Beyond the Semicircle: Free Diffusion Models with Prescribed Equilibria

Abstract

A growing class of machine-learning objects -- covariance and Gram matrices, kernel and attention matrices, MIMO channel matrices, density operators -- are naturally spectra rather than coordinate vectors. Building a denoising diffusion model for such data by corrupting eigenvalues coordinatewise is not merely elegant: it converges to the wrong limit, because eigenvalues repel rather than move independently. Free probability theory supplies the correct forward process, with free convolution replacing classical convolution and Voiculescu's conjugate variable replacing the score, but constant-coefficient free diffusions have a hidden limitation of their own: their only possible equilibrium is the semicircular law, whatever the target distribution looks like. We show that state-dependent free volatility removes this restriction. For any sufficiently regular compactly supported target law, we explicitly construct, through a closed-form Hilbert-transform drift, a free Fokker--Planck flow having that law as a stationary spectral distribution. Under an additional convex-potential condition, the construction admits a globally relaxing diffusion interpretation, reverse-time dynamics, and a trainable score. The generalization is genuine rather than cosmetic: no change of spectral variable reduces it to the constant-coefficient case, and it is a Wasserstein gradient flow only for an explicitly characterized family of coefficients that contains no bounded non-constant member. We instantiate the construction on the Marchenko--Pastur law, the limiting spectrum of isotropic sample covariance matrices, and validate it numerically: the free-versus-coordinatewise gap, an exact dimension-independent score driving reverse-time matrix dynamics at several matrix sizes, the designed Marchenko--Pastur equilibrium, and end-to-end generation with a learned score.