🤖 AI Summary
This study addresses the performance degradation of traditional PCA in small-sample clustering caused by neglecting manifold curvature. We propose SHOPCA, a geometry-aware dimensionality reduction framework that introduces a novel covariance regularization mechanism based on the mean shape operator. By employing mixing coefficients to balance variance and curvature, alongside an unsupervised spectral gap criterion for adaptive parameter selection, SHOPCA effectively preserves intrinsic geometric structures. Extensive experiments across over 50 datasets demonstrate that SHOPCA significantly outperforms PCA in clustering metrics and surpasses UMAP in small-sample scenarios. Furthermore, the method offers distinct advantages in computational efficiency and parameter parsimony, successfully achieving robust unsupervised geometric structure preservation without requiring labeled data.
📝 Abstract
In this paper, we propose SHOPCA (Shape Operator-based Principal Component Analysis), a novel method for unsupervised metric learning and dimensionality reduction that incorporates differential geometric information into the covariance structure of classical PCA. SHOPCA regularizes the global covariance matrix using the mean shape operator, defined as the average of the absolute local shape operators estimated from the data manifold, steering principal components toward directions of both maximum variance and informative curvature. A single trace-normalized mixing coefficient $α$ controls the regularization, recovering standard PCA at $α= 0$ and a curvature-driven embedding as $α\to \infty$. We further introduce a fully unsupervised criterion for selecting $α$ based on the spectral eigengap of the regularized covariance matrix, maximizing the relative separation between the top-$d$ and remaining eigenvalues without using class labels. We evaluate SHOPCA on more than 50 real-world benchmark datasets, comparing it with PCA, ISOMAP, and UMAP using Adjusted Rand Index (ARI), Normalized Mutual Information (NMI), Fowlkes-Mallows index (FM), and V-measure. Results show that SHOPCA consistently improves clustering quality over PCA across a broad range of datasets and surpasses UMAP on small-sample settings, where iterative neighborhood-based manifold estimation can degrade. SHOPCA is computationally tractable, parameter-efficient, and applicable to domains requiring fully unsupervised, geometry-aware dimensionality reduction.