State-Dependent Lyapunov Method for Rank-1 Matrix Factorization

ArXi:2604.26993v1 Announce Type: cross We study gradient descent for rank-1 matrix factorization through a certificate-based viewpoint. The central object is a parameterized quadratic certificate $I(\delta;\,\cdot)$ whose level sets shrink along the dynamics, thereby inducing a monotone state parameter $\delta_t$. In the certified regime, this mechanism yields convergence to a global minimizer; in the post-critical regime, it forces trajectories toward a terminal balanced manifold.