Gaussian processes are an effective model class for learning unknown functions, particularly in settings where accurately representing predictive uncertainty is of key importance. Motivated by applications in the physical sciences, the widely-used Mat\'ern class of Gaussian processes has recently been generalized to model functions whose domains are Riemannian manifolds, by re-expressing said processes as solutions of stochastic partial differential equations. In this work, we propose techniques for computing the kernels of these processes on compact Riemannian manifolds via spectral theory of the Laplace-Beltrami operator in a fully constructive manner, thereby allowing them to be trained via standard scalable techniques such as inducing point methods. We also extend the generalization from the Mat\'ern to the widely-used squared exponential Gaussian process. By allowing Riemannian Mat\'ern Gaussian processes to be trained using well-understood techniques, our work enables their use in mini-batch, online, and non-conjugate settings, and makes them more accessible to machine learning practitioners.
翻译:Gaussian 进程是学习未知功能的有效模型类,特别是在准确代表预测不确定性具有关键重要性的环境下。在物理科学应用的激励下,最近广泛使用的Mat\'ern类高森进程被广泛推广为模型函数,其领域为里曼尼方块,将所述进程重新表述为随机偏差部分方程式的解决方案。在这项工作中,我们提议了通过拉普尔-贝尔特拉米操作员的光谱理论,在紧凑的里曼方块上计算这些过程的内核的技术,从而能够以完全建设性的方式通过标准可伸缩技术(如导点方法)对其进行培训。我们还将通用功能从马特尔恩推广到广泛使用的平方形方形高斯方形进程。通过允许里曼尼·马特恩高斯进程使用井然技术接受培训,我们的工作使得这些流程在小型、在线和非协同环境中得以使用,并使机器学习从业者更容易使用。