Intrinsic Gaussian processes on complex constrained domains
Intrinsic Gaussian processes on complex constrained domains
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DOI：
10.1111/rssb.12320
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发表时间：
2018-01
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通讯作者：
Mu Niu;P. Cheung;Lizhen Lin;Zhenwen Dai;Neil D. Lawrence;D. Dunson
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作者：
Mu Niu;P. Cheung;Lizhen Lin;Zhenwen Dai;Neil D. Lawrence;D. Dunson
我们提出了一类内在高斯过程（GP），用于流形上的插值、回归和分类，主要关注作为 R 、 R2 、 R3 等的子集或子流形而出现的复杂约束域或不规则形状的空间。例如，内在 GP 可以容纳作为欧几里德空间的复杂子集而出现的空间域。内在 GP 尊重潜在复杂的边界或内部条件以及空间的内在几何形状。该方法的关键新颖之处在于利用热核与流形上布朗运动的转变密度之间的关系来构建和近似有效且计算上可行的协方差核。这使得内在 GP 能够在实际中广泛应用，而现有的约束域平滑方法仅限于简单的特殊情况。通过模拟研究和数据示例说明了内在 GP 方法的广泛实用性。
We propose a class of intrinsic Gaussian processes (GPs) for interpolation, regression and classification on manifolds with a primary focus on complex constrained domains or irregularly shaped spaces arising as subsets or submanifolds of R , R2 , R3 and beyond. For example, intrinsic GPs can accommodate spatial domains arising as complex subsets of Euclidean space. Intrinsic GPs respect the potentially complex boundary or interior conditions as well as the intrinsic geometry of the spaces. The key novelty of the approach proposed is to utilize the relationship between heat kernels and the transition density of Brownian motion on manifolds for constructing and approximating valid and computationally feasible covariance kernels. This enables intrinsic GPs to be practically applied in great generality, whereas existing approaches for smoothing on constrained domains are limited to simple special cases. The broad utilities of the intrinsic GP approach are illustrated through simulation studies and data examples.