The Polar Express: Optimal Matrix Sign Methods and Their Application to the Muon Algorithm

arXiv:2505.16932v4 Announce Type: replace-cross Abstract: Computing the polar decomposition and the related matrix sign function has been a well-studied problem in numerical analysis for decades. Recently, it has emerged as an important subroutine within the Muon optimizer for training deep neural networks. However, the requirements of this application differ sharply from classical settings: deep learning demands GPU-friendly algorithms that prioritize high throughput over high precision. We int