We introduce "posterior flows" - generalizations of "probability flows" to a broader class of stochastic processes not necessarily diffusion processes - and propose a systematic training-free method to transform the posterior flow of a "linear" stochastic process characterized by the equation Xt = at * X0 + st * X1 into a straight constant-speed (SC) flow, reminiscent of Rectified Flow. This transformation facilitates fast sampling along the original posterior flow without training a new model of the SC flow. The flexibility of our approach allows us to extend our transformation to inter-convert two posterior flows from distinct "linear" stochastic processes. Moreover, we can easily integrate high-order numerical solvers into the transformed SC flow, further enhancing sampling accuracy and efficiency. Rigorous theoretical analysis and extensive experimental results substantiate the advantages of our framework.

翻译：我们引入“后验流”——将“概率流”推广到更广泛的一类随机过程（不一定是扩散过程）——并提出一种系统的免训练方法，将由方程 Xt = at * X0 + st * X1 描述的“线性”随机过程的“后验流”转化为直线恒速（SC）流，类似于修正流（Rectified Flow）。这种转化使得无需训练新的SC流模型即可沿原始后验流进行快速采样。我们的方法具有灵活性，可进一步将转化扩展至在不同“线性”随机过程的两个后验流之间相互转换。此外，我们还能轻松将高阶数值求解器集成到转化后的SC流中，从而进一步提升采样精度与效率。严格的理论分析与广泛的实验结果充分证实了我们框架的优势。