# Incorporating Sum Constraints into Multitask Gaussian Processes

arXiv: https://arxiv.org/abs/2202.01793 · HTML: https://mlsanity.com/paper/2202.01793/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 44 references that are in the map.
- First posted on arXiv 3 Feb 2022.

## Builds on (most cited first, 15 of 15)

- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [GPyTorch: Blackbox Matrix-Matrix Gaussian Process Inference with GPU Acceleration](https://mlsanity.com/paper/1809.11165/) (2018)
- [Gaussian Processes for Big Data](https://mlsanity.com/paper/1309.6835/) (2013)
- [Integrating Scientific Knowledge with Machine Learning for Engineering and Environmental Systems](https://mlsanity.com/paper/2003.04919/) (2020)
- [Machine Learning of Linear Differential Equations using Gaussian Processes](https://mlsanity.com/paper/1701.02440/) (2017)
- [Numerical Gaussian Processes for Time-dependent and Non-linear Partial Differential Equations](https://mlsanity.com/paper/1703.10230/) (2017)
- [A Survey of Constrained Gaussian Process Regression: Approaches and Implementation Challenges](https://mlsanity.com/paper/2006.09319/) (2020)
- [Linearly constrained Gaussian processes](https://mlsanity.com/paper/1703.00787/) (2017)
- [The Variational Gaussian Process](https://mlsanity.com/paper/1511.06499/) (2015)
- [Modeling and interpolation of the ambient magnetic field by Gaussian processes](https://mlsanity.com/paper/1509.04634/) (2015)
- [Algorithmic Linearly Constrained Gaussian Processes](https://mlsanity.com/paper/1801.09197/) (2018)
- [Linearly Constrained Neural Networks](https://mlsanity.com/paper/2002.01600/) (2020)
- [Gaussian process emulators for computer experiments with inequality constraints](https://mlsanity.com/paper/1606.01265/) (2016)
- [Learning Constrained Dynamics with Gauss Principle adhering Gaussian Processes](https://mlsanity.com/paper/2004.11238/) (2020)
- [Know Your Boundaries: Constraining Gaussian Processes by Variational Harmonic Features](https://mlsanity.com/paper/1904.05207/) (2019)

## Built on by (most cited first, 2 of 2)

- [Time-Varying Transition Matrices with Multi-task Gaussian Processes](https://mlsanity.com/paper/2306.11772/) (2023)
- [Multi-output Gaussian process prediction of physical fields under linear equality constraints](https://mlsanity.com/paper/2608.25709/) (2026)

## Abstract (arXiv)

Machine learning models can be improved by adapting them to respect existing background knowledge. In this paper we consider multitask Gaussian processes, with background knowledge in the form of constraints that require a specific sum of the outputs to be constant. This is achieved by conditioning the prior distribution on the constraint fulfillment. The approach allows for both linear and nonlinear constraints. We demonstrate that the constraints are fulfilled with high precision and that the construction can improve the overall prediction accuracy as compared to the standard Gaussian process.