Most existing manifold dimension estimators rely on the assumption that the underlying manifold is locally flat within the neighborhoods under consideration. More recently, curvature-adjusted principal component analysis (CA-PCA) has emerged as a powerful alternative by explicitly accounting for the manifold's curvature. Motivated by these ideas, we propose a manifold dimension estimation framework that captures the local graph structure of the manifold through regression on local PCA coordinates. Within this framework, we introduce two representative estimators: quadratic embedding (QE) and total least squares (TLS). Experiments on both synthetic and real-world datasets demonstrate that these methods perform competitively with, and often outperform, state-of-the-art approaches.

翻译：现有大多数流形维数估计器依赖于一个假设，即所考虑的邻域内的底层流形是局部平坦的。近期，考虑曲率的主成分分析（CA-PCA）通过显式地计入流形的曲率，成为一种强有力的替代方法。受这些思想的启发，我们提出了一种流形维数估计框架，该框架通过对局部PCA坐标进行回归来捕捉流形的局部图结构。在此框架内，我们引入了两种代表性估计器：二次嵌入（QE）和总最小二乘（TLS）。在合成数据集和真实世界数据集上的实验表明，这些方法的性能与现有技术水平相当，且往往优于后者。