State-Dependent Lyapunov Analysis of Rank-1 Matrix Factorization

Abstract:We develop a state-dependent Lyapunov framework for gradient descent on rank-1 matrix factorization. A parameterized quadratic certificate $I(\delta;\,\cdot)$ generates strictly nested sublevel sets. Their ordering assigns each point a state $\delta$, while a boundary-inward property makes this state monotone along gradient-descent trajectories. Together with internal chain transitivity, this geometry identifies the limiting dynamics even when all relevant stationary points are unstable. For scalar and rank-1 factorization, it yields convergence to a global minimizer below the stability threshold and to a balanced period-$2$ orbit in a post-critical interval, for almost every initialization in an explicit region. We formalize the mechanism through structural and dynamical axioms. Within the origin-centered, exchange-symmetric quadratic class, the axioms determine the ordered level-set geometry uniquely up to a monotone relabeling of the state, recovering the scalar geometry identified by Liang--Mont{ú}far. Additional analytic and numerical examples suggest broader applicability.

Numerical Analysis (math.NA); Machine Learning (cs.LG); Optimization and Control (math.OC)

arXiv:2604.26993 [math.NA]

(or arXiv:2604.26993v3 [math.NA] for this version)

https://doi.org/10.48550/arXiv.2604.26993 Focus to learn more arXiv-issued DOI via DataCite

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