# The Minecraft Kernel: Modelling correlated Gaussian Processes in the Fourier domain

arXiv: https://arxiv.org/abs/2103.06950 · HTML: https://mlsanity.com/paper/2103.06950/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 19 references that are in the map.
- First posted on arXiv 11 Mar 2021.

## Builds on (most cited first, 9 of 9)

- [GPflow: A Gaussian process library using TensorFlow](https://mlsanity.com/paper/1610.08733/) (2016)
- [Gaussian Process Kernels for Pattern Discovery and Extrapolation](https://mlsanity.com/paper/1302.4245/) (2013)
- [Automatic Construction and Natural-Language Description of Nonparametric Regression Models](https://mlsanity.com/paper/1402.4304/) (2014)
- [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)
- [Differentiable Compositional Kernel Learning for Gaussian Processes](https://mlsanity.com/paper/1806.04326/) (2018)
- [Large-Scale Cox Process Inference using Variational Fourier Features](https://mlsanity.com/paper/1804.01016/) (2018)
- [Band-Limited Gaussian Processes: The Sinc Kernel](https://mlsanity.com/paper/1909.07279/) (2019)
- [Physically-Inspired Gaussian Process Models for Post-Transcriptional Regulation in Drosophila](https://mlsanity.com/paper/1808.10026/) (2018)

## Built on by (most cited first, 3 of 3)

- [Kernel Identification Through Transformers](https://mlsanity.com/paper/2106.08185/) (2021)
- [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)

## Abstract (arXiv)

In the univariate setting, using the kernel spectral representation is an appealing approach for generating stationary covariance functions. However, performing the same task for multiple-output Gaussian processes is substantially more challenging. We demonstrate that current approaches to modelling cross-covariances with a spectral mixture kernel possess a critical blind spot. For a given pair of processes, the cross-covariance is not reproducible across the full range of permitted correlations, aside from the special case where their spectral densities are of identical shape. We present a solution to this issue by replacing the conventional Gaussian components of a spectral mixture with block components of finite bandwidth (i.e. rectangular step functions). The proposed family of kernel represents the first multi-output generalisation of the spectral mixture kernel that can approximate any stationary multi-output kernel to arbitrary precision.