Convergence of Generative Deep Linear Networks Trained with Bures-Wasserstein Loss

Pierre Bréchet, Katerina Papagiannouli, Jing An, Guido Montufar

Keywords: deep linear network, low-rank approximation, Bures-Wasserstein distance, optimal transport, implicit generative model, critical points

TL;DR: We prove convergence of gradient decent optimization of generative deep linear networks trained with the Bures-Wasserstein loss.

Abstract: We consider a deep matrix factorization model of covariance matrices trained with the Bures-Wasserstein distance. While recent works have made important 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 interesting 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. For low-rank matrices the Hessian of this loss can blow up, which creates challenges to analyze convergence of optimizaton methods. We establish convergence results for gradient flow using a smooth perturbative version of the loss and convergence results for finite step size gradient descent under certain assumptions on the initial weights

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