Scientific Paper

This section highlights the scientific foundation behind the Riemannian STATS package.

Riemannian Principal Component Analysis

R-PCA is a novel extension of Principal Component Analysis designed to operate on datasets that reside on non-Euclidean spaces. Instead of assuming a flat Euclidean structure, R-PCA leverages local geometric information, derived via the UMAP algorithm, to define a Riemannian manifold over the data. This allows for more accurate dimensionality reduction and statistical analysis that respects the intrinsic structure of complex datasets.

Applications include:

  • High-dimensional structured data

  • Clustering with geometric consistency

  • Real-world datasets such as image sets (e.g., Olivetti Faces)

R-PCA improves interpretability, variance capture, and clustering performance compared to standard PCA especially in cases where local metrics vary significantly across the dataset.

📄 Link to the complete paper: "Riemannian Principal Component Analysis"

Watch the official presentation introducing the concepts behind R-PCA. (Video in Spanish)