It is often of interest to infer lower-dimensional structure underlying complex data. As a flexible class of non-linear structures, it is common to focus on Riemannian manifolds. Most existing manifold learning algorithms replace the original data with lower-dimensional coordinates without providing an estimate of the manifold or using the manifold to denoise the original data. This article proposes a new methodology to address these problems, allowing interpolation of the estimated manifold between the fitted data points. The proposed approach is motivated by the novel theoretical properties of local covariance matrices constructed from samples near a manifold. Our results enable us to turn a global manifold reconstruction problem into a local regression problem, allowing for the application of Gaussian processes for probabilistic manifold reconstruction. In addition to the theory justifying our methodology, we provide simulated and real data examples to illustrate the performance.

翻译：从复杂数据中推断其潜在的更低维结构通常具有重要意义。作为一种灵活的非线性结构类别，黎曼流形是常见的关注对象。现有的大多数流形学习算法仅将原始数据替换为低维坐标，既不提供流形的估计，也不利用流形对原始数据进行去噪。本文提出了一种新方法来解决这些问题，允许在拟合的数据点之间对估计的流形进行插值。该方法的动机源于从流形附近样本构建的局部协方差矩阵所具有的新颖理论性质。我们的研究结果使得能够将全局流形重构问题转化为局部回归问题，从而可以应用高斯过程进行概率流形重构。除了为我们的方法提供理论依据外，我们还提供了模拟和真实数据示例以说明其性能。