FRITES: Provable Diffusion Posterior Sampling via Forward-Reverse Iterations of Tilted SDEs

Score-based diffusion models have emerged as powerful generative frameworks, particularly suited for sampling from high-dimensional distributions. From a Bayesian point of view, the framework provides a potential solution to inverse problems, i.e., combining observations with prior data distributions learned by the diffusion model to obtain posterior sampling — a task of both high practical importance and significant theoretical challenge. Earlier works have achieved high generative quality, but lack evidence on provable posterior sampling. Recent work has sought theoretical guarantees, but faces a persistent dilemma: transport and particle-based methods are either restricted to linear settings or become computationally prohibitive in high dimensions, while annealing-based plug-and-play approaches stall in practice and fails to achieve posterior sampling. To tackle these issues, we introduce Forward-Reverse Iterations of Tilted SDEs (FRITES), a novel algorithm with provable convergence, scalable to high-dimensional data, adaptive to both linear and non-linear inverse problems, and easy to implement. In particular, FRITES introduces a Markov chain built on coupled forward and reverse diffusion dynamics, with a principled tilt so that the target posterior emerges as the unique stationary distribution. We establish in this paper both asymptotic and finite-time theoretical guarantees, and provide empirical evidence that FRITES achieves superior performance spanning synthetic low-dimensional benchmarks and high-dimensional imaging tasks comparing to state-of-the-art baseline methods.