In my master’s thesis project, we investigate the low-rank matrix factorization problem and its related formulations, which play a central role in diverse applications such as machine learning, digital communication, and image processing. Enforcing the low-rank constraint via matrix factorization has become the standard approach to solving these problems. Perhaps surprisingly, every asymmetric factorization problem can be reformulated as a symmetric one, making the following optimization objective a general representation of the problem:
Our work begins with an in-depth analysis of the non-convex loss landscape, focusing on the characterization of critical points. We then leverage advanced tools from random matrix theory to study the gradient flow dynamics of the matrix factorization problem. Our main result provides a deterministic equivalent for the gradient flow dynamics starting from a random initialization, offering new insights into the fundamental behavior of low-rank matrix factorization.