mlsanityData to 7 Oct 2026

Universal Joint Approximation of Manifolds and Densities by Simple Injective Flows

arXiv 2110.04227 ↗First posted 8 Oct 2021cs.LGcited by 13

Not on the map. Why?

Abstract (arXiv)

We study approximation of probability measures supported on n-dimensional manifolds embedded in ℝm by injective flows -- neural networks composed of invertible flows and injective layers. We show that in general, injective flows between ℝn and ℝm universally approximate measures supported on images of extendable embeddings, which are a subset of standard embeddings: when the embedding dimension m is small, topological obstructions may preclude certain manifolds as admissible targets. When the embedding dimension is sufficiently large, m ≥ 3n+1, we use an argument from algebraic topology known as the clean trick to prove that the topological obstructions vanish and injective flows universally approximate any differentiable embedding. Along the way we show that the studied injective flows admit efficient projections on the range, and that their optimality can be established "in reverse," resolving a conjecture made in Brehmer and Cranmer 2020.

Key facts. Cited by 13 papers in the map (data through 7 Oct 2026). Builds on 29 of its 50 references that are in the map. First posted on arXiv 8 Oct 2021.

Builds on

29 of 50 references.

References · 29

  1. 2016Density estimation using Real NVPcited by 2,029
  2. 2018Glow: Generative Flow with Invertible 1x1 Convolutionscited by 1,539
  3. 2019Normalizing Flows for Probabilistic Modeling and Inferencecited by 1,315
  4. 2014NICE: Non-linear Independent Components Estimationcited by 1,052
  5. 2016Improving Variational Inference with Inverse Autoregressive Flowcited by 636
  6. 2017Wasserstein GANcited by 635
  7. 2019Neural Spline Flowscited by 590
  8. 2018FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Modelscited by 584
  9. 2016Error bounds for approximations with deep ReLU networkscited by 567
  10. 2017Compressed Sensing using Generative Modelscited by 337
  11. 2018Analyzing Inverse Problems with Invertible Neural Networkscited by 239
  12. 2017The Reversible Residual Network: Backpropagation Without Storing Activationscited by 215
17 more
  1. 2018Neural Autoregressive Flowscited by 207
  2. 2018Optimal approximation of continuous functions by very deep ReLU networkscited by 165
  3. 2018i-RevNet: Deep Invertible Networkscited by 137
  4. 2020Flows for simultaneous manifold learning and density estimationcited by 117
  5. 2020Coupling-based Invertible Neural Networks Are Universal Diffeomorphism Approximatorscited by 85
  6. 2019Reverse-Engineering Deep ReLU Networkscited by 81
  7. 2019Sum-of-Squares Polynomial Flowcited by 62
  8. 2020Deep Probabilistic Imaging: Uncertainty Quantification and Multi-modal Solution Characterization for Computational Imagingcited by 57
  9. 2021Trumpets: Injective Flows for Inference and Inverse Problemscited by 36
  10. 2019Inverting Deep Generative models, One layer at a timecited by 33
  11. 2019HINT: Hierarchical Invertible Neural Transport for Density Estimation and Bayesian Inferencecited by 31
  12. 2017On the ability of neural nets to express distributionscited by 26
  13. 2021Benchmarking Invertible Architectures on Inverse Problemscited by 25
  14. 2020Globally Injective ReLU Networkscited by 23
  15. 2020Normalizing Flows Across Dimensionscited by 19
  16. 2020When and How Can Deep Generative Models be Inverted?cited by 8
  17. 2020A Universal Approximation Theorem of Deep Neural Networks for Expressing Probability Distributionscited by 7

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 5; 2023: 4; 2024: 3; 2025: 0; 2026: 1.

13 papers.

Sort:

By area:

  1. 2024Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connectionscited by 47
  2. 2022Universal approximation property of invertible neural networkscited by 30
  3. 2022Conditional Injective Flows for Bayesian Imagingcited by 13
  4. 2023Globally injective and bijective neural operatorscited by 13
  5. 2023Injectivity of ReLU networks: perspectives from statistical physicscited by 10
  6. 2022Do ReLU Networks Have An Edge When Approximating Compactly-Supported Functions?cited by 6
  7. 2022Continuous Generative Neural Networks: A Wavelet-Based Architecture in Function Spacescited by 6
  8. 2024Injectivity capacity of ReLU gatescited by 4
  9. 2023Deep Injective Prior for Inverse Scatteringcited by 3
  10. 2023On the Approximation of Bi-Lipschitz Maps by Invertible Neural Networkscited by 2
  11. 2022Deep Invertible Approximation of Topologically Rich Maps between Manifoldscited by 0
  12. 2024Injectivity of ReLU-layers: Tools from Frame Theorycited by 0
1 more
  1. 2026Kinks vs. Smoothness: Identifiability of Real Analytic nICA for Laplace-like Sourcescited by 0