Expressivity of Bi-Lipschitz Normalizing Flows: A Score-Based Diffusion Perspective

• URL: http://arxiv.org/abs/2605.06172v1
• Date: Thu, 07 May 2026 12:54:49 GMT
• Title: Expressivity of Bi-Lipschitz Normalizing Flows: A Score-Based Diffusion Perspective
• Abstract summary: We study the expressivity of bi-Lipschitz normalizing flows through the lens of score-based diffusion models.<n>Our key idea is to use the probability flow ODE to link regularity of the score to regularity of the induced transport maps.
• Score: 22.909795047222755
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Many normalizing flow architectures impose regularity constraints, yet their distributional approximation properties are not fully characterized. We study the expressivity of bi-Lipschitz normalizing flows through the lens of score-based diffusion models. For the probability flow ODE of a variance-preserving diffusion, Lipschitz regularity of the score induces a flow of bi-Lipschitz diffeomorphic transport maps. This ODE bridge allows us to analyze the distributional approximation power of bi-Lipschitz normalizing flows and, conversely, derive deterministic convergence guarantees for diffusion-based transport. Our key idea is to use the probability flow ODE to link regularity of the score to regularity of the induced transport maps. We verify score regularity for broad target densities, including compactly supported densities, Gaussian convolutions of compactly supported measures and finite Gaussian mixtures. We obtain a universal distributional approximation result: Gaussian pullbacks induced by bi-Lipschitz variance-preserving transport maps are $L^1$-dense among all probability densities. For Gaussian convolution targets, we further obtain convergence in Kullback-Leibler divergence without early stopping.

Related papers

• Lipschitz regularity in Flow Matching and Diffusion Models: sharp sampling rates and functional inequalities [0.0]
  We establish a sharp Lipschitz regularity theory for flow-matching vector fields and diffusion-model scores.<n>We show that the one-sided Lipschitz control yields a globally Lipschitz transport map from the standard Gaussian to $pstar$.
  arXiv  Detail & Related papers  (2026-04-07T16:46:40Z)
• Entropy-Controlled Flow Matching [0.08460698440162889]
  We propose a constrained variational principle over continuity-equation paths enforcing a global entropy-rate budget d/dt H(mu_t) >= -lambda.<n>We obtain certificate-style mode-coverage and density-floor guarantees with Lipschitz, and construct near-optimal counterexamples for unconstrained flow matching.
  arXiv  Detail & Related papers  (2026-02-25T06:07:01Z)
• On the Contractivity of Stochastic Interpolation Flow [0.0]
  We investigate, a recently introduced framework for high dimensional sampling which bears many similarities to diffusion modeling.<n>We show that for a base distribution and a strongly log-concave target distribution, the flow map is Lipschitz with a sharp constant which matches that of Caffarelli's theorem for optimal transport maps.<n>We are further able to construct Lipschitz transport maps between non-Gaussian distributions, generalizing some recent constructions in the literature on transport methods for establishing functional inequalities.
  arXiv  Detail & Related papers  (2025-04-14T19:10:22Z)
• 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)
• Theory on Score-Mismatched Diffusion Models and Zero-Shot Conditional Samplers [49.97755400231656]
  We present the first performance guarantee with explicit dimensional dependencies for general score-mismatched diffusion samplers.<n>We show that score mismatches result in an distributional bias between the target and sampling distributions, proportional to the accumulated mismatch between the target and training distributions.<n>This result can be directly applied to zero-shot conditional samplers for any conditional model, irrespective of measurement noise.
  arXiv  Detail & Related papers  (2024-10-17T16:42:12Z)
• Adaptive Annealed Importance Sampling with Constant Rate Progress [68.8204255655161]
  Annealed Importance Sampling (AIS) synthesizes weighted samples from an intractable distribution. We propose the Constant Rate AIS algorithm and its efficient implementation for $alpha$-divergences.
  arXiv  Detail & Related papers  (2023-06-27T08:15:28Z)
• 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)
• Resampling Base Distributions of Normalizing Flows [0.0]
  We introduce a base distribution for normalizing flows based on learned rejection sampling. We develop suitable learning algorithms using both maximizing the log-likelihood and the optimization of the reverse Kullback-Leibler divergence.
  arXiv  Detail & Related papers  (2021-10-29T14:44:44Z)
• On the expressivity of bi-Lipschitz normalizing flows [49.92565116246822]
  An invertible function is bi-Lipschitz if both the function and its inverse have bounded Lipschitz constants. Most Normalizing Flows are bi-Lipschitz by design or by training to limit numerical errors.
  arXiv  Detail & Related papers  (2021-07-15T10:13:46Z)
• Gaussianization Flows [113.79542218282282]
  We propose a new type of normalizing flow model that enables both efficient iteration of likelihoods and efficient inversion for sample generation. Because of this guaranteed expressivity, they can capture multimodal target distributions without compromising the efficiency of sample generation.
  arXiv  Detail & Related papers  (2020-03-04T08:15:06Z)

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.