用于缪尔型优化的平滑矩阵-极谱梯度流的连续时间分析
A Continuous-Time Analysis of Smoothed Matrix-Polar Spectral Gradient Flows for Muon-Type Optimization
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中文总结 AI 辅助
本文针对标准极分解映射的秩亏与病态问题，引入谱反馈律提出平滑矩阵-极谱梯度流，证明其适定性与全局收敛性，分析不同设置下的收敛速率，并对比其与标准梯度方向的局部下降速率优势。
中文摘要 AI 辅助
本文研究用于无约束矩阵值优化的平滑矩阵-极谱梯度流。标准极分解映射在秩亏矩阵处失去光滑性，且当奇异值趋近于零时变得病态，这会产生分析问题。因此，我们引入由光滑谱势生成的谱反馈律，并建立该反馈律的正则性、单调性、有界性和耗散性质。基于该反馈律，我们提出一种平滑谱梯度流，并证明其适定性和全局收敛性。我们推导了该谱梯度流在非凸、凸以及Polyak-Lojasiewicz（PL）设置下的收敛速率结果，且在相同谱反馈律下分析了动量增强系统的李雅普诺夫结构和收敛性。此外，我们使用基于海森矩阵的一般二次模型，对平滑谱梯度方向与标准弗罗贝尼乌斯梯度方向进行了局部下降速率比较。该分析得出了可验证的归一化下降速率优势条件，表明谱方向的局部优势既取决于与梯度矩阵的一阶对齐，也取决于海森矩阵诱导的方向曲率。
英文摘要
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.
