Extrinsic Bayesian Optimization on Manifolds
Extrinsic Bayesian Optimization on Manifolds
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DOI：
10.3390/a16020117
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发表时间：
2023-02
期刊：
影响因子：
2.3
通讯作者：
Yi-Zheng Fang;Mu Niu;P. Cheung;Lizhen Lin
中科院分区：
文献类型：
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作者：
Yi-Zheng Fang;Mu Niu;P. Cheung;Lizhen Lin
我们提出了一个外在贝叶斯优化（eBO）框架一般优化问题的流形。贝叶斯优化算法通过采用高斯过程并通过导出获取函数来利用该替代中的不确定性来构建目标函数的替代。该获取函数表示基于高斯过程的内核的改进概率，其引导优化过程中的搜索。流形上贝叶斯优化算法设计的关键挑战在于构造一般流形上高斯过程的有效协方差核的困难。我们的方法是采用外部高斯过程，首先嵌入流形到一些高维欧氏空间通过等变嵌入，然后构建一个有效的协方差嵌入后的图像流形上的内核。这导致了复杂流形上的优化的有效和可扩展的算法。模拟研究和真实的数据分析表明，我们的eBO框架的效用，通过应用eBO的各种流形上的优化问题，如球，格拉斯曼，和流形的正定矩阵。
We propose an extrinsic Bayesian optimization (eBO) framework for general optimization problems on manifolds. Bayesian optimization algorithms build a surrogate of the objective function by employing Gaussian processes and utilizing the uncertainty in that surrogate by deriving an acquisition function. This acquisition function represents the probability of improvement based on the kernel of the Gaussian process, which guides the search in the optimization process. The critical challenge for designing Bayesian optimization algorithms on manifolds lies in the difficulty of constructing valid covariance kernels for Gaussian processes on general manifolds. Our approach is to employ extrinsic Gaussian processes by first embedding the manifold onto some higher dimensional Euclidean space via equivariant embeddings and then constructing a valid covariance kernel on the image manifold after the embedding. This leads to efficient and scalable algorithms for optimization over complex manifolds. Simulation study and real data analyses are carried out to demonstrate the utilities of our eBO framework by applying the eBO to various optimization problems over manifolds such as the sphere, the Grassmannian, and the manifold of positive definite matrices.