Unraveling the Connections between Flow Matching and Diffusion Probabilistic Models in Training-free Conditional Generation

Training-free conditional generation based on flow matching aims to leverage pre-trained unconditional flow matching models to perform conditional generation without retraining. Recently, a successful training-free conditional generation approach incorporates conditions via posterior sampling, which relies on the availability of a score function in the unconditional diffusion model. However, flow matching models do not possess an explicit score function, rendering such a strategy inapplicable. Approximate posterior sampling for flow matching has been explored, but it is limited to linear inverse problems. In this paper, we propose Flow Matching-based Posterior Sampling (FMPS) to expand its application scope. We introduce a correction term by steering the velocity field. This correction term can be reformulated to incorporate a surrogate score function, thereby bridging the gap between flow matching models and score-based posterior sampling. Hence, FMPS enables the posterior sampling to be adjusted within the flow matching framework. Further, we propose two practical implementations of the correction mechanism: one aimed at improving generation quality, and the other focused on computational efficiency. Experimental results on diverse conditional generation tasks demonstrate that our method achieves superior generation quality compared to existing state-of-the-art approaches, validating the effectiveness and generality of FMPS.