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.

中文翻译：

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任意潜在流形上向量场的隐式高斯过程表示

高斯过程（GP）是流行的非参数统计模型，用于学习未知函数和量化数据中的时空不确定性。最近的工作已将 GP 扩展到对分布在非欧几里得域上的标量和向量进行建模，包括计算机视觉、动力系统和神经科学等众多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是已知的，限制了它们的实际用途。我们引入了 RVGP，它是 GP 的推广，用于学习潜在黎曼流形上的矢量信号。我们的方法使用位置编码，该位置编码具有连接拉普拉斯算子的特征函数，与切线束相关联，很容易从常见的基于图形的数据近似中导出。我们证明 RVGP 在流形上具有全局正则性，这使得它能够超解析和修复矢量场，同时保留奇点。此外，我们使用 RVGP 来重建来自健康个体和阿尔茨海默病患者的低密度脑电图记录的高密度神经动力学。我们证明矢量场奇点是重要的疾病标记，它们的重建可以使疾病状态的分类精度与高密度记录相当。因此，我们的方法克服了实验和临床应用中的重大实际限制。我们证明矢量场奇点是重要的疾病标记，它们的重建可以使疾病状态的分类精度与高密度记录相当。因此，我们的方法克服了实验和临床应用中的重大实际限制。我们证明矢量场奇点是重要的疾病标记，它们的重建可以使疾病状态的分类精度与高密度记录相当。因此，我们的方法克服了实验和临床应用中的重大实际限制。