On the Geometry of Low Rank Correlation Matrices

Abstract

Correlation matrices of bounded rank form a stratified space whose factor representations lie in a product of spheres. Using the orbit-type description of the rank stratification, we prove that padding factors preserves the Riemannian geometry of each fixed-rank stratum and the intrinsic length metric of every rank-bounded subset. In the reduced model, a Sylvester equation describes the mechanical connection and its horizontal lifts. We derive sectional-curvature formulas and bounds, characterize zero-curvature directions, and determine the connection's holonomy groups. In particular, a flat rank-two connection can have nontrivial dihedral holonomy. For geodesics subject to a rank bound, rank cannot decrease in the interior of a minimizing segment with maximal-rank endpoints; the local excess over the unconstrained quotient distance is controlled by a dimension-independent cubic estimate. Finally, we compute the exact change of measure between quotient Riemannian volume and Euclidean surface volume in the off-diagonal entries. Coarea yields explicit factor densities for both normalized uniform laws.

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BibTeX RIS

BibTeX

@misc{bisson2026geometry,
  title = {{On the Geometry of Low Rank Correlation Matrices}},
  author = {Olivier Bisson and David Loiseaux and Xavier Pennec},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.28475},
  eprint = {2610.28475},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.28475},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - On the Geometry of Low Rank Correlation Matrices
AU  - Olivier Bisson
AU  - David Loiseaux
AU  - Xavier Pennec
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2610.28475
UR  - https://arxiv.org/abs/2610.28475
AB  - Correlation matrices of bounded rank form a stratified space whose factor representations lie in a product of spheres. Using the orbit-type description of the rank stratification, we prove that padding factors preserves the Riemannian geometry of each fixed-rank stratum and the intrinsic length metric of every rank-bounded subset. In the reduced model, a Sylvester equation describes the mechanical connection and its horizontal lifts. We derive sectional-curvature formulas and bounds, characterize zero-curvature directions, and determine the connection's holonomy groups. In particular, a flat rank-two connection can have nontrivial dihedral holonomy. For geodesics subject to a rank bound, rank cannot decrease in the interior of a minimizing segment with maximal-rank endpoints; the local excess over the unconstrained quotient distance is controlled by a dimension-independent cubic estimate. Finally, we compute the exact change of measure between quotient Riemannian volume and Euclidean surface volume in the off-diagonal entries. Coarea yields explicit factor densities for both normalized uniform laws.
ER  -

Abstract

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