Beyond PCA: Manifold Dimension Estimation via Local Graph Structure

Abstract: 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.