State-Dependent Lyapunov Analysis of Rank-1 Matrix Factorization
Researchers developed a state-dependent Lyapunov analysis framework to study gradient descent behavior in rank-1 matrix factorization using a parameterized quadratic certificate. The analysis proves convergence to global minimizers for certified initializations below a critical step size, and characterizes period-2 behavior and edge-of-stability phenomena above it. This work provides theoretical foundations for understanding gradient descent dynamics in a fundamental optimization problem.
The paper introduces a state-dependent Lyapunov perspective for analyzing gradient descent in rank-1 matrix factorization, centered on a parameterized quadratic certificate whose boundary-inward property creates a monotone state parameter that confines trajectories to shrinking level sets. For step sizes below a critical threshold with certified initializations, this mechanism provably guarantees convergence to global minimizers. Above the critical step size, the same monotonicity mechanism produces a balanced terminal regime exhibiting period-2 behavior consistent with observed edge-of-stability phenomena in neural network training. The authors demonstrate that the scalar certificate is uniquely determined by their monotonicity axioms and normalization conditions, rather than being an ad hoc construction. Numerical experiments suggest the mechanism extends beyond the theoretical guarantees to higher-dimensional cases and quartic augmentations.
What's missing
The paper does not discuss computational complexity or practical implications for large-scale applications. The relationship between these theoretical findings and empirical observations in deep learning beyond the rank-1 setting remains largely unexplored in the abstract.
What different sources said
- arXiv cs.LGCenter
State-Dependent Lyapunov Analysis of Rank-1 Matrix Factorization
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