Statistical Properties of Rectified Flow

• URL: http://arxiv.org/abs/2511.03193v2
• Date: Thu, 06 Nov 2025 01:42:53 GMT
• Title: Statistical Properties of Rectified Flow
• Authors: Gonzalo Mena, Arun Kumar Kuchibhotla, Larry Wasserman,
• Abstract summary: Rectified flow is a method for defining a transport map between two distributions.<n>We study structural properties of the rectified flow, including existence, uniqueness, and regularity.<n>We are able to establish convergence at faster rates than the ones for the usual nonparametric regression and density estimation.
• Score: 0.7136933021609079
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Rectified flow (Liu et al., 2022; Liu, 2022; Wu et al., 2023) is a method for defining a transport map between two distributions, and enjoys popularity in machine learning, although theoretical results supporting the validity of these methods are scant. The rectified flow can be regarded as an approximation to optimal transport, but in contrast to other transport methods that require optimization over a function space, computing the rectified flow only requires standard statistical tools such as regression or density estimation. Because of this, one can leverage standard data analysis tools for regression and density estimation to develop empirical versions of transport maps. We study some structural properties of the rectified flow, including existence, uniqueness, and regularity, as well as the related statistical properties, such as rates of convergence and central limit theorems, for some selected estimators. To do so, we analyze separately the bounded and unbounded cases as each presents unique challenges. In both cases, we are able to establish convergence at faster rates than the ones for the usual nonparametric regression and density estimation.

Related papers

• Flow-Based Density Ratio Estimation for Intractable Distributions with Applications in Genomics [80.05951561886123]
  We leverage condition-aware flow matching to derive a single dynamical formulation for tracking density ratios along generative trajectories.<n>We demonstrate competitive performance on simulated benchmarks for closed-form ratio estimation, and show that our method supports versatile tasks in single-cell genomics data analysis.
  arXiv  Detail & Related papers  (2026-02-27T17:27:55Z)
• Flow Matching is Adaptive to Manifold Structures [32.55405572762157]
  Flow matching is a simulation-based alternative to diffusion-based generative modeling.<n>We show how flow matching adapts to data geometry and circumvents the curse of dimensionality.
  arXiv  Detail & Related papers  (2026-02-25T23:52:32Z)
• Profiling systematic uncertainties in Simulation-Based Inference with Factorizable Normalizing Flows [0.0]
  We propose a general framework for Simulation-Based Inference that efficiently profiles nuisance parameters.<n>We introduce Factorizable Normalizing Flows to model systematic variations as a parametrics of a nominal density.<n>We develop an amortized training strategy that learns the conditional dependence of the DoI on nuisance parameters in a single optimization process.<n>This allows for the simultaneous extraction of the underlying distribution and the robust profiling of nuisances.
  arXiv  Detail & Related papers  (2026-02-13T18:48:12Z)
• Generative Modeling with Continuous Flows: Sample Complexity of Flow Matching [60.37045080890305]
  We provide the first analysis of the sample complexity for flow-matching based generative models.<n>We decompose the velocity field estimation error into neural-network approximation error, statistical error due to the finite sample size, and optimization error due to the finite number of optimization steps for estimating the velocity field.
  arXiv  Detail & Related papers  (2025-12-01T05:14:25Z)
• Minibatch Optimal Transport and Perplexity Bound Estimation in Discrete Flow Matching [0.0]
  Outperforming autoregressive models on categorical data distributions, such as textual data, remains challenging for continuous diffusion and flow models. We propose a dynamic-optimal-transport-like minimization objective for discrete flows with convex interpolants. Unlike continuous flows, wherein the instantaneous change of variables enables density estimation, discrete models lack a similar mechanism due to the inherent non-determinism and discontinuity of their paths.
  arXiv  Detail & Related papers  (2024-11-01T17:35:09Z)
• On the Wasserstein Convergence and Straightness of Rectified Flow [54.580605276017096]
  Rectified Flow (RF) is a generative model that aims to learn straight flow trajectories from noise to data.<n>We provide a theoretical analysis of the Wasserstein distance between the sampling distribution of RF and the target distribution.<n>We present general conditions guaranteeing uniqueness and straightness of 1-RF, which is in line with previous empirical findings.
  arXiv  Detail & Related papers  (2024-10-19T02:36:11Z)
• Flow-based Distributionally Robust Optimization [23.232731771848883]
  We present a framework, called $textttFlowDRO$, for solving flow-based distributionally robust optimization (DRO) problems with Wasserstein uncertainty sets. We aim to find continuous worst-case distribution (also called the Least Favorable Distribution, LFD) and sample from it. We demonstrate its usage in adversarial learning, distributionally robust hypothesis testing, and a new mechanism for data-driven distribution perturbation differential privacy.
  arXiv  Detail & Related papers  (2023-10-30T03:53:31Z)
• Statistical Efficiency of Score Matching: The View from Isoperimetry [96.65637602827942]
  We show a tight connection between statistical efficiency of score matching and the isoperimetric properties of the distribution being estimated. We formalize these results both in the sample regime and in the finite regime.
  arXiv  Detail & Related papers  (2022-10-03T06:09:01Z)
• Efficient CDF Approximations for Normalizing Flows [64.60846767084877]
  We build upon the diffeomorphic properties of normalizing flows to estimate the cumulative distribution function (CDF) over a closed region. Our experiments on popular flow architectures and UCI datasets show a marked improvement in sample efficiency as compared to traditional estimators.
  arXiv  Detail & Related papers  (2022-02-23T06:11:49Z)
• Near-optimal estimation of smooth transport maps with kernel sums-of-squares [81.02564078640275]
  Under smoothness conditions, the squared Wasserstein distance between two distributions could be efficiently computed with appealing statistical error upper bounds. The object of interest for applications such as generative modeling is the underlying optimal transport map. We propose the first tractable algorithm for which the statistical $L2$ error on the maps nearly matches the existing minimax lower-bounds for smooth map estimation.
  arXiv  Detail & Related papers  (2021-12-03T13:45:36Z)
• Comparing Probability Distributions with Conditional Transport [63.11403041984197]
  We propose conditional transport (CT) as a new divergence and approximate it with the amortized CT (ACT) cost. ACT amortizes the computation of its conditional transport plans and comes with unbiased sample gradients that are straightforward to compute. On a wide variety of benchmark datasets generative modeling, substituting the default statistical distance of an existing generative adversarial network with ACT is shown to consistently improve the performance.
  arXiv  Detail & Related papers  (2020-12-28T05:14:22Z)
• Variational Transport: A Convergent Particle-BasedAlgorithm for Distributional Optimization [106.70006655990176]
  A distributional optimization problem arises widely in machine learning and statistics. We propose a novel particle-based algorithm, dubbed as variational transport, which approximately performs Wasserstein gradient descent. We prove that when the objective function satisfies a functional version of the Polyak-Lojasiewicz (PL) (Polyak, 1963) and smoothness conditions, variational transport converges linearly.
  arXiv  Detail & Related papers  (2020-12-21T18:33:13Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.