ICLR 2025

Published as a conference paper at ICLR 2025
TOWARDS HIERARCHICAL RECTIFIED FLOW
Yichi Zhang1,
Yici Yan1,
Alex Schwing1,
Zhizhen Zhao1
1University of Illinois Urbana-Champaign
ABSTRACT
We formulate a hierarchical rectified flow to model data distributions. It hierarchi-
cally 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 recti-
fied 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 loca-
tion 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. Inter-
secting 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/.
1
INTRODUCTION
Diffusion models (Ho et al., 2020; Song et al., 2021a;b) and particularly also flow matching (Liu
et al., 2023; Lipman et al., 2023; Albergo & Vanden-Eijnden, 2023; Albergo et al., 2023) have
gained significant attention recently. This is partly due to impressive results that have been reported
across domains from computer vision (Ho et al., 2020) and medical imaging (Song et al., 2022) to
robotics (Kapelyukh et al., 2023) and computational biology...