Riemannian Networks over Full-Rank Correlation Matrices

Representations on the Symmetric Positive Definite (SPD) manifold have garnered significant attention across different applications. In contrast, the manifold of full-rank correlation matrices, a normalized alternative to SPD matrices, remains largely underexplored. This paper introduces Riemannian networks over the correlation manifold, leveraging five recently developed correlation geometries. We systematically extend basic layers, including Multinomial Logistic Regression (MLR), Fully Connected (FC), and convolutional layers, to these geometries. Besides, we present methods for accurate backpropagation for two correlation geometries. Experiments comparing our approach against existing SPD and Grassmannian networks demonstrate its effectiveness.

Empirical Inference

Bernhard Schölkopf

Author(s):	Chen, Z. and Wu, X.-J. and Schölkopf, B. and Sebe, N.
Links:	arXiv
Book Title:	Proceedings of the 43rd International Conference on Machine Learning (ICML)
Volume:	306
Pages:	17467--17507
Year:	2026
Month:	July

Series:	Proceedings of Machine Learning Research
Editors:	Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto
Publisher:	PMLR

BibTeX Type:	Conference Paper (inproceedings)

Event Name:	International Conference on Machine Learning
Event Place:	Seoul, South Korea
State:	Published
URL:	https://proceedings.mlr.press/v306/chen26fb.html

BibTeX

@inproceedings{CheWuSchSeb26,
  title = {Riemannian Networks over Full-Rank Correlation Matrices},
  booktitle = {Proceedings of the 43rd International Conference on Machine Learning (ICML)},
  abstract = {Representations on the Symmetric Positive Definite (SPD) manifold have garnered significant attention across different applications. In contrast, the manifold of full-rank correlation matrices, a normalized alternative to SPD matrices, remains largely underexplored. This paper introduces Riemannian networks over the correlation manifold, leveraging five recently developed correlation geometries. We systematically extend basic layers, including Multinomial Logistic Regression (MLR), Fully Connected (FC), and convolutional layers, to these geometries. Besides, we present methods for accurate backpropagation for two correlation geometries. Experiments comparing our approach against existing SPD and Grassmannian networks demonstrate its effectiveness.},
  volume = {306},
  pages = {17467--17507},
  series = {Proceedings of Machine Learning Research},
  editors = {Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto},
  publisher = {PMLR},
  month = jul,
  year = {2026},
  author = {Chen, Z. and Wu, X.-J. and Sch{\"o}lkopf, B. and Sebe, N.},
  url = {https://proceedings.mlr.press/v306/chen26fb.html},
  month_numeric = {7}
}