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Extrinsic Bayesian Optimizations on Manifolds

Yihao FangMu NiuPokman CheungLizhen Lin
Dec 2022
摘要
We propose an extrinsic Bayesian optimization (eBO) framework for generaloptimization problems on manifolds. Bayesian optimization algorithms build asurrogate of the objective function by employing Gaussian processes andquantify the uncertainty in that surrogate by deriving an acquisition function.This acquisition function represents the probability of improvement based onthe kernel of the Gaussian process, which guides the search in the optimizationprocess. The critical challenge for designing Bayesian optimization algorithmson manifolds lies in the difficulty of constructing valid covariance kernelsfor Gaussian processes on general manifolds. Our approach is to employextrinsic Gaussian processes by first embedding the manifold onto some higherdimensional Euclidean space via equivariant embeddings and then constructing avalid covariance kernel on the image manifold after the embedding. This leadsto efficient and scalable algorithms for optimization over complex manifolds.Simulation study and real data analysis are carried out to demonstrate theutilities of our eBO framework by applying the eBO to various optimizationproblems over manifolds such as the sphere, the Grassmannian, and the manifoldof positive definite matrices.
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