# Cubic-Spline Flows

arXiv: https://arxiv.org/abs/1906.02145 · HTML: https://mlsanity.com/paper/1906.02145/

## Key facts

- Cited by 35 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Jun 2019.

## Built on by (most cited first, 30 of 35)

- [Deep Generative Modelling: A Comparative Review of VAEs, GANs, Normalizing Flows, Energy-Based and Autoregressive Models](https://mlsanity.com/paper/2103.04922/) (2021)
- [Generative Networks for Precision Enthusiasts](https://mlsanity.com/paper/2110.13632/) (2021)
- [An unfolding method based on conditional Invertible Neural Networks (cINN) using iterative training](https://mlsanity.com/paper/2212.08674/) (2022)
- [The Landscape of Unfolding with Machine Learning](https://mlsanity.com/paper/2404.18807/) (2024)
- [On the Universality of Volume-Preserving and Coupling-Based Normalizing Flows](https://mlsanity.com/paper/2402.06578/) (2024)
- [Multi-Asset Spot and Option Market Simulation](https://mlsanity.com/paper/2112.06823/) (2021)
- [A Review of Change of Variable Formulas for Generative Modeling](https://mlsanity.com/paper/2308.02652/) (2023)
- [Variational Bayesian Optimal Experimental Design with Normalizing Flows](https://mlsanity.com/paper/2404.13056/) (2024)
- [Learning Neural Contracting Dynamics: Extended Linearization and Global Guarantees](https://mlsanity.com/paper/2402.08090/) (2024)
- [Normalizing Flow Ensembles for Rich Aleatoric and Epistemic Uncertainty Modeling](https://mlsanity.com/paper/2302.01312/) (2023)
- [On the Robustness of Normalizing Flows for Inverse Problems in Imaging](https://mlsanity.com/paper/2212.04319/) (2022)
- [Continuous and Distribution-free Probabilistic Wind Power Forecasting: A Conditional Normalizing Flow Approach](https://mlsanity.com/paper/2206.02433/) (2022)
- [Bounded KRnet and its applications to density estimation and approximation](https://mlsanity.com/paper/2305.09063/) (2023)
- [Deep Generative Models, Synthetic Tabular Data, and Differential Privacy: An Overview and Synthesis](https://mlsanity.com/paper/2307.15424/) (2023)
- [DA-Flow: Dual Attention Normalizing Flow for Skeleton-based Video Anomaly Detection](https://mlsanity.com/paper/2406.02976/) (2024)
- [Robust normalizing flows using Bernstein-type polynomials](https://mlsanity.com/paper/2102.03509/) (2021)
- [Short-Term Density Forecasting of Low-Voltage Load using Bernstein-Polynomial Normalizing Flows](https://mlsanity.com/paper/2204.13939/) (2022)
- [Whitening Convergence Rate of Coupling-based Normalizing Flows](https://mlsanity.com/paper/2210.14032/) (2022)
- [Bootstrap Your Flow](https://mlsanity.com/paper/2111.11510/) (2021)
- [Conformalized Deep Splines for Optimal and Efficient Prediction Sets](https://mlsanity.com/paper/2311.00774/) (2023)
- [One flow to correct them all: improving simulations in high-energy physics with a single normalising flow and a switch](https://mlsanity.com/paper/2403.18582/) (2024)
- [Flows on convex polytopes](https://mlsanity.com/paper/2503.10232/) (2025)
- [Risk-Neutral Market Simulation](https://mlsanity.com/paper/2202.13996/) (2022)
- [Learning to Bound: A Generative Cram'er-Rao Bound](https://mlsanity.com/paper/2203.03695/) (2022)
- [Neural Diffeomorphic Non-uniform B-spline Flows](https://mlsanity.com/paper/2304.04555/) (2023)
- [Improved Auto-Encoding using Deterministic Projected Belief Networks](https://mlsanity.com/paper/2309.07481/) (2023)
- [Deep generative modelling of canonical ensemble with differentiable thermal properties](https://mlsanity.com/paper/2404.18404/) (2024)
- [Efficient Epistemic Uncertainty Estimation in Regression Ensemble Models Using Pairwise-Distance Estimators](https://mlsanity.com/paper/2308.13498/) (2023)
- [BitHEP -- The Limits of Low-Precision ML in HEP](https://mlsanity.com/paper/2504.03387/) (2025)
- [Generative flow-based warm start of the variational quantum eigensolver](https://mlsanity.com/paper/2507.01726/) (2025)

## 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.