EAGER: Reverse Discrete Time Diffusions: Transforming Generative AI

Generative AI using diffusion models has proved to be very effective in creating synthetic images, videos, text, speech and other data. Its impact on recent AI algorithms has been revolutionary. Impressive algorithms have emerged on the Generative AI landscape over the last decade. In these models, synthetic data are progressively refined by the time reversal of certain forward diffusion processes. A key bottleneck in this technology is the lack of a theory for directly reversing these diffusion processes and current implementation deploys indirect approaches. These indirect time reversals are inaccurate, inefficient, slow and computationally onerous. A theory for direct reversal of these diffusion processes is sorely needed and will not only significantly improve Generative AI algorithms but will also constitute a foundational advance to the theory of diffusions. The diffusion models to be time reversed in Generative AI are stochastic difference equations. While the theory of reversing a stochastic differential equation (SDE) was formulated by Anderson in 1982, no comparable theory exists for difference equations. Instead, an indirect approach comprising three steps is employed. A stochastic difference equation is first approximated by a differential equation. It is then reversed using Anderson’s theory. The final step discretizes this reverse differential equation. This use of an approximation of another approximation has many issues. Approximations always induce errors. Discretizing a reverse SDE to sufficient fidelity is computationally onerous. A state space of as modest a dimension as 256 may take as many as 48 hours to accomplish reversal using standard packages for obtaining reverse diffusions. More fundamentally, there are SDEs with bounded solutions whose discretization, however fine-grained, always lead to unbounded solutions. This project will develop a fundamental theory for the direct time-reversal of a stochastic difference equation. Both Linear Time Varying and Nonlinear forward diffusions will be considered. What makes the development of this theory particularly challenging is that many elegant results for stochastic differential equations do not apply to stochastic difference equations. The project will characterize all reverse processes for a stochastic difference equation and identify those that are analytically tractable. The theory will be refined to permit implementation of the time reversal through empirical data used in Generative AI. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.