Training-free conditional generation aims to leverage the unconditional diffusion models to implement the conditional generation, where flow-matching (FM) and diffusion probabilistic models (DPMs) are two mature unconditional diffusion models that achieve high-quality generation. Two questions were asked in this paper: What are the underlying connections between FM and DPMs in training-free conditional generation? Can we leverage DPMs to improve the training-free conditional generation for FM? We first show that a probabilistic diffusion path can be associated with the FM and DPMs. Then, we reformulate the ordinary differential equation (ODE) of FM based on the score function of DPMs, and thus, the conditions in FM can be incorporated as those in DPMs. Finally, we propose two posterior sampling methods to estimate the conditional term and achieve a training-free conditional generation of FM. Experimental results show that our proposed method could be implemented for various conditional generation tasks. Our method can generate higher-quality results than the state-of-the-art methods.
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