mlsanityData to 7 Oct 2026

On the expressivity of bi-Lipschitz normalizing flows

arXiv 2107.07232 ↗First posted 15 Jul 2021cs.LG stat.MLcited by 15

Not on the map. Why?

Abstract (arXiv)

An invertible function is bi-Lipschitz if both the function and its inverse have bounded Lipschitz constants. Nowadays, most Normalizing Flows are bi-Lipschitz by design or by training to limit numerical errors (among other things). In this paper, we discuss the expressivity of bi-Lipschitz Normalizing Flows and identify several target distributions that are difficult to approximate using such models. Then, we characterize the expressivity of bi-Lipschitz Normalizing Flows by giving several lower bounds on the Total Variation distance between these particularly unfavorable distributions and their best possible approximation. Finally, we discuss potential remedies which include using more complex latent distributions.

Key facts. Cited by 15 papers in the map (data through 7 Oct 2026). Builds on 24 of its 37 references that are in the map. First posted on arXiv 15 Jul 2021.

Builds on

24 of 37 references.

References · 24

  1. 2013Intriguing properties of neural networkscited by 5,114
  2. 2014Deep Learning Face Attributes in the Wildcited by 3,679
  3. 2018Spectral Normalization for Generative Adversarial Networkscited by 1,618
  4. 2018Glow: Generative Flow with Invertible 1x1 Convolutionscited by 1,539
  5. 2019Improved Precision and Recall Metric for Assessing Generative Modelscited by 925
  6. 2017Spectrally-normalized margin bounds for neural networkscited by 681
  7. 2017Wasserstein GANcited by 635
  8. 2018Lipschitz regularity of deep neural networks: analysis and efficient estimationcited by 413
  9. 2018Assessing Generative Models via Precision and Recallcited by 401
  10. 2018Invertible Residual Networkscited by 364
  11. 2019Residual Flows for Invertible Generative Modelingcited by 256
  12. 2020Coupling-based Invertible Neural Networks Are Universal Diffeomorphism Approximatorscited by 85
12 more
  1. 2019Semi-Supervised Learning with Normalizing Flowscited by 76
  2. 2020Understanding and Mitigating Exploding Inverses in Invertible Neural Networkscited by 71
  3. 2018Generalizable Adversarial Training via Spectral Normalizationcited by 68
  4. 2019Approximation Capabilities of Neural ODEs and Invertible Residual Networkscited by 56
  5. 2019Lipschitz Certificates for Layered Network Structures Driven by Averaged Activation Operatorscited by 54
  6. 2020The Expressive Power of a Class of Normalizing Flow Modelscited by 35
  7. 2018Disconnected Manifold Learning for Generative Adversarial Networkscited by 27
  8. 2020Learning disconnected manifolds: a no GANs landcited by 25
  9. 2020Representational aspects of depth and conditioning in normalizing flowscited by 25
  10. 2019A RAD approach to deep mixture modelscited by 18
  11. 2020Variational Mixture of Normalizing Flowscited by 15
  12. 2021Universal Approximation of Residual Flows in Maximum Mean Discrepancycited by 7

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 1; 2022: 3; 2023: 5; 2024: 1; 2025: 4; 2026: 1.

15 papers.

Sort:

By area:

  1. 2021A Dynamical System Perspective for Lipschitz Neural Networkscited by 59
  2. 2023A mean-field games laboratory for generative modelingcited by 28
  3. 2022Out-of-Distribution Detection with Reconstruction Error and Typicality-based Penaltycited by 11
  4. 2023Simulation-based Inference for Cardiovascular Modelscited by 11
  5. 2023Precision-Recall Divergence Optimization for Generative Modeling with GANs and Normalizing Flowscited by 9
  6. 2025Intention-Conditioned Flow Occupancy Modelscited by 7
  7. 2022Vec2Face-v2: Unveil Human Faces from their Blackbox Features via Attention-based Network in Face Recognitioncited by 4
  8. 2023NF-ULA: Langevin Monte Carlo with Normalizing Flow Prior for Imaging Inverse Problemscited by 4
  9. 2022Learning to Bound: A Generative Cram'er-Rao Boundcited by 2
  10. 2025Ergodic Generative Flowscited by 2
  11. 2023Master's Thesis: Out-of-distribution Detection with Energy-based Modelscited by 1
  12. 2025Rare event modeling with self-regularized normalizing flows: what can we learn from a single failure?cited by 1
3 more
  1. 2024Distance-Preserving Representations for Genomic Spatial Reconstructioncited by 0
  2. 2025Conditional Distribution Quantization in Machine Learningcited by 0
  3. 2026Expressivity of Bi-Lipschitz Normalizing Flows: A Score-Based Diffusion Perspectivecited by 0