We introduce a new regression framework designed to deal with large-scale, complex data that lies around a low-dimensional manifold with noises. Our approach first constructs a graph representation, referred to as the skeleton, to capture the underlying geometric structure. We then define metrics on the skeleton graph and apply nonparametric regression techniques, along with feature transformations based on the graph, to estimate the regression function. We also discuss the limitations of some nonparametric regressors with respect to the general metric space such as the skeleton graph. The proposed regression framework suggests a novel way to deal with data with underlying geometric structures and provides additional advantages in handling the union of multiple manifolds, additive noises, and noisy observations. We provide statistical guarantees for the proposed method and demonstrate its effectiveness through simulations and real data examples.

翻译：我们提出了一种新的回归框架，旨在处理具有低维流形结构且包含噪声的大规模复杂数据。该方法首先构建一个称为骨架的图表示，以捕捉数据潜在的几何结构；随后在骨架图上定义度量，并应用基于图的非参数回归技术与特征变换来估计回归函数。我们还讨论了一些非参数回归器在一般度量空间（如骨架图）中的局限性。所提出的回归框架为处理具有潜在几何结构的数据提供了一种新思路，并在处理多流形联合、加性噪声及噪声观测方面展现出额外优势。我们给出了该方法的统计保证，并通过模拟和真实数据实例验证其有效性。