🤖 AI Summary
This paper addresses the limited accuracy of intrinsic dimension estimation on manifolds caused by neglecting local curvature and geometric structure. To overcome this, we propose a unified framework integrating principal component analysis (PCA) with regression modeling. Methodologically, we introduce two novel estimators—quadratic embedding and total least squares—that explicitly encode local curvature and graph-topological structure, thereby mitigating bias inherent in conventional PCA-based approaches, especially in highly curved regions. Theoretically grounded and empirically validated, our framework exhibits both robustness and interpretability. Experiments on synthetic and real-world datasets demonstrate that it matches or surpasses state-of-the-art methods in estimation accuracy, with particularly pronounced improvements in strongly curved manifold regions.
📝 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.