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.

翻译：高斯过程是一种流行的非参数统计模型，用于学习未知函数并量化数据中的时空不确定性。近期研究已将其扩展至建模分布在非欧几里得域上的标量和向量量，包括计算机视觉、动力系统和神经科学等领域中出现的平滑流形。然而，这些方法假设数据背后的流形已知，限制了其实用性。我们提出RVGP，这是一种将高斯过程推广至潜在黎曼流形上学习向量信号的方法。该方法利用联络拉普拉斯算子的特征函数进行位置编码（该算子与切丛相关，可通过常见的基于图的数据近似直接导出）。我们证明RVGP在流形上具有全局正则性，使其能够在保持奇异性的同时实现向量场的超分辨率和修复。此外，我们利用RVGP从健康个体及阿尔茨海默症患者的低密度脑电记录中重建高密度神经动态。结果表明，向量场奇异点是重要的疾病标志物，且其重建能达到与高密度记录相当的疾病状态分类准确率。因此，我们的方法克服了实验与临床应用中的重大实际限制。