mlsanityData to 14 Oct 2026

Tractable Density Estimation on Learned Manifolds with Conformal Embedding Flows

arXiv 2106.05275 ↗First posted 9 Jun 2021stat.ML cs.LGcited by 31

Not on the map. Why?

Abstract (arXiv)

Normalizing flows are generative models that provide tractable density estimation via an invertible transformation from a simple base distribution to a complex target distribution. However, this technique cannot directly model data supported on an unknown low-dimensional manifold, a common occurrence in real-world domains such as image data. Recent attempts to remedy this limitation have introduced geometric complications that defeat a central benefit of normalizing flows: exact density estimation. We recover this benefit with Conformal Embedding Flows, a framework for designing flows that learn manifolds with tractable densities. We argue that composing a standard flow with a trainable conformal embedding is the most natural way to model manifold-supported data. To this end, we present a series of conformal building blocks and apply them in experiments with synthetic and real-world data to demonstrate that flows can model manifold-supported distributions without sacrificing tractable likelihoods.

Key facts. Cited by 31 papers in the map (data through 14 Oct 2026). Builds on 49 of its 75 references that are in the map. First posted on arXiv 9 Jun 2021.

Builds on

49 of 75 references.

References · 49

  1. 2014Adam: A Method for Stochastic Optimizationcited by 37,240
  2. 2020Denoising Diffusion Probabilistic Modelscited by 21,076
  3. 2015Deep Unsupervised Learning using Nonequilibrium Thermodynamicscited by 7,058
  4. 2018A Style-Based Generator Architecture for Generative Adversarial Networkscited by 6,013
  5. 2016SGDR: Stochastic Gradient Descent with Warm Restartscited by 5,045
  6. 2019Generative Modeling by Estimating Gradients of the Data Distributioncited by 4,101
  7. 2013Auto-Encoding Variational Bayescited by 4,053
  8. 2014Deep Learning Face Attributes in the Wildcited by 3,693
  9. 2018Large Scale GAN Training for High Fidelity Natural Image Synthesiscited by 2,626
  10. 2017Improved Training of Wasserstein GANscited by 2,235
  11. 2015Variational Inference with Normalizing Flowscited by 2,170
  12. 2015Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networkscited by 2,054
37 more
  1. 2016Density estimation using Real NVPcited by 2,042
  2. 2018Glow: Generative Flow with Invertible 1x1 Convolutionscited by 1,545
  3. 2019Normalizing Flows for Probabilistic Modeling and Inferencecited by 1,323
  4. 2020Training Generative Adversarial Networks with Limited Datacited by 1,186
  5. 2014NICE: Non-linear Independent Components Estimationcited by 1,058
  6. 2016Pixel Recurrent Neural Networkscited by 816
  7. 2016Improving Variational Inference with Inverse Autoregressive Flowcited by 640
  8. 2020NVAE: A Deep Hierarchical Variational Autoencodercited by 637
  9. 2019Neural Spline Flowscited by 596
  10. 2018Do Deep Generative Models Know What They Don't Know?cited by 443
  11. 2020Reliable Fidelity and Diversity Metrics for Generative Modelscited by 434
  12. 2018Assessing Generative Models via Precision and Recallcited by 406
  13. 2017Wasserstein Auto-Encoderscited by 373
  14. 2020Very Deep VAEs Generalize Autoregressive Models and Can Outperform Them on Imagescited by 264
  15. 2019Diagnosing and Enhancing VAE Modelscited by 193
  16. 2018Dynamical Isometry and a Mean Field Theory of CNNs: How to Train 10,000-Layer Vanilla Convolutional Neural Networkscited by 176
  17. 2017Generalization and Equilibrium in Generative Adversarial Nets (GANs)cited by 166
  18. 2018The Malicious Use of Artificial Intelligence: Forecasting, Prevention, and Mitigationcited by 165
  19. 2020Riemannian Continuous Normalizing Flowscited by 145
  20. 2019Cheap Orthogonal Constraints in Neural Networks: A Simple Parametrization of the Orthogonal and Unitary Groupcited by 134
  21. 2020Flows for simultaneous manifold learning and density estimationcited by 118
  22. 2020Normalizing Flows on Tori and Spherescited by 117
  23. 2020Image Representations Learned With Unsupervised Pre-Training Contain Human-like Biasescited by 107
  24. 2017Orthogonal Weight Normalization: Solution to Optimization over Multiple Dependent Stiefel Manifolds in Deep Neural Networkscited by 92
  25. 2019Trivializations for Gradient-Based Optimization on Manifoldscited by 79
  26. 2020Understanding and Mitigating Exploding Inverses in Invertible Neural Networkscited by 71
  27. 2016Normalizing Flows on Riemannian Manifoldscited by 68
  28. 2021Rectangular Flows for Manifold Learningcited by 50
  29. 2016Improving Variational Auto-Encoders using Householder Flowcited by 42
  30. 2020Deep Isometric Learning for Visual Recognitioncited by 39
  31. 2021Trumpets: Injective Flows for Inference and Inverse Problemscited by 36
  32. 2016Improving training of deep neural networks via Singular Value Boundingcited by 26
  33. 2020Regularized Autoencoders via Relaxed Injective Probability Flowcited by 23
  34. 2020LOCA: LOcal Conformal Autoencoder for standardized data coordinatescited by 19
  35. 2020Normalizing Flows Across Dimensionscited by 19
  36. 2016Generalized BackPropagation, 'Etude De Cas: Orthogonalitycited by 17
  37. 2016Optimization on Submanifolds of Convolution Kernels in CNNscited by 15

