Local principal component analysis (Local PCA) has proven to be an effective tool for estimating the intrinsic dimension of a manifold. More recently, curvature-adjusted PCA (CA-PCA) has improved upon this approach by explicitly accounting for the curvature of the underlying manifold, rather than assuming local flatness. Building on these insights, we propose a general framework for manifold dimension estimation that captures the manifold's local graph structure by integrating PCA with regression-based techniques. 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 alternatives.

翻译：局部主成分分析（Local PCA）已被证明是估计流形内在维度的有效工具。近年来，曲率调整主成分分析（CA-PCA）通过显式考虑底层流形的曲率而非假设局部平坦性，进一步改进了该方法。基于这些见解，我们提出了一种通用的流形维度估计框架，通过将主成分分析与基于回归的技术相结合，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）和总体最小二乘法（TLS）。在合成数据集和真实数据集上的实验表明，这些方法在性能上与当前最先进的替代方案相当，且往往表现更优。