Gaussian processes (GPs) are popular nonparametric statistical models for learning unknown functions and quantifying the spatiotemporal uncertainty in data. Recent works have extended GPs to model scalar and vector quantities distributed over non-Euclidean domains, including smooth manifolds appearing in numerous fields such as computer vision, dynamical systems, and neuroscience. However, these approaches assume that the manifold underlying the data is known, limiting their practical utility. We introduce RVGP, a generalisation of GPs for learning vector signals over latent Riemannian manifolds. Our method uses positional encoding with eigenfunctions of the connection Laplacian, associated with the tangent bundle, readily derived from common graph-based approximation of data. We demonstrate that RVGP possesses global regularity over the manifold, which allows it to super-resolve and inpaint vector fields while preserving singularities. Furthermore, we use RVGP to reconstruct high-density neural dynamics derived from low-density EEG recordings in healthy individuals and Alzheimer's patients. We show that vector field singularities are important disease markers and that their reconstruction leads to a comparable classification accuracy of disease states to high-density recordings. Thus, our method overcomes a significant practical limitation in experimental and clinical applications.

RVGP是一种推广的高斯过程（GPs），用于学习潜在黎曼流形上的向量信号，通过与数据的常见基于图形的逼近相关的切空间束联结Laplacian的特征函数进行位置编码，具有全局规律性，可以在保留奇异点的同时对向量场进行超分辨率重建和内插，用于重构健康人和阿尔茨海默病患者低密度脑电图记录的高密度神经动力学，并且我们发现向量场的奇异点是重要的疾病标记，并且其重建方法在疾病状态的分类精度上与高密度记录相当，因此克服了实验和临床应用中的一个重要限制。

任意潜在流形上的向量场的隐式高斯过程表示