# State-Dependent Lyapunov Analysis of Rank-1 Matrix Factorization

arXiv: https://arxiv.org/abs/2604.26993 · HTML: https://mlsanity.com/paper/2604.26993/

## 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 (most cited first, 6 of 6)

- [Gradient Descent on Neural Networks Typically Occurs at the Edge of Stability](https://mlsanity.com/paper/2103.00065/) (2021)
- [Understanding the unstable convergence of gradient descent](https://mlsanity.com/paper/2204.01050/) (2022)
- [Understanding Edge-of-Stability Training Dynamics with a Minimalist Example](https://mlsanity.com/paper/2210.03294/) (2022)
- [Large Learning Rate Tames Homogeneity: Convergence and Balancing Effect](https://mlsanity.com/paper/2110.03677/) (2021)
- [Gradient Descent with Large Step Sizes: Chaos and Fractal Convergence Region](https://mlsanity.com/paper/2509.25351/) (2025)
- [On the Convergence of Gradient Descent for Large Learning Rates](https://mlsanity.com/paper/2402.13108/) (2024)

## Built on by (most cited first, 2 of 2)

- [Unbiased Gradients, Moving Stability Boundaries: Exact Mini-Batch Geometry in Linear Self-Attention](https://mlsanity.com/paper/2605.21292/) (2026)
- [Adam at the Edge of Stability: Adaptive Feedback, Provable Oscillation, and Gradient Reversal](https://mlsanity.com/paper/2608.20638/) (2026)

## 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.