Finite-dimensional Gaussian approximation with linear inequality constraints

TL;DR

This paper extends finite-dimensional Gaussian process models to incorporate general linear inequality constraints, using MCMC methods for posterior approximation and analyzing their properties for improved uncertainty quantification.

Contribution

It introduces a generalized approach for linear inequality constraints in Gaussian processes, along with MCMC techniques for posterior sampling and theoretical analysis of constrained likelihood.

Findings

Efficient Hamiltonian Monte Carlo sampling for constrained GPs.

Improved uncertainty quantification with inequality constraints.

Successful application to artificial and real data.

Abstract

Introducing inequality constraints in Gaussian process (GP) models can lead to more realistic uncertainties in learning a great variety of real-world problems. We consider the finite-dimensional Gaussian approach from Maatouk and Bay (2017) which can satisfy inequality conditions everywhere (either boundedness, monotonicity or convexity). Our contributions are threefold. First, we extend their approach in order to deal with general sets of linear inequalities. Second, we explore several Markov Chain Monte Carlo (MCMC) techniques to approximate the posterior distribution. Third, we investigate theoretical and numerical properties of the constrained likelihood for covariance parameter estimation. According to experiments on both artificial and real data, our full framework together with a Hamiltonian Monte Carlo-based sampler provides efficient results on both data fitting and uncertainty quantification.