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Coupling-based Invertible Neural Networks Are Universal Diffeomorphism Approximators

arXiv 2006.11469 ↗First posted 20 Jun 2020cs.LG cs.NE math.CAcited by 85

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Abstract (arXiv)

Invertible neural networks based on coupling flows (CF-INNs) have various machine learning applications such as image synthesis and representation learning. However, their desirable characteristics such as analytic invertibility come at the cost of restricting the functional forms. This poses a question on their representation power: are CF-INNs universal approximators for invertible functions? Without a universality, there could be a well-behaved invertible transformation that the CF-INN can never approximate, hence it would render the model class unreliable. We answer this question by showing a convenient criterion: a CF-INN is universal if its layers contain affine coupling and invertible linear functions as special cases. As its corollary, we can affirmatively resolve a previously unsolved problem: whether normalizing flow models based on affine coupling can be universal distributional approximators. In the course of proving the universality, we prove a general theorem to show the equivalence of the universality for certain diffeomorphism classes, a theoretical insight that is of interest by itself.

Key facts. Cited by 85 papers in the map (data through 7 Oct 2026). First posted on arXiv 20 Jun 2020.

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Citing papers per year: 2021: 20; 2022: 20; 2023: 19; 2024: 12; 2025: 5; 2026: 9.

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  1. 2021Neural Flows: Efficient Alternative to Neural ODEscited by 99
  2. 2023On the Stability of Iterative Retraining of Generative Models on their own Datacited by 98
  3. 2023Learning Linear Causal Representations from Interventions under General Nonlinear Mixingcited by 92
  4. 2022Identifiability of deep generative models without auxiliary informationcited by 70
  5. 2021Trumpets: Injective Flows for Inference and Inverse Problemscited by 36
  6. 2025Scalable Equilibrium Sampling with Sequential Boltzmann Generatorscited by 36
  7. 2021Memory-Efficient Network for Large-scale Video Compressive Sensingcited by 30
  8. 2022Universal approximation property of invertible neural networkscited by 30
  9. 2023Free-form Flows: Make Any Architecture a Normalizing Flowcited by 26
  10. 2022Reliable amortized variational inference with physics-based latent distribution correctioncited by 25
  11. 2022PatchNR: Learning from Very Few Images by Patch Normalizing Flow Regularizationcited by 22
  12. 2024Deep Learning in Medical Image Registration: Magic or Mirage?cited by 20
38 more
  1. 2023Efficient Neural Network Approaches for Conditional Optimal Transport with Applications in Bayesian Inferencecited by 19
  2. 2024On the Universality of Volume-Preserving and Coupling-Based Normalizing Flowscited by 18
  3. 2022Bridging Mean-Field Games and Normalizing Flows with Trajectory Regularizationcited by 17
  4. 2022Flow-based Recurrent Belief State Learning for POMDPscited by 16
  5. 2021On the expressivity of bi-Lipschitz normalizing flowscited by 15
  6. 2022Invertible Rescaling Network and Its Extensionscited by 15
  7. 2021Sparse Flows: Pruning Continuous-depth Modelscited by 13
  8. 2021Universal Joint Approximation of Manifolds and Densities by Simple Injective Flowscited by 13
  9. 2021Diffeomorphically Learning Stable Koopman Operatorscited by 13
  10. 2023Minimum Width of Leaky-ReLU Neural Networks for Uniform Universal Approximationcited by 13
  11. 2023Globally injective and bijective neural operatorscited by 13
  12. 2023Conditional Generative Models are Provably Robust: Pointwise Guarantees for Bayesian Inverse Problemscited by 12
  13. 2021Rethinking Importance Weighting for Transfer Learningcited by 11
  14. 2022Learning reversible symplectic dynamicscited by 11
  15. 2022Gradual Domain Adaptation via Normalizing Flowscited by 11
  16. 2024Learning Neural Contracting Dynamics: Extended Linearization and Global Guaranteescited by 11
  17. 2024ASPIRE: Iterative Amortized Posterior Inference for Bayesian Inverse Problemscited by 11
  18. 2024Flexible Tails for Normalizing Flowscited by 11
  19. 2021Learning the Koopman Eigendecomposition: A Diffeomorphic Approachcited by 9
  20. 2022C^*-algebra Net: A New Approach Generalizing Neural Network Parameters to C^*-algebracited by 9
  21. 2021Jacobian Determinant of Normalizing Flowscited by 8
  22. 2023LU-Net: Invertible Neural Networks Based on Matrix Factorizationcited by 8
  23. 2021Universal Approximation of Residual Flows in Maximum Mean Discrepancycited by 7
  24. 2021Universal Approximation for Log-concave Distributions using Well-conditioned Normalizing Flowscited by 7
  25. 2023Bounded KRnet and its applications to density estimation and approximationcited by 7
  26. 2021EMFlow: Data Imputation in Latent Space via EM and Deep Flow Modelscited by 6
  27. 2022Heavy-tailed Sampling via Transformed Unadjusted Langevin Algorithmcited by 6
  28. 2022Approximation of nearly-periodic symplectic maps via structure-preserving neural networkscited by 6
  29. 2022Invertible Monotone Operators for Normalizing Flowscited by 6
  30. 2022Whitening Convergence Rate of Coupling-based Normalizing Flowscited by 6
  31. 2024Enhancing Accuracy in Generative Models via Knowledge Transfercited by 6
  32. 2025Efficient Regression-Based Training of Normalizing Flows for Boltzmann Generatorscited by 6
  33. 2023On the Convergence Rate of Gaussianization with Random Rotationscited by 5
  34. 2025Manifold Learning with Normalizing Flows: Towards Regularity, Expressivity and Iso-Riemannian Geometrycited by 5
  35. 2021On the Generative Utility of Cyclic Conditionalscited by 4
  36. 2022Vanilla Feedforward Neural Networks as a Discretization of Dynamical Systemscited by 4
  37. 2023A Flow-Based Generative Model for Rare-Event Simulationcited by 4
  38. 2023Finding emergence in data by maximizing effective informationcited by 4

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