This presentation introduces score-based diffusion models in function space, extending denoising score matching and Langevin dynamics from finite-dimensional settings to infinite-dimensional functional spaces. It first reviews the basic score matching objective and its denoising interpretation, then develops the corresponding formulation in function space along with approximation theory and pre-conditioned or annealed Langevin sampling methods. The talk also emphasizes conditional sampling and suggests that diffusion posterior sampling can be naturally applied in this framework. To demonstrate the approach, it presents experiments on Gaussian mixtures, the Navier–Stokes equation, volcano datasets, MNIST signed distance functions, and a Bayesian inverse problem on Darcy flow. The presentation concludes by highlighting the significance of diffusion modeling on infinite-dimensional spaces and proposing future directions such as extending the framework to stochastic differential equations, improving noise scheduling, enhancing sampling methods, and exploring Crank–Nicolson and second-order discretization schemes.
Summary:
This presentation introduces score-based diffusion models in function space, extending denoising score matching and Langevin dynamics from finite-dimensional settings to infinite-dimensional functional spaces. It first reviews the basic score matching objective and its denoising interpretation, then develops the corresponding formulation in function space along with approximation theory and pre-conditioned or annealed Langevin sampling methods. The talk also emphasizes conditional sampling and suggests that diffusion posterior sampling can be naturally applied in this framework. To demonstrate the approach, it presents experiments on Gaussian mixtures, the Navier–Stokes equation, volcano datasets, MNIST signed distance functions, and a Bayesian inverse problem on Darcy flow. The presentation concludes by highlighting the significance of diffusion modeling on infinite-dimensional spaces and proposing future directions such as extending the framework to stochastic differential equations, improving noise scheduling, enhancing sampling methods, and exploring Crank–Nicolson and second-order discretization schemes.