Po Chen

Po Chen, Rujun Jiang, Peng Wang

Despite its wide range of applications across various domains, the optimization foundations of deep matrix factorization (DMF) remain largely open. In this work, we aim to fill this gap by conducting a comprehensive study of the loss landscape of the regularized DMF problem. Toward this goal, we first provide a closed-form characterization of all critical points of the problem. Building on this, we establish precise conditions under which a critical point is a local minimizer, a global minimizer, a strict saddle point, or a non-strict saddle point. Leveraging these results, we derive a necessary and sufficient condition under which every critical point is either a local minimizer or a strict saddle point. This provides insights into why gradient-based methods almost always converge to a local minimizer of the regularized DMF problem. Finally, we conduct numerical experiments to visualize its loss landscape to support our theory.

Po Chen, Rujun Jiang, Peng Wang

The optimization foundations of deep linear networks have recently received significant attention. However, due to their inherent non-convexity and hierarchical structure, analyzing the loss functions of deep linear networks remains a challenging task. In this work, we study the local geometric landscape of the regularized squared loss of deep linear networks around each critical point. Specifically, we derive a closed-form characterization of the critical point set and establish an error bound for the regularized loss under mild conditions on network width and regularization parameters. Notably, this error bound quantifies the distance from a point to the critical point set in terms of the current gradient norm, which can be used to derive linear convergence of first-order methods. To support our theoretical findings, we conduct numerical experiments and demonstrate that gradient descent converges linearly to a critical point when optimizing the regularized loss of deep linear networks.