提出分层修正流，用多层级微分方程更真实建模数据生成路径。
Towards Hierarchical Rectified Flow
- 通过分层耦合多个微分方程，建模速度场、加速度场等随机变量的完整分布。
- 生成路径可相交，轨迹更直，生成数据所需神经网络计算量减少。
- 适用于高维数据生成，尤其在图像数据上验证了效率优势。
本文提出一种分层修正流（Hierarchical Rectified Flow），用于建模数据分布。该方法通过分层耦合多个常微分方程（ODE），构建一个时间可微的随机过程，从已知源分布生成目标数据分布。每个ODE对应不同物理量域（如位置、速度、加速度），与经典修正流仅在位置域定义单个ODE并仅捕捉期望速度场不同，本方法完整建模了多模态的随机速度场、加速度场等。这种更精确的建模使生成过程中积分路径可相交，从而获得比经典方法更直的轨迹，显著降低生成数据所需的神经网络函数评估次数。我们在一维、二维合成数据以及MNIST、CIFAR-10和ImageNet-32数据上进行了实证验证。代码已公开：https://riccizz.github.io/HRF/
原文摘要 · Abstract (English)
We formulate a hierarchical rectified flow to model data distributions. It hierarchically couples multiple ordinary differential equations (ODEs) and defines a time-differentiable stochastic process that generates a data distribution from a known source distribution. Each ODE resembles the ODE that is solved in a classic rectified flow, but differs in its domain, i.e., location, velocity, acceleration, etc. Unlike the classic rectified flow formulation, which formulates a single ODE in the location domain and only captures the expected velocity field (sufficient to capture a multi-modal data distribution), the hierarchical rectified flow formulation models the multi-modal random velocity field, acceleration field, etc., in their entirety. This more faithful modeling of the random velocity field enables integration paths to intersect when the underlying ODE is solved during data generation. Intersecting paths in turn lead to integration trajectories that are more straight than those obtained in the classic rectified flow formulation, where integration paths cannot intersect. This leads to modeling of data distributions with fewer neural function evaluations. We empirically verify this on synthetic 1D and 2D data as well as MNIST, CIFAR-10, and ImageNet-32 data. Our code is available at: https://riccizz.github.io/HRF/.
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