# PRISM: Structured Optimization via Anisotropic Spectral Shaping

arXiv: https://arxiv.org/abs/2602.03096 · HTML: https://mlsanity.com/paper/2602.03096/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 13 references that are in the map.
- First posted on arXiv 3 Feb 2026.

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

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Qwen2 Technical Report](https://mlsanity.com/paper/2407.10671/) (2024)
- [The FineWeb Datasets: Decanting the Web for the Finest Text Data at Scale](https://mlsanity.com/paper/2406.17557/) (2024)
- [Muon is Scalable for LLM Training](https://mlsanity.com/paper/2502.16982/) (2025)
- [Shampoo: Preconditioned Stochastic Tensor Optimization](https://mlsanity.com/paper/1802.09568/) (2018)
- [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)
- [A New Perspective on Shampoo's Preconditioner](https://mlsanity.com/paper/2406.17748/) (2024)
- [Structured Preconditioners in Adaptive Optimization: A Unified Analysis](https://mlsanity.com/paper/2503.10537/) (2025)

## Built on by (most cited first, 2 of 2)

- [Adam Improves Muon: Adaptive Moment Estimation with Orthogonalized Momentum](https://mlsanity.com/paper/2602.17080/) (2026)
- [Dimension-Free Saddle-Point Escape in Muon](https://mlsanity.com/paper/2605.09331/) (2026)

## Abstract (arXiv)

We propose PRISM, an optimizer that enhances first-order spectral descent methods like Muon with partial second-order information. It constructs an efficient, low-rank quasi-second-order preconditioner via innovation-augmented polar decomposition. This mechanism enables PRISM to perform anisotropic spectral shaping, which adaptively suppresses updates in high-variance subspaces while preserving update strength in signal-dominated directions. Crucially, this is achieved with minimal computational overhead and zero additional memory compared to first-order baselines. PRISM demonstrates a practical strategy for integrating curvature-adaptive properties into the spectral optimization paradigm.