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.

翻译：受欧氏空间中贝叶斯优化算法成功的启发，我们提出了一种新颖方法，在流形上构建内在贝叶斯优化，重点关注作为R²、R³及更高维子流形而出现的复杂约束域或不规则形状空间。数据可能收集于空间域，但受限于对应于地理特征（如湖泊）的复杂或精细结构区域。传统贝叶斯优化使用径向基函数核，无法适应这些复杂约束条件。内在贝叶斯优化采用稀疏内在高斯过程代理模型，以考虑流形的几何结构。稀疏内在高斯过程利用流形上的热核构建，该热核估计为流形上布朗运动的转移密度。通过U形域、咬环面以及咸海真实数据集的模拟研究，展示了内在贝叶斯优化的效率，并将其性能与定义在欧氏空间中的传统贝叶斯优化进行了比较。