Covariance and Principal Component Analysis on Riemannian Manifolds and Graphs
Abstract
We develop notions of covariance and principal component analysis (PCA) for probability measures on Riemannian manifolds of bounded geometry and a discrete counterpart for distributions on the vertex sets of weighted simple graphs. Rather than anchoring variation at the Fréchet mean, whose uniqueness and usefulness can fail in this context, we consider variation about every point. This leads to a representation of each point on the manifold by a vector field derived from the heat kernel and thus to a map of the underlying manifold into a Hilbert space of vector fields, in which covariance and PCA carry over from the Euclidean setting. A distinguishing feature of this formulation is that one typically obtains infinitely many principal components, capable of capturing highly nonlinear geometric features and patterns of variation. We prove an embedding theorem for the mapping into the space of vector fields and develop two computational reductions of the theory: RiePCA, a finite-dimensional reduction based on vector fields supported on finitely many points, and GraphPCA, a discrete formulation for measures on weighted graphs in which vector fields assign orientations and magnitudes to edges. Numerical examples and experiments illustrate the behavior of the proposed methods on both synthetic and real data.
Related
- Online Covariance Estimation for Averaged Stochastic Gradient Descent on Riemannian Manifolds
- Principal component analysis for a mix of stationary and nonstationary variables
- Principal component analysis in Φ α-linear spaces
- Robust Multi-Task Learning for Principal Component Analysis
- Functional Principal Component Analysis for Manifold-Indexed Data
BibTeX
@misc{hartmann2026covariance,
title = {{Covariance and Principal Component Analysis on Riemannian Manifolds and Graphs}},
author = {Luiz Hartmann and Wenwen Li and Washington Mio},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.15753},
eprint = {2609.15753},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15753},
}
RIS
TY - UNPB
TI - Covariance and Principal Component Analysis on Riemannian Manifolds and Graphs
AU - Luiz Hartmann
AU - Wenwen Li
AU - Washington Mio
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2609.15753
UR - https://arxiv.org/abs/2609.15753
AB - We develop notions of covariance and principal component analysis (PCA) for probability measures on Riemannian manifolds of bounded geometry and a discrete counterpart for distributions on the vertex sets of weighted simple graphs. Rather than anchoring variation at the Fréchet mean, whose uniqueness and usefulness can fail in this context, we consider variation about every point. This leads to a representation of each point on the manifold by a vector field derived from the heat kernel and thus to a map of the underlying manifold into a Hilbert space of vector fields, in which covariance and PCA carry over from the Euclidean setting. A distinguishing feature of this formulation is that one typically obtains infinitely many principal components, capable of capturing highly nonlinear geometric features and patterns of variation. We prove an embedding theorem for the mapping into the space of vector fields and develop two computational reductions of the theory: RiePCA, a finite-dimensional reduction based on vector fields supported on finitely many points, and GraphPCA, a discrete formulation for measures on weighted graphs in which vector fields assign orientations and magnitudes to edges. Numerical examples and experiments illustrate the behavior of the proposed methods on both synthetic and real data.
ER -