# Low-Dimensional Adaptation of Rectified Flow: A Diffusion and Stochastic Localization Perspective arXiv: https://arxiv.org/abs/2601.15500 ยท HTML: https://mlsanity.com/paper/2601.15500/ ## Key facts - Cited by 3 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 41 of its 58 references that are in the map. - First posted on arXiv 21 Jan 2026. ## 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, 30 of 41) - [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020) - [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) - [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022) - [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) - [SiT: Exploring Flow and Diffusion-based Generative Models with Scalable Interpolant Transformers](https://mlsanity.com/paper/2401.08740/) (2024) - [Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions](https://mlsanity.com/paper/2209.11215/) (2022) - [InstaFlow: One Step is Enough for High-Quality Diffusion-Based Text-to-Image Generation](https://mlsanity.com/paper/2309.06380/) (2023) - [The Intrinsic Dimension of Images and Its Impact on Learning](https://mlsanity.com/paper/2104.08894/) (2021) - [Improving Video Generation with Human Feedback](https://mlsanity.com/paper/2501.13918/) (2025) - [Rectified Flow: A Marginal Preserving Approach to Optimal Transport](https://mlsanity.com/paper/2209.14577/) (2022) - [Intrinsic dimension of data representations in deep neural networks](https://mlsanity.com/paper/1905.12784/) (2019) - [Estimating the intrinsic dimension of datasets by a minimal neighborhood information](https://mlsanity.com/paper/1803.06992/) (2018) - [Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localization](https://mlsanity.com/paper/2308.03686/) (2023) - [Semantic Image Inversion and Editing using Rectified Stochastic Differential Equations](https://mlsanity.com/paper/2410.10792/) (2024) - [Improving the Training of Rectified Flows](https://mlsanity.com/paper/2405.20320/) (2024) - [PeRFlow: Piecewise Rectified Flow as Universal Plug-and-Play Accelerator](https://mlsanity.com/paper/2405.07510/) (2024) - [Character-Aware Models Improve Visual Text Rendering](https://mlsanity.com/paper/2212.10562/) (2022) - [Goku: Flow Based Video Generative Foundation Models](https://mlsanity.com/paper/2502.04896/) (2025) - [A Sharp Convergence Theory for The Probability Flow ODEs of Diffusion Models](https://mlsanity.com/paper/2408.02320/) (2024) - [Convergence of Diffusion Models Under the Manifold Hypothesis in High-Dimensions](https://mlsanity.com/paper/2409.18804/) (2024) - [Estimating High Order Gradients of the Data Distribution by Denoising](https://mlsanity.com/paper/2111.04726/) (2021) - [Adapting to Unknown Low-Dimensional Structures in Score-Based Diffusion Models](https://mlsanity.com/paper/2405.14861/) (2024) - [Frieren: Efficient Video-to-Audio Generation Network with Rectified Flow Matching](https://mlsanity.com/paper/2406.00320/) (2024) - [Linear Convergence of Diffusion Models Under the Manifold Hypothesis](https://mlsanity.com/paper/2410.09046/) (2024) - [Denoising diffusion probabilistic models are optimally adaptive to unknown low dimensionality](https://mlsanity.com/paper/2410.18784/) (2024) - [Sampling, Diffusions, and Stochastic Localization](https://mlsanity.com/paper/2305.10690/) (2023) - [Low-dimensional adaptation of diffusion models: Convergence in total variation](https://mlsanity.com/paper/2501.12982/) (2025) - [Unifying Bayesian Flow Networks and Diffusion Models through Stochastic Differential Equations](https://mlsanity.com/paper/2404.15766/) (2024) ## Built on by (most cited first, 3 of 3) - [Total Variation Rates for Riemannian Flow Matching](https://mlsanity.com/paper/2602.05174/) (2026) - [Flow Matching under Noisy Latent Structure: Beyond Exact Low-Dimensional Support](https://mlsanity.com/paper/2609.38918/) (2026) - [Conditional Flow Matching for Transport Between Markov Processes](https://mlsanity.com/paper/2610.07229/) (2026) ## Abstract (arXiv) In recent years, Rectified flow (RF) has gained considerable popularity largely due to its generation efficiency and state-of-the-art performance. In this paper, we investigate the degree to which RF automatically adapts to the intrinsic low dimensionality of the support of the target distribution to accelerate sampling. We show that, using a carefully designed choice of the time-discretization scheme and with sufficiently accurate drift estimates, the RF sampler enjoys an iteration complexity of order O(k/varepsilon) (up to log factors), where varepsilon is the precision in total variation distance and k is the intrinsic dimension of the target distribution. In addition, we show that the denoising diffusion probabilistic model (DDPM) procedure is equivalent to a stochastic version of RF by establishing a novel connection between these processes and stochastic localization. Building on this connection, we further design a stochastic RF sampler that also adapts to the low-dimensionality of the target distribution under milder requirements on the accuracy of the drift estimates, and also with a specific time schedule. We illustrate with simulations on the synthetic data and text-to-image data experiments the improved performance of the proposed samplers implementing the newly designed time-discretization schedules.