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2021: 1; 2022: 12; 2023: 8; 2024: 5; 2025: 2; 2026: 3.

31 papers.

Sort:

By area:

  1. 2022Verifying the Union of Manifolds Hypothesis for Image Datacited by 64
  2. 2024Deep Generative Models through the Lens of the Manifold Hypothesis: A Survey and New Connectionscited by 47
  3. 2022LIDL: Local Intrinsic Dimension Estimation Using Approximate Likelihoodcited by 30
  4. 2022Diagnosing and Fixing Manifold Overfitting in Deep Generative Modelscited by 28
  5. 2022CaloMan: Fast generation of calorimeter showers with density estimation on learned manifoldscited by 28
  6. 2024A Geometric Explanation of the Likelihood OOD Detection Paradoxcited by 22
  7. 2023Lifting Architectural Constraints of Injective Flowscited by 18
  8. 2023Canonical normalizing flows for manifold learningcited by 15
  9. 2022Conditional Injective Flows for Bayesian Imagingcited by 13
  10. 2023A Review of Change of Variable Formulas for Generative Modelingcited by 13
  11. 2022Gradual Domain Adaptation via Normalizing Flowscited by 11
  12. 2022nu-Flows: Conditional Neutrino Regressioncited by 11
19 more
  1. 2022VQ-Flows: Vector Quantized Local Normalizing Flowscited by 10
  2. 2023Injectivity of ReLU networks: perspectives from statistical physicscited by 10
  3. 2022Denoising Deep Generative Modelscited by 6
  4. 2022Flowification: Everything is a Normalizing Flowcited by 5
  5. 2022Neural Implicit Manifold Learning for Topology-Aware Density Estimationcited by 5
  6. 2021Spread Flows for Manifold Modellingcited by 4
  7. 2022Deterministic training of generative autoencoders using invertible layerscited by 4
  8. 2024Injectivity capacity of ReLU gatescited by 4
  9. 2023Deep Injective Prior for Inverse Scatteringcited by 3
  10. 2025On Convolutions, Intrinsic Dimension, and Diffusion Modelscited by 3
  11. 2023TR0N: Translator Networks for 0-Shot Plug-and-Play Conditional Generationcited by 2
  12. 2024Analyzing Generative Models by Manifold Entropic Metricscited by 2
  13. 2023Conformal Generative Modeling on Triangulated Surfacescited by 1
  14. 2024Injective flows for star-like manifoldscited by 1
  15. 2026Tunable Latent Generative Priors for Compressed Sensing and Inverse Problemscited by 1
  16. 2023Out-of-distribution detection using normalizing flows on the data manifoldcited by 0
  17. 2025Assessing local deformation and computing scalar curvature with nonlinear conformal regularization of decoderscited by 0
  18. 2026Complex normalizing flows can almost be information K"ahler-Ricci flowscited by 0
  19. 2026NAE: Normalizing AutoEncodercited by 0