mlsanityData to 7 Oct 2026

Principal Manifold Flows

arXiv 2202.07037 ↗First posted 14 Feb 2022stat.ML cs.LGcited by 4

Not on the map. Why?

Abstract (arXiv)

Normalizing flows map an independent set of latent variables to their samples using a bijective transformation. Despite the exact correspondence between samples and latent variables, their high level relationship is not well understood. In this paper we characterize the geometric structure of flows using principal manifolds and understand the relationship between latent variables and samples using contours. We introduce a novel class of normalizing flows, called principal manifold flows (PF), whose contours are its principal manifolds, and a variant for injective flows (iPF) that is more efficient to train than regular injective flows. PFs can be constructed using any flow architecture, are trained with a regularized maximum likelihood objective and can perform density estimation on all of their principal manifolds. In our experiments we show that PFs and iPFs are able to learn the principal manifolds over a variety of datasets. Additionally, we show that PFs can perform density estimation on data that lie on a manifold with variable dimensionality, which is not possible with existing normalizing flows.

Key facts. Cited by 4 papers in the map (data through 7 Oct 2026). Builds on 26 of its 47 references that are in the map. First posted on arXiv 14 Feb 2022.

Builds on

26 of 47 references.

References · 26

  1. 2018A Style-Based Generator Architecture for Generative Adversarial Networkscited by 5,986
  2. 2015Variational Inference with Normalizing Flowscited by 2,161
  3. 2016Density estimation using Real NVPcited by 2,029
  4. 2018Glow: Generative Flow with Invertible 1x1 Convolutionscited by 1,539
  5. 2019Normalizing Flows for Probabilistic Modeling and Inferencecited by 1,315
  6. 2016InfoGAN: Interpretable Representation Learning by Information Maximizing Generative Adversarial Netscited by 858
  7. 2019Neural Spline Flowscited by 590
  8. 2020AdaBelief Optimizer: Adapting Stepsizes by the Belief in Observed Gradientscited by 282
  9. 2019Flow++: Improving Flow-Based Generative Models with Variational Dequantization and Architecture Designcited by 258
  10. 2019Residual Flows for Invertible Generative Modelingcited by 256
  11. 2020Normalizing Flows on Tori and Spherescited by 117
  12. 2020Flows for simultaneous manifold learning and density estimationcited by 117
14 more
  1. 2021Independent mechanism analysis, a new concept?cited by 96
  2. 2020Convex Potential Flows: Universal Probability Distributions with Optimal Transport and Convex Optimizationcited by 89
  3. 2019NeuTra-lizing Bad Geometry in Hamiltonian Monte Carlo Using Neural Transportcited by 73
  4. 2018Sylvester Normalizing Flows for Variational Inferencecited by 71
  5. 2016Normalizing Flows on Riemannian Manifoldscited by 68
  6. 2020Latent Variable Modelling with Hyperbolic Normalizing Flowscited by 49
  7. 2021Rectangular Flows for Manifold Learningcited by 49
  8. 2021Trumpets: Injective Flows for Inference and Inverse Problemscited by 36
  9. 2021Orthogonal Jacobian Regularization for Unsupervised Disentanglement in Image Generationcited by 35
  10. 2020Regularized Autoencoders via Relaxed Injective Probability Flowcited by 23
  11. 2020Normalizing Flows Across Dimensionscited by 19
  12. 2020Isometric Autoencoderscited by 18
  13. 2021Principal Component Density Estimation for Scenario Generation Using Normalizing Flowscited by 10
  14. 2021Generative Locally Linear Embeddingcited by 1

Built on by

Cited by year

Citing papers per year: 2022: 3; 2023: 1.

4 papers.

  1. 2022Verifying the Union of Manifolds Hypothesis for Image Datacited by 62
  2. 2023Independent Mechanism Analysis and the Manifold Hypothesiscited by 7
  3. 2022Nonlinear Isometric Manifold Learning for Injective Normalizing Flowscited by 5
  4. 2022Probing the Robustness of Independent Mechanism Analysis for Representation Learningcited by 5