Gaussian process regression is widely used because of its ability to provide well-calibrated uncertainty estimates and handle small or sparse datasets. However, it struggles with high-dimensional data. One possible way to scale this technique to higher dimensions is to leverage the implicit low-dimensional manifold upon which the data actually lies, as postulated by the manifold hypothesis. Prior work ordinarily requires the manifold structure to be explicitly provided though, i.e. given by a mesh or be known to be one of the well-known manifolds like the sphere. In contrast, in this paper we propose a Gaussian process regression technique capable of inferring implicit structure directly from data (labeled and unlabeled) in a fully differentiable way. For the resulting model, we discuss its convergence to the Mat\'ern Gaussian process on the assumed manifold. Our technique scales up to hundreds of thousands of data points, and may improve the predictive performance and calibration of the standard Gaussian process regression in high-dimensional~settings.

高斯过程回归是一种用于提供准确的不确定性估计和处理小型或稀疏数据集的方法，然而在高维数据上存在困难，本文提出了一种能够在实际数据中直接推断隐含结构的高斯过程回归技术，并讨论了该模型收敛到假设流形上的Matern高斯过程的情况，该技术能够扩展到数十万个数据点，并提高高维情况下标准高斯过程回归的预测性能和校准性。

隐含流形高斯过程回归