mlsanity.com/paper/2407.02476/) (2024)
- [Fed-Joint: Joint Modeling of Nonlinear Degradation Signals and Failure Events for Remaining Useful Life Prediction using Federated Learning](https://mlsanity.com/paper/2503.13404/) (2025)
- [Multi-Output Robust and Conjugate Gaussian Processes](https://mlsanity.com/paper/2510.26401/) (2025)
- [Shallow and Deep Nonparametric Convolutions for Gaussian Processes](https://mlsanity.com/paper/2206.08972/) (2022)
- [CatBOX: A Categorical-Continuous Bayesian Optimization with Spectral Mixture Kernels for Accelerated Catalysis Experiments](https://mlsanity.com/paper/2505.17393/) (2025)
- [Transformed Latent Variable Multi-Output Gaussian Processes](https://mlsanity.com/paper/2605.05133/) (2026)
- [Adapting Nonstationary Multi-output Gaussian Processes to Bayesian Optimization](https://mlsanity.com/paper/2609.32464/) (2026)

## Abstract (arXiv)

Kernel design for Multi-output Gaussian Processes (MOGP) has received increased attention recently. In particular, the Multi-Output Spectral Mixture kernel (MOSM) arXiv:1709.01298 approach has been praised as a general model in the sense that it extends other approaches such as Linear Model of Corregionalization, Intrinsic Corregionalization Model and Cross-Spectral Mixture. MOSM relies on Cram'er's theorem to parametrise the power spectral densities (PSD) as a Gaussian mixture, thus, having a structural restriction: by assuming the existence of a PSD, the method is only suited for multi-output stationary applications. We develop a nonstationary extension of MOSM by proposing the family of harmonizable kernels for MOGPs, a class of kernels that contains both stationary and a vast majority of non-stationary processes. A main contribution of the proposed harmonizable kernels is that they automatically identify a possible nonstationary behaviour meaning that practitioners do not need to choose between stationary or non-stationary kernels. The proposed method is first validated on synthetic data with the purpose of illustrating the key properties of our approach, and then compared to existing MOGP methods on two real-world settings from finance and electroencephalography.