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Gaussian Interpolation Flows

Yuan Gao, Jian Huang, Yuling Jiao

TL;DR

This work analyzes Gaussian denoising-based, simulation-free continuous normalizing flows by introducing Gaussian interpolation flows (GIFs), an ODE-flow framework driven by a velocity field $v(t,x)$ tied to the score of the marginal $p_t$. It proves well-posedness and Lipschitz regularity for the flow and its inverse across rich target distributions using variance inequalities (e.g., Brascamp–Lieb, Cramér–Rao) and a detailed analysis of $ abla_x v(t,x)$ via conditional covariances, ensuring a stable transport map. The authors establish auto-encoding and cycle-consistency properties at the population level when the flow is Lipschitz, and provide stability bounds in Wasserstein distance under perturbations of the source or velocity, laying a solid theoretical foundation for learning GIFs with neural networks. Collectively, these results offer end-to-end error analyses and connect GIFs to score-based diffusion, CNFs, and consistency-model frameworks, with implications for scalable, theory-backed generative modeling.

Abstract

Gaussian denoising has emerged as a powerful method for constructing simulation-free continuous normalizing flows for generative modeling. Despite their empirical successes, theoretical properties of these flows and the regularizing effect of Gaussian denoising have remained largely unexplored. In this work, we aim to address this gap by investigating the well-posedness of simulation-free continuous normalizing flows built on Gaussian denoising. Through a unified framework termed Gaussian interpolation flow, we establish the Lipschitz regularity of the flow velocity field, the existence and uniqueness of the flow, and the Lipschitz continuity of the flow map and the time-reversed flow map for several rich classes of target distributions. This analysis also sheds light on the auto-encoding and cycle consistency properties of Gaussian interpolation flows. Additionally, we study the stability of these flows in source distributions and perturbations of the velocity field, using the quadratic Wasserstein distance as a metric. Our findings offer valuable insights into the learning techniques employed in Gaussian interpolation flows for generative modeling, providing a solid theoretical foundation for end-to-end error analyses of learning Gaussian interpolation flows with empirical observations.

Gaussian Interpolation Flows

TL;DR

This work analyzes Gaussian denoising-based, simulation-free continuous normalizing flows by introducing Gaussian interpolation flows (GIFs), an ODE-flow framework driven by a velocity field tied to the score of the marginal . It proves well-posedness and Lipschitz regularity for the flow and its inverse across rich target distributions using variance inequalities (e.g., Brascamp–Lieb, Cramér–Rao) and a detailed analysis of via conditional covariances, ensuring a stable transport map. The authors establish auto-encoding and cycle-consistency properties at the population level when the flow is Lipschitz, and provide stability bounds in Wasserstein distance under perturbations of the source or velocity, laying a solid theoretical foundation for learning GIFs with neural networks. Collectively, these results offer end-to-end error analyses and connect GIFs to score-based diffusion, CNFs, and consistency-model frameworks, with implications for scalable, theory-backed generative modeling.

Abstract

Gaussian denoising has emerged as a powerful method for constructing simulation-free continuous normalizing flows for generative modeling. Despite their empirical successes, theoretical properties of these flows and the regularizing effect of Gaussian denoising have remained largely unexplored. In this work, we aim to address this gap by investigating the well-posedness of simulation-free continuous normalizing flows built on Gaussian denoising. Through a unified framework termed Gaussian interpolation flow, we establish the Lipschitz regularity of the flow velocity field, the existence and uniqueness of the flow, and the Lipschitz continuity of the flow map and the time-reversed flow map for several rich classes of target distributions. This analysis also sheds light on the auto-encoding and cycle consistency properties of Gaussian interpolation flows. Additionally, we study the stability of these flows in source distributions and perturbations of the velocity field, using the quadratic Wasserstein distance as a metric. Our findings offer valuable insights into the learning techniques employed in Gaussian interpolation flows for generative modeling, providing a solid theoretical foundation for end-to-end error analyses of learning Gaussian interpolation flows with empirical observations.
Paper Structure (18 sections, 31 theorems, 115 equations, 6 figures, 1 table)

This paper contains 18 sections, 31 theorems, 115 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Our main contributions
  3. Preliminaries
  4. Assumptions
  5. Variance inequalities
  6. Gaussian interpolation flows
  7. Spatial Lipschitz estimates for the velocity field
  8. Well-posedness and Lipschtiz flow maps
  9. Applications to generative modeling
  10. Related work
  11. Conclusions and discussion
  12. Proofs of Theorem \ref{['thm:flow-gif']} and Lemma \ref{['lm:cond-cov']}
  13. Auxiliary lemmas for Lipschitz flow maps
  14. Proofs of spatial Lipschitz estimates for the velocity field
  15. Proofs of well-posedness and Lipschitz flow maps
  16. ...and 3 more sections

Key Result

Lemma 4

Let $\mu(\mathrm{d} x) = \exp(-U(x)) \mathrm{d} x$ be a probability measure on a convex set $\Omega \subseteq {\mathbb{R}}^d$ whose potential function $U: \Omega \to {\mathbb{R}}$ is of class $C^2$ and strictly convex. Then for every locally Lipschitz function $f \in L^2(\Omega, \mu)$,

Figures (6)

  • Figure 1: Roadmap of the main results.
  • Figure 2: Snapshots of a Gaussian interpolation flow based on the Föllmer interpolant. The source distribution is the standard two-dimensional Gaussian distribution $\gamma_2$, and the target distribution is a mixture of six two-dimensional Gaussian distributions as the shape of a circle. The image panels are placed sequentially from time $t = 0$ to time $t = 1$.
  • Figure 3: An illustration of auto-encoding using DDIBs. The Concentric Rings data in the source domain (the first panel) is encoded into the latent domain (the second panel), and then decoded into the source domain (the third panel). According to the consistent color pattern and pointwise correspondences across the domains, both the learned encoder mapping and the learned decoder mapping exhibit approximate Lipschitz continuity with respect to the space variable. One justification of such auto-encoding observation is presented in Corollary \ref{['cor:ae']} where we prove that the composition of the encoder map and the decoder map yields an identity map.
  • Figure 4: An illustration of cycle consistency using DDIBs. The cycle consistency property is manifested through the consistency of color patterns across the transformations. We transform the Moons data in the source domain onto the Concentric Squares data in the target domain, and then complete the cycle by mapping the target data back to the source domain. The latent spaces play a central role in the bidirectional translation. We provide a proof in Corollary \ref{['cor:cycle']} accounting for the cycle consistency property.
  • Figure 5: An approximately linear relation between $b_0$ and the Wasserstein-2 distance.
  • ...and 1 more figures

Theorems & Definitions (49)

  • Definition 1: cattiaux2014semi
  • Definition 2: eldan2018regularization
  • Definition 3
  • Lemma 4: Brascamp-Lieb inequality
  • Lemma 5: Cramér-Rao inequality
  • Definition 6: Vector interpolation
  • Remark 7
  • Remark 8
  • Remark 9
  • Definition 10: Measure interpolation
  • ...and 39 more