<ul><li>A new algorithm called Polar Express has been introduced for computing the polar decomposition and matrix sign function, which are crucial in deep learning applications like the Muon optimization framework.</li><li>The Polar Express algorithm is designed to be GPU-friendly, highly efficient, and compatible with GPUs, addressing the specific requirements of deep learning where high accuracy may not be necessary.</li><li>Unlike traditional numerical analysis methods like Newton-Schulz or rational functions, Polar Express uses matrix-matrix multiplications, ensuring GPU compatibility and efficient computation.</li><li>The algorithm adapts the polynomial update rule by solving a minimax optimization problem, leading to rapid early convergence and fast asymptotic convergence, while also addressing finite-precision issues for stability in practice.</li></ul>

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

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