Building Neural Networks on Matrix Manifolds: A Gyrovector Space Approach

Xuan Son Nguyen, Shuo Yang
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:26031-26062, 2023.

Abstract

Matrix manifolds, such as manifolds of Symmetric Positive Definite (SPD) matrices and Grassmann manifolds, appear in many applications. Recently, by applying the theory of gyrogroups and gyrovector spaces that is a powerful framework for studying hyperbolic geometry, some works have attempted to build principled generalizations of Euclidean neural networks on matrix manifolds. However, due to the lack of many concepts in gyrovector spaces for the considered manifolds, e.g., the inner product and gyroangles, techniques and mathematical tools provided by these works are still limited compared to those developed for studying hyperbolic geometry. In this paper, we generalize some notions in gyrovector spaces for SPD and Grassmann manifolds, and propose new models and layers for building neural networks on these manifolds. We show the effectiveness of our approach in two applications, i.e., human action recognition and knowledge graph completion.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-nguyen23f, title = {Building Neural Networks on Matrix Manifolds: A Gyrovector Space Approach}, author = {Nguyen, Xuan Son and Yang, Shuo}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {26031--26062}, year = {2023}, editor = {Krause, Andreas and Brunskill, Emma and Cho, Kyunghyun and Engelhardt, Barbara and Sabato, Sivan and Scarlett, Jonathan}, volume = {202}, series = {Proceedings of Machine Learning Research}, month = {23--29 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v202/nguyen23f/nguyen23f.pdf}, url = {https://proceedings.mlr.press/v202/nguyen23f.html}, abstract = {Matrix manifolds, such as manifolds of Symmetric Positive Definite (SPD) matrices and Grassmann manifolds, appear in many applications. Recently, by applying the theory of gyrogroups and gyrovector spaces that is a powerful framework for studying hyperbolic geometry, some works have attempted to build principled generalizations of Euclidean neural networks on matrix manifolds. However, due to the lack of many concepts in gyrovector spaces for the considered manifolds, e.g., the inner product and gyroangles, techniques and mathematical tools provided by these works are still limited compared to those developed for studying hyperbolic geometry. In this paper, we generalize some notions in gyrovector spaces for SPD and Grassmann manifolds, and propose new models and layers for building neural networks on these manifolds. We show the effectiveness of our approach in two applications, i.e., human action recognition and knowledge graph completion.} }
Endnote
%0 Conference Paper %T Building Neural Networks on Matrix Manifolds: A Gyrovector Space Approach %A Xuan Son Nguyen %A Shuo Yang %B Proceedings of the 40th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2023 %E Andreas Krause %E Emma Brunskill %E Kyunghyun Cho %E Barbara Engelhardt %E Sivan Sabato %E Jonathan Scarlett %F pmlr-v202-nguyen23f %I PMLR %P 26031--26062 %U https://proceedings.mlr.press/v202/nguyen23f.html %V 202 %X Matrix manifolds, such as manifolds of Symmetric Positive Definite (SPD) matrices and Grassmann manifolds, appear in many applications. Recently, by applying the theory of gyrogroups and gyrovector spaces that is a powerful framework for studying hyperbolic geometry, some works have attempted to build principled generalizations of Euclidean neural networks on matrix manifolds. However, due to the lack of many concepts in gyrovector spaces for the considered manifolds, e.g., the inner product and gyroangles, techniques and mathematical tools provided by these works are still limited compared to those developed for studying hyperbolic geometry. In this paper, we generalize some notions in gyrovector spaces for SPD and Grassmann manifolds, and propose new models and layers for building neural networks on these manifolds. We show the effectiveness of our approach in two applications, i.e., human action recognition and knowledge graph completion.
APA
Nguyen, X.S. & Yang, S.. (2023). Building Neural Networks on Matrix Manifolds: A Gyrovector Space Approach. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:26031-26062 Available from https://proceedings.mlr.press/v202/nguyen23f.html.

Related Material