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:高斯過程是學習未知函數的有效模型類,特別是在需要準確表示預測不確定性的情況下。受物理科學應用的啟發,最近將廣泛使用的瑪特恩高斯過程推廣到用於描述定義在黎曼流形上的函數,通過重新表示為隨機偏微分方程的解來實現。在這項工作中,我們提出了在緊湊黎曼流形上計算這些過程內核的技術,通過拉普拉斯-貝爾特拉米算子的譜理論以完全建構的方式,從而使它們能夠通過標準的可擴展技術。例如感覺點方法進行訓練。我們還將將瑪特恩廣泛使用的平方指數高斯過程推廣到該流程。通過允許使用熟悉的技術來訓練黎曼瑪特南高斯過程,我們的工作使這些過程能夠在小批量、在線和非共軛設置中使用,並使機器學習從業人員更容易使用。