# Orthogonal Witness Control for Muon Optimization via Sigmoid Spectral Reshaping

arXiv: https://arxiv.org/abs/2609.33194 · HTML: https://mlsanity.com/paper/2609.33194/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 31 references that are in the map.
- First posted on arXiv 27 Sep 2026.

## Builds on (most cited first, 14 of 14)

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- [TruthfulQA: Measuring How Models Mimic Human Falsehoods](https://mlsanity.com/paper/2109.07958/) (2021)
- [Enhancing Chat Language Models by Scaling High-quality Instructional Conversations](https://mlsanity.com/paper/2305.14233/) (2023)
- [Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Condition](https://mlsanity.com/paper/1608.04636/) (2016)
- [UltraFeedback: Boosting Language Models with Scaled AI Feedback](https://mlsanity.com/paper/2310.01377/) (2023)
- [Relatively-Smooth Convex Optimization by First-Order Methods, and Applications](https://mlsanity.com/paper/1610.05708/) (2016)
- [HTMuon: Improving Muon via Heavy-Tailed Spectral Correction](https://mlsanity.com/paper/2603.10067/) (2026)
- [Delving into Muon and Beyond: Deep Analysis and Extensions](https://mlsanity.com/paper/2602.04669/) (2026)

## Abstract (arXiv)

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.