A Reverse-BSDE Diffusion Sampler

Diffusion-based generative models have renewed interest in stochastic differential equation methods for sampling from complex distributions. We study a setting in which the target density is known only up to a normalizing constant and reformulate the reverse-time diffusion sampler as a forward-backward stochastic differential equation (FBSDE). This formulation replaces the separate pre-estimation of the time-dependent score with the solution of a coupled stochastic system. We prove the equivalence of these formulations and provide a decomposition of the approximation error arising from initializing the sampler with a standard Gaussian, applying Euler discretization to the dynamics, and solving the FBSDE approximately. We then evaluate the proposed algorithm on synthetic targets, including separated mixtures, anisotropic Gaussian distributions, and banana-shaped and ring-shaped distributions. The results demonstrate the promise of the method, particularly for targets with complex global structure.