# IFNSO: Iteration-Free Newton-Schulz Orthogonalization

arXiv: https://arxiv.org/abs/2602.02500 · HTML: https://mlsanity.com/paper/2602.02500/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 23 references that are in the map.
- First posted on arXiv 18 Jan 2026.

## Builds on (most cited first, 9 of 9)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Old Optimizer, New Norm: An Anthology](https://mlsanity.com/paper/2409.20325/) (2024)
- [Training Deep Learning Models with Norm-Constrained LMOs](https://mlsanity.com/paper/2502.07529/) (2025)
- [Practical Efficiency of Muon for Pretraining](https://mlsanity.com/paper/2505.02222/) (2025)
- [MuonBP: Faster Muon via Block-Periodic Orthogonalization](https://mlsanity.com/paper/2510.16981/) (2025)
- [ROOT: Robust Orthogonalized Optimizer for Neural Network Training](https://mlsanity.com/paper/2511.20626/) (2025)
- [Riemannian Optimization on the Symplectic Stiefel Manifold](https://mlsanity.com/paper/2006.15226/) (2020)
- [Decentralized Riemannian Conjugate Gradient Method on the Stiefel Manifold](https://mlsanity.com/paper/2308.10547/) (2023)
- [MuonAll: Muon Variant for Efficient Finetuning of Large Language Models](https://mlsanity.com/paper/2511.06086/) (2025)

## Built on by (most cited first, 1 of 1)

- [Muonp: Muon with Fractional Spectral Powers](https://mlsanity.com/paper/2606.13867/) (2026)

## Abstract (arXiv)

The Newton-Schulz (NS) iteration has become a key technique for orthogonalization in optimizers such as Muon and for optimization on the Stiefel manifold. Despite its effectiveness, the conventional NS iteration incurs significant computational overhead due to repeated high-dimensional matrix multiplications. To overcome these limitations, we propose Iteration-Free Newton-Schulz Orthogonalization (IFNSO), a novel framework that consolidates the traditional iterative structure into a unified and Iteration-Free formulation. By analyzing the contribution of individual matrix powers, we streamline the process by removing insignificant terms and introducing a polynomial with learnable coefficients. These coefficients are optimized to ensure both superior computational efficiency and stable convergence. Extensive experiments demonstrate that IFNSO achieves superior performance compared to existing methods. Our code is available at: https://github.com/greekinRoma/Unified_Newton_Schulz_Orthogonalization.