Abstract
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.
To explain the origin of these certificates, we formulate a state-dependent Lyapunov framework based on structural axioms. Within this framework, the scalar certificate is uniquely determined, and the same local Lagrange analysis constrains the signal and noise blocks of rank-1 extensions. Thus, the certificates arise from the monotonicity structure of the dynamics, rather than from ad hoc algebraic constructions.
We also provide numerical evidence beyond the proved cases. For the 2-dimensional rank-1 approximation problem X=\mathrm{diag}(1,\sigma) with \sigma\in(0,1), the experiments are consistent with the existence of a C^1 admissible certificate branch. For the quartic-augmented scalar loss \frac12(ab-1)^2+\mu(ab-1)^4, the same scalar certificate remains predictive for several values of \mu after choosing an empirical threshold. These experiments suggest that the state-dependent Lyapunov method may extend beyond the settings proved in this paper.
