A Free Probabilistic Framework for Denoising Diffusion Models: Entropy, Transport, and Reverse Processes
Swagatam Das
TL;DR
This work extends diffusion-based generative modeling to the noncommutative realm using free probability. By modeling the forward dynamics as a free Ornstein–Uhlenbeck process and deriving a reverse-time SDE via free Malliavin calculus (Clark–Ocone), it establishes a gradient-flow interpretation of spectral-entropy evolution in the free Wasserstein geometry. The authors develop a free Jordan–Kinderlehrer–Otto scheme, prove Γ-convergence and discrete energy–dissipation, and derive free analogues of de Bruijn, logarithmic Sobolev, Talagrand, and HWI inequalities, yielding exponential entropy decay and Wasserstein contraction toward a semicircular equilibrium. This framework unifies stochastic diffusion, operator-valued information geometry, and noncommutative optimal transport, providing rigorous foundations for generative modeling on structured, matrix- or operator-valued data. It opens avenues for learning and sampling in noncommutative settings and connects diffusion theory with quantum-information-inspired geometry and random-matrix theory.
Abstract
This paper develops a rigorous probabilistic framework that extends denoising diffusion models to the setting of noncommutative random variables. Building on Voiculescu's theory of free entropy and free Fisher information, we formulate diffusion and reverse processes governed by operator-valued stochastic dynamics whose spectral measures evolve by additive convolution. Using tools from free stochastic analysis -- including a Malliavin calculus and a Clark--Ocone representation -- we derive the reverse-time stochastic differential equation driven by the conjugate variable, the analogue of the classical score function. The resulting dynamics admit a gradient-flow structure in the noncommutative Wasserstein space, establishing an information-geometric link between entropy production, transport, and deconvolution. We further construct a variational scheme analogous to the Jordan--Kinderlehrer--Otto (JKO) formulation and prove convergence toward the semicircular equilibrium. The framework provides functional inequalities (free logarithmic Sobolev, Talagrand, and HWI) that quantify entropy dissipation and Wasserstein contraction. These results unify diffusion-based generative modeling with the geometry of operator-valued information, offering a mathematical foundation for generative learning on structured and high-dimensional data.