aiqbench analysis
This paper introduces SHOPCA, an unsupervised dimensionality reduction technique that integrates differential geometric curvature information into the covariance structure of classical PCA. By regularizing with mean shape operators, the method steers principal components to capture both maximum variance and informative manifold geometry without requiring class labels or iterative neighborhood estimation. The approach demonstrates improved clustering performance over standard methods like UMAP in small-sample regimes while maintaining computational tractability for real-world datasets.
Shape Operator PCA: Curvature-Aware Projections for Geometric Machine Learning
arXiv:2608.15313v1 Announce Type: new 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 $\alpha$ controls the regularization, recovering standard PCA at $\alpha = 0$ and a curvature-driven embedding as $\alpha \to \infty$. We further introduce a fully unsupervised criterion for selecting $\alpha$ 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.
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 $\alpha$ controls the regularization, recovering standard PCA at $\alpha = 0$ and a curvature-driven embedding as $\alpha \to \infty$. We further introduce a fully unsupervised criterion for selecting $\alpha$ 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.
Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Computer Vision and Pattern Recognition (cs.CV); Machine Learning (stat.ML)
arXiv:2608.15313 [cs.LG]
(or arXiv:2608.15313v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2608.15313 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.
Full text under the publisher's open licence · 310 words
Related coverage
Curvature-Aware Radius Shrinkage for Adaptive Nearest Neighbor Classification
arXiv · Sep 1, 2026
A Geometry-Aware Triplane Field Network for Vehicle Aerodynamic Prediction
arXiv · Jun 9, 2026
A Geometry-Aware Triplane Field Network for Vehicle Aerodynamic Prediction
arXiv · Sep 3, 2026
Introduction to optimization methods for training SciML models
arXiv · Jun 3, 2026