# Towards Hierarchical Rectified Flow

arXiv: https://arxiv.org/abs/2502.17436 · HTML: https://mlsanity.com/paper/2502.17436/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 29 of its 44 references that are in the map.
- First posted on arXiv 24 Feb 2025.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 29 of 29)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding](https://mlsanity.com/paper/2205.11487/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Equivariant Diffusion for Molecule Generation in 3D](https://mlsanity.com/paper/2203.17003/) (2022)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [Solving Inverse Problems in Medical Imaging with Score-Based Generative Models](https://mlsanity.com/paper/2111.08005/) (2021)
- [Tackling the Generative Learning Trilemma with Denoising Diffusion GANs](https://mlsanity.com/paper/2112.07804/) (2021)
- [Multisample Flow Matching: Straightening Flows with Minibatch Couplings](https://mlsanity.com/paper/2304.14772/) (2023)
- [Cold Diffusion: Inverting Arbitrary Image Transforms Without Noise](https://mlsanity.com/paper/2208.09392/) (2022)
- [How to train your neural ODE: the world of Jacobian and kinetic regularization](https://mlsanity.com/paper/2002.02798/) (2020)
- [Neural Autoregressive Flows](https://mlsanity.com/paper/1804.00779/) (2018)
- [OT-Flow: Fast and Accurate Continuous Normalizing Flows via Optimal Transport](https://mlsanity.com/paper/2006.00104/) (2020)
- [DALL-E-Bot: Introducing Web-Scale Diffusion Models to Robotics](https://mlsanity.com/paper/2210.02438/) (2022)
- [Maximum Likelihood Training for Score-Based Diffusion ODEs by High-Order Denoising Score Matching](https://mlsanity.com/paper/2206.08265/) (2022)
- [Moser Flow: Divergence-based Generative Modeling on Manifolds](https://mlsanity.com/paper/2108.08052/) (2021)
- [Matching Normalizing Flows and Probability Paths on Manifolds](https://mlsanity.com/paper/2207.04711/) (2022)
- [Variational Rectified Flow Matching](https://mlsanity.com/paper/2502.09616/) (2025)

## Built on by (most cited first, 20 of 20)

- [The Curse of Conditions: Analyzing and Improving Optimal Transport for Conditional Flow-Based Generation](https://mlsanity.com/paper/2503.10636/) (2025)
- [Beyond Optimal Transport: Model-Aligned Coupling for Flow Matching](https://mlsanity.com/paper/2505.23346/) (2025)
- [Transition Matching: Scalable and Flexible Generative Modeling](https://mlsanity.com/paper/2506.23589/) (2025)
- [Learning Straight Flows: Variational Flow Matching for Efficient Generation](https://mlsanity.com/paper/2511.17583/) (2025)
- [Hierarchical Rectified Flow Matching with Mini-Batch Couplings](https://mlsanity.com/paper/2507.13350/) (2025)
- [Weighted Conditional Flow Matching](https://mlsanity.com/paper/2507.22270/) (2025)
- [Variational Masked Diffusion Models](https://mlsanity.com/paper/2510.23606/) (2025)
- [Training-Free Refinement of Flow Matching with Divergence-based Sampling](https://mlsanity.com/paper/2604.04646/) (2026)
- [Transition Flow Matching](https://mlsanity.com/paper/2603.15689/) (2026)
- [DiReCT: Disentangled Regularization of Contrastive Trajectories for Physics-Refined Video Generation](https://mlsanity.com/paper/2603.25931/) (2026)
- [Drift Flow Matching](https://mlsanity.com/paper/2605.17244/) (2026)
- [TracingFlow: A Simulation-Free Trajectory Inference Framework Based on Second-Order Dynamics](https://mlsanity.com/paper/2608.21070/) (2026)
- [Risk-Entropic Flow Matching](https://mlsanity.com/paper/2512.03078/) (2025)
- [Exploring the Design Space of Transition Matching](https://mlsanity.com/paper/2512.12465/) (2025)
- [TRACER: Texture-Robust Affordance Chain-of-Thought for Deformable-Object Refinement](https://mlsanity.com/paper/2601.20208/) (2026)
- [Structured Coupling for Flow Matching](https://mlsanity.com/paper/2605.07676/) (2026)
- [The Reverse Telescoping Coordinate System for Positive Definite Matrices: Geometry, Computation, and Generative Modeling](https://mlsanity.com/paper/2606.15442/) (2026)
- [QuacamFM: Quaternion-Constrained Flow Matching for Camera Pose Estimation](https://mlsanity.com/paper/2609.32455/) (2026)
- [VTV-FM: Flow Matching through Variational Terminal-Velocity Closure](https://mlsanity.com/paper/2610.00785/) (2026)
- [Posterior Flow Matching](https://mlsanity.com/paper/2610.13994/) (2026)

## Abstract (arXiv)

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/.