Free Denoising Diffusion Models

We develop a free-probabilistic framework for denoising diffusion, in which the data is a self-adjoint operator and its law a spectral distribution. The forward process is the free Ornstein--Uhlenbeck diffusion, whose spectral marginals solve a nonlocal Fokker--Planck equation of complex Burgers type carried by the Hilbert transform. Organising the analysis around the 1-convexity of the free energy along free Wasserstein geodesics, we recover, in the form required by the diffusion schedule, the free logarithmic Sobolev, Talagrand and HWI inequalities of Biane, Hiai--Petz--Ueda and Ledoux--Popescu, together with a dissipation identity and the convergence of a free Jordan--Kinderlehrer--Otto scheme. Specialising Dabrowski's time reversal of free diffusions, we obtain the reverse-time equation, whose drift involves Voiculescu's conjugate variable; the latter is square-integrable at every positive time, with an explicit bound along the schedule. A free Tweedie identity yields a consistent score-matching objective and a finite-dimensional algorithm. For operator-valued volatility, we compute the spectral velocity field, match it to a Hermitian matrix model, and show that the resulting diffusion admits neither a spectral Lamperti reduction nor a Wasserstein gradient-flow structure unless the coefficient is constant. We further give a threshold for the survival of a spectral gap along the flow, in terms of the second-order Cauchy kernel of the initial law, and report numerical experiments.