PRISM: Structured Optimization via Anisotropic Spectral Shaping

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.

💡 Research Summary

The paper introduces PRISM, a novel optimizer that augments first‑order spectral descent methods such as Muon with a lightweight, low‑rank approximation of second‑order curvature information. The authors observe that Muon’s preconditioner P = (MᵀM)⁻¹ᐟ² is built solely from the first‑moment estimate of the gradients, i.e., E