The task of conditional generation is one of the most important applications of generative models, and numerous methods have been developed to date based on the celebrated diffusion models, with the guidance-based classifier-free method taking the lead. However, the theory of the guidance-based method not only requires the user to fine-tune the "guidance strength," but its target vector field does not necessarily correspond to the conditional distribution used in training. In this paper, we develop the theory of conditional generation based on Flow Matching, a current strong contender of diffusion methods. Motivated by the interpretation of a probability path as a distribution on path space, we establish a novel theory of flow-based generation of conditional distribution by employing the mathematical framework of generalized continuity equation instead of the continuity equation in flow matching. This theory naturally derives a method that aims to match the matrix field as opposed to the vector field. Our framework ensures the continuity of the generated conditional distribution through the existence of flow between conditional distributions. We will present our theory through experiments and mathematical results.

本文基于流匹配理论开发了基于流匹配的条件生成理论，通过广义连续性方程数学框架，匹配矩阵场而非向量场，确保生成的条件分布的连续性，并通过实验和数学结果验证了我们的理论。

扩展流匹配：一种具有广义连续方程的条件生成方法