# Tensorion: A Tensor-Aware Generalization of the Muon Optimizer

arXiv: https://arxiv.org/abs/2606.25975 · HTML: https://mlsanity.com/paper/2606.25975/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 54 references that are in the map.
- First posted on arXiv 24 Jun 2026.

## Builds on (most cited first, 23 of 23)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Swin Transformer: Hierarchical Vision Transformer using Shifted Windows](https://mlsanity.com/paper/2103.14030/) (2021)
- [Optimizing Neural Networks with Kronecker-factored Approximate Curvature](https://mlsanity.com/paper/1503.05671/) (2015)
- [signSGD: Compressed Optimisation for Non-Convex Problems](https://mlsanity.com/paper/1802.04434/) (2018)
- [Muon is Scalable for LLM Training](https://mlsanity.com/paper/2502.16982/) (2025)
- [Early Convolutions Help Transformers See Better](https://mlsanity.com/paper/2106.14881/) (2021)
- [Tensorizing Neural Networks](https://mlsanity.com/paper/1509.06569/) (2015)
- [Old Optimizer, New Norm: An Anthology](https://mlsanity.com/paper/2409.20325/) (2024)
- [SOAP: Improving and Stabilizing Shampoo using Adam](https://mlsanity.com/paper/2409.11321/) (2024)
- [Training Deep Learning Models with Norm-Constrained LMOs](https://mlsanity.com/paper/2502.07529/) (2025)
- [A Kronecker-factored approximate Fisher matrix for convolution layers](https://mlsanity.com/paper/1602.01407/) (2016)
- [Squeeze, Recover and Relabel: Dataset Condensation at ImageNet Scale From A New Perspective](https://mlsanity.com/paper/2306.13092/) (2023)
- [Muon Optimizes Under Spectral Norm Constraints](https://mlsanity.com/paper/2506.15054/) (2025)
- [Stable Low-rank Tensor Decomposition for Compression of Convolutional Neural Network](https://mlsanity.com/paper/2008.05441/) (2020)
- [Gluon: Making Muon & Scion Great Again! (Bridging Theory and Practice of LMO-based Optimizers for LLMs)](https://mlsanity.com/paper/2505.13416/) (2025)
- [A New Perspective on Shampoo's Preconditioner](https://mlsanity.com/paper/2406.17748/) (2024)
- [Kronecker-Factored Approximate Curvature for Modern Neural Network Architectures](https://mlsanity.com/paper/2311.00636/) (2023)
- [Multilinear Mixture of Experts: Scalable Expert Specialization through Factorization](https://mlsanity.com/paper/2402.12550/) (2024)
- [LoRA meets Riemannion: Muon Optimizer for Parametrization-independent Low-Rank Adapters](https://mlsanity.com/paper/2507.12142/) (2025)
- [TensorLLM: Tensorising Multi-Head Attention for Enhanced Reasoning and Compression in LLMs](https://mlsanity.com/paper/2501.15674/) (2025)
- [Tight and Efficient Upper Bound on Spectral Norm of Convolutional Layers](https://mlsanity.com/paper/2409.11859/) (2024)
- [Fantastic Four: Differentiable Bounds on Singular Values of Convolution Layers](https://mlsanity.com/paper/1911.10258/) (2019)

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

- [Muon on the Stiefel Manifold Admits an Exact Closed-Form Update](https://mlsanity.com/paper/2608.06218/) (2026)

## Abstract (arXiv)

Common first-order optimizers, such as Adam, implicitly treat each parameter block as an unstructured vector, which disregards the multilinear weight structure present in many modern machine learning models. Recent work has shown that exploiting matrix structure can improve optimization dynamics. A notable example is Muon, which performs steepest descent under the spectral norm constraint. We take the next step and introduce Tensorion, a tensor-aware optimizer that extends Muon's constrained optimization perspective from matrices to higher-order tensors. Tensorion is built around a linear minimization oracle (LMO) over a tensor norm ball. The norm is carefully chosen to balance two objectives: tightly bounding the tensor spectral norm, while still keeping the LMO tractable. This LMO becomes computable because it reduces to operations on adaptively selected unfolding matrices. Notably, when restricted to order-2 tensors (i.e., matrices), Tensorion recovers Muon exactly. Experiments on tensor-based computer vision problems suggest that Tensorion can offer improved convergence behavior and more stable gradient updates compared with Adam-based and existing tensor-aware baselines in the evaluated settings.