# Gaussian processes with linear operator inequality constraints

arXiv: https://arxiv.org/abs/1901.03134 · HTML: https://mlsanity.com/paper/1901.03134/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Jan 2019.

## Built on by (most cited first, 16 of 16)

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- [A spectrum of physics-informed Gaussian processes for regression in engineering](https://mlsanity.com/paper/2309.10656/) (2023)
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- [Learning Energy Conserving Dynamics Efficiently with Hamiltonian Gaussian Processes](https://mlsanity.com/paper/2303.01925/) (2023)
- [Handling Hard Affine SDP Shape Constraints in RKHSs](https://mlsanity.com/paper/2101.01519/) (2021)
- [Conditioning Gaussian Processes on Almost Anything](https://mlsanity.com/paper/2605.21041/) (2026)
- [Posterior Regularized Bayesian Neural Network Incorporating Soft and Hard Knowledge Constraints](https://mlsanity.com/paper/2210.08608/) (2022)
- [Monotonic Gaussian process for physics-constrained machine learning with materials science applications](https://mlsanity.com/paper/2209.00628/) (2022)
- [LatentFlow: A General Framework for Conditioning Stochastic Processes](https://mlsanity.com/paper/2607.12922/) (2026)
- [Hierarchical Non-Stationary Temporal Gaussian Processes With L1-Regularization](https://mlsanity.com/paper/2105.09695/) (2021)
- [BALLAST: Bayesian Active Learning with Look-ahead Amendment for Sea-drifter Trajectories under Spatio-Temporal Vector Fields](https://mlsanity.com/paper/2509.26005/) (2025)
- [Deep Bayesian Reinforcement Learning for Spacecraft Proximity Maneuvers and Docking](https://mlsanity.com/paper/2311.03680/) (2023)
- [Gaussian Process Regression with Soft Inequality and Monotonicity Constraints](https://mlsanity.com/paper/2404.02873/) (2024)
- [Fast Gaussian Processes under Monotonicity Constraints](https://mlsanity.com/paper/2507.06677/) (2025)

## Abstract (arXiv)

This paper presents an approach for constrained Gaussian Process (GP) regression where we assume that a set of linear transformations of the process are bounded. It is motivated by machine learning applications for high-consequence engineering systems, where this kind of information is often made available from phenomenological knowledge. We consider a GP f over functions on X subset ℝn taking values in ℝ, where the process Lf is still Gaussian when L is a linear operator. Our goal is to model f under the constraint that realizations of Lf are confined to a convex set of functions. In particular, we require that a ≤ Lf ≤ b, given two functions a and b where a < b pointwise. This formulation provides a consistent way of encoding multiple linear constraints, such as shape-constraints based on e.g. boundedness, monotonicity or convexity. We adopt the approach of using a sufficiently dense set of virtual observation locations where the constraint is required to hold, and derive the exact posterior for a conjugate likelihood. The results needed for stable numerical implementation are derived, together with an efficient sampling scheme for estimating the posterior process.