State-Dependent Lyapunov Analysis of Rank-1 Matrix Factorization
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Abstract (arXiv)
We develop a state-dependent Lyapunov framework for gradient descent on rank-1 matrix factorization. A parameterized quadratic certificate I(δ;,·) generates strictly nested sublevel sets. Their ordering assigns each point a state δ, 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'ufar. Additional analytic and numerical examples suggest broader applicability.
Key facts. Cited by 2 papers in the map (data through 14 Oct 2026). Builds on 6 of its 32 references that are in the map. First posted on arXiv 28 Apr 2026.
Builds on
6 of 32 references.
References · 6
- 2021Gradient Descent on Neural Networks Typically Occurs at the Edge of Stabilitycited by 427
- 2022Understanding the unstable convergence of gradient descentcited by 86
- 2022Understanding Edge-of-Stability Training Dynamics with a Minimalist Examplecited by 56
- 2021Large Learning Rate Tames Homogeneity: Convergence and Balancing Effectcited by 50
- 2025Gradient Descent with Large Step Sizes: Chaos and Fractal Convergence Regioncited by 7
- 2024On the Convergence of Gradient Descent for Large Learning Ratescited by 3
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Citing papers per year: 2026: 2.
2 papers.