# Low-rank computation of the posterior mean in Multi-Output Gaussian Processes

arXiv: https://arxiv.org/abs/2504.21527 · HTML: https://mlsanity.com/paper/2504.21527/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 51 references that are in the map.
- First posted on arXiv 30 Apr 2025.

## Builds on (most cited first, 7 of 7)

- [Mat'ern Gaussian Processes on Graphs](https://mlsanity.com/paper/2010.15538/) (2020)
- [Convergence of Sparse Variational Inference in Gaussian Processes Regression](https://mlsanity.com/paper/2008.00323/) (2020)
- [Gaussian Processes on Graphs via Spectral Kernel Learning](https://mlsanity.com/paper/2006.07361/) (2020)
- [Gaussian Processes Over Graphs](https://mlsanity.com/paper/1803.05776/) (2018)
- [Graph Based Gaussian Processes on Restricted Domains](https://mlsanity.com/paper/2010.07242/) (2020)
- [Blitzkriging: Kronecker-structured Stochastic Gaussian Processes](https://mlsanity.com/paper/1510.07965/) (2015)
- [On Detecting Low-pass Graph Signals under Partial Observations](https://mlsanity.com/paper/2405.10001/) (2024)

## Abstract (arXiv)

Gaussian processes (GP) are a versatile tool in machine learning and computational science. We here consider the case of multi-output Gaussian processes (MOGP) and present low-rank approaches for efficiently computing the posterior mean of a MOGP. Starting from low-rank spatio-temporal data we consider a structured covariance function, assuming separability across space and time. This separability, in turn, gives a decomposition of the covariance matrix into a Kronecker product of individual covariance matrices. Incorporating the typical noise term to the model then requires the solution of a large-scale Stein equation for computing the posterior mean. For this, we propose efficient low-rank methods based on a combination of a LRPCG method with the Sylvester equation solver KPIK adjusted for solving Stein equations. We test the developed method on real world street network graphs by using graph filters as covariance matrices. Moreover, we propose a degree-weighted average covariance matrix, which can be employed under specific assumptions to achieve more efficient convergence.