Beyond PCA: Manifold Dimension Estimation via Local Graph Structure
This work addresses the challenge of accurate manifold dimension estimation for data analysis, but it is incremental as it builds on existing Local PCA and CA-PCA methods.
The paper tackled the problem of estimating the intrinsic dimension of a manifold by proposing a framework that integrates PCA with regression-based techniques to capture local graph structure, resulting in methods that perform competitively or outperform state-of-the-art alternatives on synthetic and real-world datasets.
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.