Orthogonal Witness Control for Muon Optimization via Sigmoid Spectral Reshaping

Matrix-valued optimizers such as Muon exploit the spectral structure of neural network updates through Newton–Schulz orthogonalization, but their near-flattening of the singular spectrum discards relative magnitude information across gradient modes. We introduce Soren (Spectral Orthogonal Reshaping), a matrix-valued optimizer that preserves the singular subspaces of the gradient while applying a bounded, monotone sigmoid transformation to its singular values. This smoothly compresses dominant modes without fully flattening the spectrum. We interpret Soren as a positive-definite preconditioned gradient method and establish convergence guarantees under relative smoothness and metric Polyak–\Lojasiewicz geometry. To avoid explicit singular value decomposition, we further develop a finite-depth Soft Newton–Schulz (SNS) polynomial realization of the sigmoid spectral map and characterize how its spectral approximation affects the induced convergence geometry. Experiments across LLM pre-training, supervised fine-tuning, and direct preference optimization demonstrate the effectiveness and robustness of Soren against established optimizers.