Intrinsic Bayesian Optimisation on Complex Constrained Domain

TL;DR

This paper introduces Intrinsic Bayesian Optimisation (In-BO), a novel method for optimizing functions on complex constrained manifolds using heat kernel-based Gaussian processes, outperforming traditional Euclidean BO in irregular domains.

Contribution

The paper develops a new In-BO framework utilizing heat kernel-based Gaussian processes for optimization on complex manifolds, addressing limitations of traditional Euclidean BO in constrained spaces.

Findings

In-BO effectively handles complex constrained domains like lakes and tori.

In-BO outperforms traditional Euclidean BO in simulation studies.

The method is validated on real-world geographic data from the Aral Sea.

Abstract

Motivated by the success of Bayesian optimisation algorithms in the Euclidean space, we propose a novel approach to construct Intrinsic Bayesian optimisation (In-BO) on manifolds with a primary focus on complex constrained domains or irregular-shaped spaces arising as submanifolds of R2, R3 and beyond. Data may be collected in a spatial domain but restricted to a complex or intricately structured region corresponding to a geographic feature, such as lakes. Traditional Bayesian Optimisation (Tra-BO) defined with a Radial basis function (RBF) kernel cannot accommodate these complex constrained conditions. The In-BO uses the Sparse Intrinsic Gaussian Processes (SIn-GP) surrogate model to take into account the geometric structure of the manifold. SInGPs are constructed using the heat kernel of the manifold which is estimated as the transition density of the Brownian Motion on manifolds. The efficiency of In-BO is demonstrated through simulation studies on a U-shaped domain, a Bitten torus, and a real dataset from the Aral sea. Its performance is compared to that of traditional BO, which is defined in Euclidean space.