State-Dependent Lyapunov Analysis of Rank-1 Matrix Factorization
Researchers have developed a state-dependent Lyapunov analysis to rigorously characterize gradient descent behavior in rank-1 matrix factorization. The framework introduces a parameterized quadratic certificate whose monotone properties certify convergence to global minimizers below a critical step size, and reveal period-2 oscillatory dynamics above it. This provides a principled theoretical foundation for understanding edge-of-stability phenomena in a fundamental optimization setting.
A preprint posted to arXiv presents a state-dependent Lyapunov approach to analyzing gradient descent for rank-1 matrix factorization, a core problem in machine learning and numerical linear algebra. The central contribution is a parameterized quadratic certificate whose boundary-inward property forces a monotonically evolving state parameter, confining optimization trajectories to a shrinking family of level sets. Below a critical step size, this mechanism guarantees convergence to global minimizers for certified initializations. Above the critical step size, the same framework predicts a balanced terminal regime exhibiting period-2 behavior, consistent with the so-called edge-of-stability phenomenon observed empirically in neural network training. The authors further establish that the scalar certificate is uniquely determined under structural axioms and a natural normalization condition, ruling out it being an ad hoc construction. Numerical experiments suggest the mechanism generalizes beyond the proved cases, including two-dimensional rank-1 approximation and quartic augmentations of scalar factorization.
What's missing
The work is a preprint and has not yet undergone peer review. The theoretical results are currently proved only for specific low-dimensional settings; the extent to which the state-dependent Lyapunov mechanism formally generalizes to higher-rank or higher-dimensional factorizations remains an open question. The relationship to edge-of-stability phenomena in full-scale deep learning is suggested by numerical experiments but not yet theoretically established.
What different sources said
- arXiv cs.LGCenter
State-Dependent Lyapunov Analysis of Rank-1 Matrix Factorization
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