Shape Operator PCA: Curvature-Aware Projections for Geometric Machine Learning
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 \inft...
| Authors | Alexandre L. M. Levada |
|---|---|
| Primary category | cs.LG |
| Published | 15 August 2026 |
| Last updated at source | 15 August 2026 |
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