  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.
