A Continuous-Time Analysis of Smoothed Matrix-Polar Spectral Gradient Flows for Muon-Type Optimization

Jinlin Liu; Song Chen; Jiaxu Liu; Chao Xu

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arXiv · 2608.01911

A Continuous-Time Analysis of Smoothed Matrix-Polar Spectral Gradient Flows for Muon-Type Optimization

Jinlin Liu·Song Chen·Jiaxu Liu·Chao Xu

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Abstract

This paper studies smoothed matrix-polar spectral gradient flows for unconstrained matrix-valued optimization.The canonical polar-factor map loses smoothness at rank-deficient matrices and becomes ill-conditioned as singular values approach zero, creating analytical difficulties.We therefore introduce a spectral feedback law generated by a smooth spectral potential and establish the regularity, monotonicity, boundedness, and dissipation properties of the feedback.Based on this feedback law, we propose a smoothed spectral gradient flow and prove well-posedness and global convergence of the flow.We derive convergence-rate results for the spectral gradient flow in nonconvex, convex, and Polyak--Lojasiewicz (PL) settings and analyze the Lyapunov structure and convergence of a momentum-augmented system under the same spectral feedback law. Furthermore, we provide a local descent-rate comparison between the smoothed spectral-gradient direction and the standard Frobenius-gradient direction using a general Hessian-based quadratic model. This analysis yields a verifiable normalized descent-rate advantage condition, showing that the local benefit of the spectral direction depends on both first-order alignment with the gradient matrix and the directional curvature induced by the Hessian.

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Jinlin Liu, Song Chen, Jiaxu Liu, Chao Xu. 2026-08-03. A Continuous-Time Analysis of Smoothed Matrix-Polar Spectral Gradient Flows for Muon-Type Optimization. https://arxiv.org/abs/2608.01911

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