Critical Points and Convergence Analysis of Generative Deep Linear Networks Trained with Bures-Wasserstein Loss

Pierre Bréchet, Katerina Papagiannouli, Jing An, Guido Montufar
Proceedings of the 40th International Conference on Machine Learning, PMLR 202:3106-3147, 2023.

Abstract

We consider a deep matrix factorization model of covariance matrices trained with the Bures-Wasserstein distance. While recent works have made advances in the study of the optimization problem for overparametrized low-rank matrix approximation, much emphasis has been placed on discriminative settings and the square loss. In contrast, our model considers another type of loss and connects with the generative setting. We characterize the critical points and minimizers of the Bures-Wasserstein distance over the space of rank-bounded matrices. The Hessian of this loss at low-rank matrices can theoretically blow up, which creates challenges to analyze convergence of gradient optimization methods. We establish convergence results for gradient flow using a smooth perturbative version of the loss as well as convergence results for finite step size gradient descent under certain assumptions on the initial weights.

Cite this Paper

BibTeX
@InProceedings{pmlr-v202-brechet23a, title = {Critical Points and Convergence Analysis of Generative Deep Linear Networks Trained with Bures-{W}asserstein Loss}, author = {Br\'{e}chet, Pierre and Papagiannouli, Katerina and An, Jing and Montufar, Guido}, booktitle = {Proceedings of the 40th International Conference on Machine Learning}, pages = {3106--3147}, 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/brechet23a/brechet23a.pdf}, url = {https://proceedings.mlr.press/v202/brechet23a.html}, abstract = {We consider a deep matrix factorization model of covariance matrices trained with the Bures-Wasserstein distance. While recent works have made advances in the study of the optimization problem for overparametrized low-rank matrix approximation, much emphasis has been placed on discriminative settings and the square loss. In contrast, our model considers another type of loss and connects with the generative setting. We characterize the critical points and minimizers of the Bures-Wasserstein distance over the space of rank-bounded matrices. The Hessian of this loss at low-rank matrices can theoretically blow up, which creates challenges to analyze convergence of gradient optimization methods. We establish convergence results for gradient flow using a smooth perturbative version of the loss as well as convergence results for finite step size gradient descent under certain assumptions on the initial weights.} }
Endnote
%0 Conference Paper %T Critical Points and Convergence Analysis of Generative Deep Linear Networks Trained with Bures-Wasserstein Loss %A Pierre Bréchet %A Katerina Papagiannouli %A Jing An %A Guido Montufar %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-brechet23a %I PMLR %P 3106--3147 %U https://proceedings.mlr.press/v202/brechet23a.html %V 202 %X We consider a deep matrix factorization model of covariance matrices trained with the Bures-Wasserstein distance. While recent works have made advances in the study of the optimization problem for overparametrized low-rank matrix approximation, much emphasis has been placed on discriminative settings and the square loss. In contrast, our model considers another type of loss and connects with the generative setting. We characterize the critical points and minimizers of the Bures-Wasserstein distance over the space of rank-bounded matrices. The Hessian of this loss at low-rank matrices can theoretically blow up, which creates challenges to analyze convergence of gradient optimization methods. We establish convergence results for gradient flow using a smooth perturbative version of the loss as well as convergence results for finite step size gradient descent under certain assumptions on the initial weights.
APA
Bréchet, P., Papagiannouli, K., An, J. & Montufar, G.. (2023). Critical Points and Convergence Analysis of Generative Deep Linear Networks Trained with Bures-Wasserstein Loss. Proceedings of the 40th International Conference on Machine Learning, in Proceedings of Machine Learning Research 202:3106-3147 Available from https://proceedings.mlr.press/v202/brechet23a.html.

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