State-Dependent Lyapunov Analysis of Rank-1 Matrix Factorization
We develop a state-dependent Lyapunov framework for gradient descent on rank-1 matrix factorization. A parameterized quadratic certificate $I(δ;\,\cdot)$ 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 e...
| Authors | Jaehong Moon |
|---|---|
| Primary category | math.NA |
| Published | 28 April 2026 |
| Last updated at source | 6 September 2026 |
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