# Spectral Mixture Kernels for Multi-Output Gaussian Processes

arXiv: https://arxiv.org/abs/1709.01298 · HTML: https://mlsanity.com/paper/1709.01298/

## Key facts

- Cited by 29 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Sep 2017.

## Built on by (most cited first, 29 of 29)

- [A spectrum of physics-informed Gaussian processes for regression in engineering](https://mlsanity.com/paper/2309.10656/) (2023)
- [Kernel Neural Operators (KNOs) for Scalable, Memory-efficient, Geometrically-flexible Operator Learning](https://mlsanity.com/paper/2407.00809/) (2024)
- [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)
- [Learning Multi-Task Gaussian Process Over Heterogeneous Input Domains](https://mlsanity.com/paper/2202.12636/) (2022)
- [Transferring model structure in Bayesian transfer learning for Gaussian process regression](https://mlsanity.com/paper/2101.06884/) (2021)
- [Recent Advances in Data-Driven Wireless Communication Using Gaussian Processes: A Comprehensive Survey](https://mlsanity.com/paper/2103.10134/) (2021)
- [Scalable Multi-Task Gaussian Processes with Neural Embedding of Coregionalization](https://mlsanity.com/paper/2109.09261/) (2021)
- [Learning Time-Varying Multi-Region Brain Communications via Scalable Markovian Gaussian Processes](https://mlsanity.com/paper/2407.00397/) (2024)
- [The Minecraft Kernel: Modelling correlated Gaussian Processes in the Fourier domain](https://mlsanity.com/paper/2103.06950/) (2021)
- [Gaussian Process Latent Variable Modeling for Few-shot Time Series Forecasting](https://mlsanity.com/paper/2212.10306/) (2022)
- [Federated Automatic Latent Variable Selection in Multi-output Gaussian Processes](https://mlsanity.com/paper/2407.16935/) (2024)
- [Bayesian Convolutional Deep Sets with Task-Dependent Stationary Prior](https://mlsanity.com/paper/2210.12363/) (2022)
- [Scalable Multi-Output Gaussian Processes with Stochastic Variational Inference](https://mlsanity.com/paper/2407.02476/) (2024)
- [Multi-Output Robust and Conjugate Gaussian Processes](https://mlsanity.com/paper/2510.26401/) (2025)
- [Geometry-Aware Hierarchical Bayesian Learning on Manifolds](https://mlsanity.com/paper/2111.00184/) (2021)
- [Shallow and Deep Nonparametric Convolutions for Gaussian Processes](https://mlsanity.com/paper/2206.08972/) (2022)
- [Gaussian process deconvolution](https://mlsanity.com/paper/2305.04871/) (2023)
- [PolyMicros: Bootstrapping a Foundation Model for Polycrystalline Material Structure](https://mlsanity.com/paper/2506.11055/) (2025)
- [Leveraging Gauge Freedom for Learning Non-Gradient Population Dynamics of Stochastic Systems](https://mlsanity.com/paper/2605.25107/) (2026)
- [Multi-Task Neural Processes](https://mlsanity.com/paper/2110.14953/) (2021)
- [A Principled Method for the Creation of Synthetic Multi-fidelity Data Sets](https://mlsanity.com/paper/2208.05667/) (2022)
- [CatBOX: A Categorical-Continuous Bayesian Optimization with Spectral Mixture Kernels for Accelerated Catalysis Experiments](https://mlsanity.com/paper/2505.17393/) (2025)
- [Universal Modelling of Autocovariance Functions via Spline Kernels](https://mlsanity.com/paper/2506.21953/) (2025)
- [Deep Intrinsic Coregionalization Multi-Output Gaussian Process Surrogate with Active Learning](https://mlsanity.com/paper/2508.16434/) (2025)
- [Mixture-of-experts Wishart model for covariance matrices with an application to Cancer drug screening](https://mlsanity.com/paper/2602.13888/) (2026)
- [Transformed Latent Variable Multi-Output Gaussian Processes](https://mlsanity.com/paper/2605.05133/) (2026)
- [Sharper Guarantees for Misspecified Kernelized Bandit Optimization](https://mlsanity.com/paper/2605.05967/) (2026)
- [Adapting Nonstationary Multi-output Gaussian Processes to Bayesian Optimization](https://mlsanity.com/paper/2609.32464/) (2026)

## Abstract (arXiv)

Early approaches to multiple-output Gaussian processes (MOGPs) relied on linear combinations of independent, latent, single-output Gaussian processes (GPs). This resulted in cross-covariance functions with limited parametric interpretation, thus conflicting with the ability of single-output GPs to understand lengthscales, frequencies and magnitudes to name a few. On the contrary, current approaches to MOGP are able to better interpret the relationship between different channels by directly modelling the cross-covariances as a spectral mixture kernel with a phase shift. We extend this rationale and propose a parametric family of complex-valued cross-spectral densities and then build on Cram'er's Theorem (the multivariate version of Bochner's Theorem) to provide a principled approach to design multivariate covariance functions. The so-constructed kernels are able to model delays among channels in addition to phase differences and are thus more expressive than previous methods, while also providing full parametric interpretation of the relationship across channels. The proposed method is first validated on synthetic data and then compared to existing MOGP methods on two real-world examples.