Gaussian Process Landmarking on Manifolds
Gaussian Process Landmarking on Manifolds
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DOI：
10.1137/18m1184035
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发表时间：
2019-01-01
影响因子：
3.6
通讯作者：
Daubechies, Ingrid
中科院分区：
文献类型：
--
作者：
Gao, Tingran;Kovalsky, Shahar Z.;Daubechies, Ingrid
作为改进生物形状分析的一种方法，我们提出了一种通过在高斯过程模型下顺序选择具有最大不确定性的点来采样黎曼流形的算法。这种贪婪策略在实验设计文献中被认为是接近最优的，并且在我们的应用程序中表示生物对象的几何形状时，它似乎优于使用用户放置的地标。在无噪声情况下，我们根据样本数量和流形的几何量建立了均方预测误差（MSPE）的上限，证明我们提出的顺序设计的 MSPE 衰减速度与任何顺序或非顺序优化设计可实现的预言率相当；据我们所知，这是此类序贯实验设计的第一个结果。关键是在偏微分方程 (PDE) 模型简化的背景下将贪婪算法与简化基方法联系起来。我们预计这种方法将在其他研究领域找到更多应用。
As a means of improving analysis of biological shapes, we propose an algorithm for sampling a Riemannian manifold by sequentially selecting points with maximum uncertainty under a Gaussian process model. This greedy strategy is known to be near-optimal in the experimental design literature, and it appears to outperform the use of user-placed landmarks in representing the geometry of biological objects in our application. In the noiseless regime, we establish an upper bound for the mean squared prediction error (MSPE) in terms of the number of samples and geometric quantities of the manifold, demonstrating that the MSPE for our proposed sequential design decays at a rate comparable to the oracle rate achievable by any sequential or nonsequential optimal design; to the best of our knowledge this is the first result of this type for sequential experimental design. The key is to link the greedy algorithm to reduced basis methods in the context of model reduction for partial differential equations (PDEs). We expect this approach will find additional applications in other fields of research.