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

arXiv: https://arxiv.org/abs/2604.18816 · HTML: https://mlsanity.com/paper/2604.18816/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 45 references that are in the map.
- First posted on arXiv 20 Apr 2026.

## Builds on (most cited first, 15 of 15)

- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Understanding over-squashing and bottlenecks on graphs via curvature](https://mlsanity.com/paper/2111.14522/) (2021)
- [Optimal Transport for Domain Adaptation](https://mlsanity.com/paper/1507.00504/) (2015)
- [Hyperbolic Graph Convolutional Neural Networks](https://mlsanity.com/paper/1910.12933/) (2019)
- [Graph Learning: A Survey](https://mlsanity.com/paper/2105.00696/) (2021)
- [Learning graphs from data: A signal representation perspective](https://mlsanity.com/paper/1806.00848/) (2018)
- [Community Detection on Networks with Ricci Flow](https://mlsanity.com/paper/1907.03993/) (2019)
- [Sliced Wasserstein Distance for Learning Gaussian Mixture Models](https://mlsanity.com/paper/1711.05376/) (2017)
- [Tree-Sliced Variants of Wasserstein Distances](https://mlsanity.com/paper/1902.00342/) (2019)
- [Beyond Euclid: An Illustrated Guide to Modern Machine Learning with Geometric, Topological, and Algebraic Structures](https://mlsanity.com/paper/2407.09468/) (2024)
- [Curvature-aware Manifold Learning](https://mlsanity.com/paper/1706.07167/) (2017)
- [Deep Learning as Ricci Flow](https://mlsanity.com/paper/2404.14265/) (2024)

## Abstract (arXiv)

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 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.