Muon, Spectral Transitions, and Phase Retrieval
A guide to the mathematical program: fixed spectral powers, dynamic BBP transitions, adaptive control, proofs, experiments, and open limits.
This research program asks how a spectral optimizer changes the order in which a high-dimensional model learns a hierarchy of hidden directions.
The setting is quadratic multi-index phase retrieval. The teacher has ordered strengths $\mu_i$, often with a power-law tail, and the student is trained from Gaussian data. The optimizer is Muon with a singular-value power $a$. Ordinary gradient descent corresponds to $a=1$; the polar or SignSVD endpoint is approached as $a\to0$.
The central physical picture is a moving spectral front. Strong teacher modes are learned first. Weak modes remain buried in random-matrix bulks until a finite-rank branch crosses an edge. Changing $a$ changes both the speed of this front and the stochastic cost paid near the hard edge.
The program follows the chain
\[\text{spectral update} \longrightarrow \text{reduced training state} \longrightarrow \text{Volterra learning curve} \longrightarrow \text{gradient, weight, and Hessian outliers} \longrightarrow \text{control of }a(t).\]Reading order
I. A fixed spectral power
Muon for Phase Retrieval I: A Spectral Front Through the Teacher begins with the quadratic teacher-student model. The population gradient and the Muon update remain in the finite span generated by the student and teacher directions. Under a fresh projected-gradient channel hypothesis, the high-dimensional spectral recursion in Phases of Muon: When Muon Eclipses SignSGD then yields a nonlinear Volterra equation.
This equation predicts the learning curve and supplies the state from which the three evolving spectra are read. A power-law asymptotic balance gives a preferred fixed exponent, while the finite experiments test risk curves, bulk edges, outlier trajectories, overlaps, and transition times.
II. Dynamic Hessian visibility
Dynamic BBP Transitions in Multi-Index Phase Retrieval is a standalone Hessian article. It is not a Muon theorem. It separates two events that need not coincide: population capture of a teacher direction and spectral visibility of that direction in an empirical Hessian.
For a fresh test sample, a matrix Dyson equation describes the bulk and a finite Schur equation describes the outliers. The derivative of the Schur matrix gives the teacher overlap of an outlier eigenvector. The article also formulates the energy-conditioned Kac-Rice problem and explains why conditioning on zero gradient changes the Hessian ensemble.
III. A power that moves with the front
Muon for Phase Retrieval III: Choosing the Spectral Power During Training treats $a$ as a control. The instantaneous term compares the current gradient channel with an ideal spectral denoiser. The future term is the directional derivative of a value function along the controlled reduced dynamics.
The Hamilton-Jacobi-Bellman Hamiltonian is
\[F_t(a)+\nabla_XV(t,X_t)\cdot b(t,X_t,a).\]Entropy regularization produces a Gibbs distribution over powers. This law is exact for the relaxed reduced control problem. The relation to AMP is narrower: when the ideal local likelihood-ratio filter is log-linear on the active spectral window, its projection onto the Muon family is a power. Muon does not thereby acquire an Onsager correction or become identical to AMP.
IV. Proof architecture
Muon for Phase Retrieval IV: From Exact Algebra to Random-Matrix Limits identifies the precise dependencies of the argument. It proves the finite-frame and Schur identities, derives the Volterra equation from the conditional mode law, and records the deterministic sensitivity and adjoint calculations for a moving power.
It also isolates the main open probabilistic bridge: the resolvent built from data reused during training must be compared uniformly with the fresh-sample resolvent, including its spectral derivative and smoothed BBP contacts.
How to interpret the claims
An exact statement is a finite-dimensional algebraic identity or a theorem inside the stated deterministic reduced model. Examples are the population loss, finite-frame closure, Schur complement, Volterra variation of constants, and deterministic adjoint.
A conditional asymptotic statement uses a specified high-dimensional input: a projected-gradient channel, a deterministic bulk resolvent, a local law, or regular-variation asymptotics. The condition is part of the theorem and is not removed by numerical agreement.
Numerical evidence compares a prediction with finite-dimensional risk curves, spectral densities, branch trajectories, residues, and exit times. It tests a conditional implication; it does not prove a uniform limit.
An open problem is stated where the required probabilistic transport has not been established. The principal examples are the nonlinear fresh channel, the same-sample resolvent law through dense contacts, and the conditioned Kac-Rice matrix Dyson equation.
Mathematical documents
- Main manuscript: Muon as Dynamic Spectral Denoising for Power-Law Phase Retrieval
- Reader’s guide
- Full mathematical details
- Fixed-power spectral companion
- Dynamic Hessian BBP and energy-conditioned spectra
- Adaptive power, HJB, and Gibbs policies
- Markovian spectral control
- Controlled Volterra sensitivity and adjoint
The experiment directory contains the simulations and figure-generation code. The papers define the assumptions and metrics under which each figure should be read.
External foundations
The projected stochastic dynamics are based on Phases of Muon: When Muon Eclipses SignSGD. The power-law phase-retrieval comparison is Fast Escape, Slow Convergence. Fresh-sample bulk and outlier limits use Local Geometry of High-Dimensional Mixture Models. The unrestricted spectral-denoising comparison uses Optimal Spectral Transitions in High-Dimensional Multi-Index Models.
These papers provide distinct inputs. None of them alone proves the combined phase-retrieval control program, and the public manuscripts state explicitly where each input enters.