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Flow Diverse and Efficient: Learning Momentum Flow Matching via Stochastic Velocity Field Sampling

Recently, the rectified flow (RF) has emerged as the new state-of-the-art among flow-based diffusion models due to its high efficiency advantage in straight path sampling, especially with the amazing images generated by a series of RF models such as Flux 1.0 and SD 3.0. Although a straight-line connection between the noisy and natural data distributions is intuitive, fast, and easy to optimize, it still inevitably leads to: 1) Diversity concerns, which arise since straight-line paths only cover a fairly restricted sampling space. 2) Multi-scale noise modeling concerns, since the straight line flow only needs to optimize the constant velocity field $\bm v$ between the two distributions $\bm\pi_0$ and $\bm\pi_1$. In this work, we present Discretized-RF, a new family of rectified flow (also called momentum flow models since they refer to the previous velocity component and the random velocity component in each diffusion step), which discretizes the straight path into a series of variable velocity field sub-paths (namely ``momentum fields'') to expand the search space, especially when close to the distribution $p_\text{noise}$. Different from the previous case where noise is directly superimposed on $\bm x$, we introduce noise on the velocity $\bm v$ of the sub-path to change its direction in order to improve the diversity and multi-scale noise modeling abilities. Experimental results on several representative datasets demonstrate that learning momentum flow matching by sampling random velocity fields will produce trajectories that are both diverse and efficient, and can consistently generate high-quality and diverse results. Code is available at https://github.com/liuruixun/momentum-fm.

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