mlsanityData to 7 Oct 2026

Cubic-Spline Flows

arXiv 1906.02145 ↗First posted 5 Jun 2019stat.ML cs.LGcited by 34

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Abstract (arXiv)

A normalizing flow models a complex probability density as an invertible transformation of a simple density. The invertibility means that we can evaluate densities and generate samples from a flow. In practice, autoregressive flow-based models are slow to invert, making either density estimation or sample generation slow. Flows based on coupling transforms are fast for both tasks, but have previously performed less well at density estimation than autoregressive flows. We stack a new coupling transform, based on monotonic cubic splines, with LU-decomposed linear layers. The resulting cubic-spline flow retains an exact one-pass inverse, can be used to generate high-quality images, and closes the gap with autoregressive flows on a suite of density-estimation tasks.

Key facts. Cited by 34 papers in the map (data through 7 Oct 2026). First posted on arXiv 5 Jun 2019.

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Citing papers per year: 2021: 5; 2022: 7; 2023: 9; 2024: 8; 2025: 4; 2026: 1.

34 papers.

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  1. 2021Deep Generative Modelling: A Comparative Review of VAEs, GANs, Normalizing Flows, Energy-Based and Autoregressive Modelscited by 251
  2. 2021Generative Networks for Precision Enthusiastscited by 21
  3. 2022An unfolding method based on conditional Invertible Neural Networks (cINN) using iterative trainingcited by 20
  4. 2024The Landscape of Unfolding with Machine Learningcited by 20
  5. 2024On the Universality of Volume-Preserving and Coupling-Based Normalizing Flowscited by 18
  6. 2021Multi-Asset Spot and Option Market Simulationcited by 17
  7. 2024Variational Bayesian Optimal Experimental Design with Normalizing Flowscited by 13
  8. 2023A Review of Change of Variable Formulas for Generative Modelingcited by 12
  9. 2024Learning Neural Contracting Dynamics: Extended Linearization and Global Guaranteescited by 11
  10. 2023Normalizing Flow Ensembles for Rich Aleatoric and Epistemic Uncertainty Modelingcited by 10
  11. 2022On the Robustness of Normalizing Flows for Inverse Problems in Imagingcited by 8
  12. 2022Continuous and Distribution-free Probabilistic Wind Power Forecasting: A Conditional Normalizing Flow Approachcited by 7
22 more
  1. 2023Bounded KRnet and its applications to density estimation and approximationcited by 7
  2. 2023Deep Generative Models, Synthetic Tabular Data, and Differential Privacy: An Overview and Synthesiscited by 7
  3. 2024DA-Flow: Dual Attention Normalizing Flow for Skeleton-based Video Anomaly Detectioncited by 7
  4. 2021Robust normalizing flows using Bernstein-type polynomialscited by 6
  5. 2022Short-Term Density Forecasting of Low-Voltage Load using Bernstein-Polynomial Normalizing Flowscited by 6
  6. 2022Whitening Convergence Rate of Coupling-based Normalizing Flowscited by 6
  7. 2021Bootstrap Your Flowcited by 5
  8. 2023Conformalized Deep Splines for Optimal and Efficient Prediction Setscited by 5
  9. 2024One flow to correct them all: improving simulations in high-energy physics with a single normalising flow and a switchcited by 4
  10. 2025Flows on convex polytopescited by 3
  11. 2022Risk-Neutral Market Simulationcited by 2
  12. 2022Learning to Bound: A Generative Cram'er-Rao Boundcited by 2
  13. 2023Neural Diffeomorphic Non-uniform B-spline Flowscited by 2
  14. 2023Improved Auto-Encoding using Deterministic Projected Belief Networkscited by 2
  15. 2024Deep generative modelling of canonical ensemble with differentiable thermal propertiescited by 2
  16. 2023Efficient Epistemic Uncertainty Estimation in Regression Ensemble Models Using Pairwise-Distance Estimatorscited by 1
  17. 2025BitHEP -- The Limits of Low-Precision ML in HEPcited by 1
  18. 2025Generative flow-based warm start of the variational quantum eigensolvercited by 1
  19. 2026Differentiable free energy surface: a variational approach to directly observing rare events using generative deep-learning modelscited by 1
  20. 2023Sig-Splines: universal approximation and convex calibration of time series generative modelscited by 0
  21. 2024An AI-powered Technology Stack for Solving Many-Electron Field Theorycited by 0
  22. 2025Hybrid Bernstein Normalizing Flows for Flexible Multivariate Density Regression with Interpretable Marginalscited by 0