A state-dependent Lyapunov method with quadratic certificates proves global convergence for gradient descent on rank-1 matrix factorization by deriving the certificates from structural axioms rather than ad hoc constructions.
A machine-rendered reading of the paper's core claim, the
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Gradient descent is commonly used to factorize a matrix into lower-rank components by minimizing a loss. For the simple rank-1 case, the paper introduces a certificate function that is quadratic and depends on a state parameter called delta. As the algorithm takes steps, the level sets of this certificate shrink, forcing delta to change monotonically. When the certificate is valid, this guarantees the iterates reach the global minimum. Beyond that point, the dynamics are pushed toward a terminal balanced manifold. The authors show that the certificate is not invented arbitrarily but follows uniquely from basic structural axioms of the dynamics combined with a local Lagrange multiplier analysis that separates signal and noise components. They also run numerical tests on a two-dimensional diagonal matrix example and on a loss function augmented with a quartic term, finding that the same certificate idea remains predictive after choosing an empirical threshold.
Core claim
In the certified regime, this mechanism yields convergence to a global minimizer; in the post-critical regime, it forces trajectories toward a terminal balanced manifold. The certificates arise from the monotonicity structure of the dynamics, rather than from ad hoc algebraic constructions.
Load-bearing premise
The structural axioms of the state-dependent Lyapunov framework hold, allowing the scalar certificate to be uniquely determined by local Lagrange analysis that constrains the signal and noise blocks of rank-1 extensions.
read the original abstract
We study gradient descent for rank-1 matrix factorization through a state-dependent Lyapunov perspective. The central object is a parameterized quadratic certificate $I(\delta;\,\cdot)$ whose boundary-inward property induces a monotone state parameter $\delta_t$, 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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The central claim rests on a set of structural axioms for the Lyapunov framework and on local Lagrange analysis; no explicit free parameters or new physical entities are introduced in the abstract.
axioms (1)
domain assumptionStructural axioms of the state-dependent Lyapunov framework Invoked to guarantee that the scalar certificate is uniquely determined.
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