Curvature-Aware PCA with Geodesic Tangent Space Aggregation for Semi-Supervised Learning

TL;DR

GTSA-PCA is a novel geometric extension of PCA that incorporates curvature and geodesic information to improve manifold learning, especially with limited data.

Contribution

It introduces a curvature-aware local covariance and a geodesic alignment operator to unify spectral PCA with manifold geometry in a semi-supervised framework.

Findings

Outperforms PCA, Kernel PCA, and Supervised PCA on real datasets.

Shows robustness in small sample size and high-curvature scenarios.

Provides a geometry-aware embedding that enhances discriminative structure.

Abstract

Principal Component Analysis (PCA) is a fundamental tool for representation learning, but its global linear formulation fails to capture the structure of data supported on curved manifolds. In contrast, manifold learning methods model nonlinearity but often sacrifice the spectral structure and stability of PCA. We propose \emph{Geodesic Tangent Space Aggregation PCA (GTSA-PCA)}, a geometric extension of PCA that integrates curvature awareness and geodesic consistency within a unified spectral framework. Our approach replaces the global covariance operator with curvature-weighted local covariance operators defined over a $k$-nearest neighbor graph, yielding local tangent subspaces that adapt to the manifold while suppressing high-curvature distortions. We then introduce a geodesic alignment operator that combines intrinsic graph distances with subspace affinities to globally synchronize these local representations. The resulting operator admits a spectral decomposition whose leading components define a geometry-aware embedding. We further incorporate semi-supervised information to guide the alignment, improving discriminative structure with minimal supervision. Experiments on real datasets show consistent improvements over PCA, Kernel PCA, Supervised PCA and strong graph-based baselines such as UMAP, particularly in small sample size and high-curvature regimes. Our results position GTSA-PCA as a principled bridge between statistical and geometric approaches to dimensionality reduction.