用状态依赖李雅普诺夫分析揭示矩阵分解梯度下降的收敛与周期行为
State-Dependent Lyapunov Analysis of Rank-1 Matrix Factorization
- 构建参数化二次证书，通过状态单调性约束轨迹范围
- 低于临界步长时收敛至全局最优，高于时呈现周期2现象
- 证书唯一性由机制决定，数值实验支持推广到高维情形
我们从状态依赖李雅普诺夫视角研究秩1矩阵分解的梯度下降。核心是参数化二次证书 $I(δ; \cdot)$，其边界向内性质诱导出单调状态参数 $δ_t$，从而证明轨迹被限制在不断收缩的层级集族中。对于低于临界步长的初始化，该机制确保收敛至全局最小值；高于临界步长时，相同单调机制导致平衡终端态；在一定超临界步长范围内，简化动力学表现出周期2行为，与边缘稳定性现象一致。进一步表明，该标量证书并非人为构造：在结构公理和自然状态参数归一化下，它由单调性机制唯一确定。数值实验显示，此状态依赖李雅普诺夫机制在证明范围外仍成立，包括二维秩1近似及标量因子化的四次扩展。
原文摘要 · Abstract (English)
We study gradient descent for rank-1 matrix factorization through a state-dependent Lyapunov perspective. The central object is a parameterized quadratic certificate $I(δ;\,\cdot)$ whose boundary-inward property induces a monotone state parameter $δ_t$, thereby certifying that the trajectory is confined to a shrinking family of level sets. For certified initializations below the critical step size, this mechanism proves convergence to global minimizers. Above the critical step size, the same monotone-state mechanism instead leads to a balanced terminal regime; for a range of post-critical step sizes, the reduced dynamics exhibit period-2 behavior consistent with edge-of-stability phenomena. We further show that the scalar certificate is not an ad hoc algebraic construction: under structural axioms and a natural state-parameter normalization, it is uniquely determined by the monotonicity mechanism. Numerical experiments suggest that this state-dependent Lyapunov mechanism persists beyond the proved cases, including two-dimensional rank-1 approximation and quartic augmentations of scalar factorization.
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