# Neural Spline Flows

arXiv: https://arxiv.org/abs/1906.04032 · HTML: https://mlsanity.com/paper/1906.04032/

## Key facts

- Cited by 596 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Jun 2019.

## Built on by (most cited first, 30 of 596)

- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Deep Generative Modelling: A Comparative Review of VAEs, GANs, Normalizing Flows, Energy-Based and Autoregressive Models](https://mlsanity.com/paper/2103.04922/) (2021)
- [Continuous diffusion for categorical data](https://mlsanity.com/paper/2211.15089/) (2022)
- [From data to functa: Your data point is a function and you can treat it like one](https://mlsanity.com/paper/2201.12204/) (2022)
- [Benchmarking Simulation-Based Inference](https://mlsanity.com/paper/2101.04653/) (2021)
- [Liquid Structural State-Space Models](https://mlsanity.com/paper/2209.12951/) (2022)
- [Flow Annealed Importance Sampling Bootstrap](https://mlsanity.com/paper/2208.01893/) (2022)
- [Nonparametric Identifiability of Causal Representations from Unknown Interventions](https://mlsanity.com/paper/2306.00542/) (2023)
- [Flow Matching for Scalable Simulation-Based Inference](https://mlsanity.com/paper/2305.17161/) (2023)
- [Stochastic interpolants with data-dependent couplings](https://mlsanity.com/paper/2310.03725/) (2023)
- [Deep End-to-end Causal Inference](https://mlsanity.com/paper/2202.02195/) (2022)
- [Annealed Flow Transport Monte Carlo](https://mlsanity.com/paper/2102.07501/) (2021)
- [Real-time gravitational-wave science with neural posterior estimation](https://mlsanity.com/paper/2106.12594/) (2021)
- [Diffusion Models for Causal Discovery via Topological Ordering](https://mlsanity.com/paper/2210.06201/) (2022)
- [Membership Inference Attacks against Synthetic Data through Overfitting Detection](https://mlsanity.com/paper/2302.12580/) (2023)
- [Verifying the Union of Manifolds Hypothesis for Image Data](https://mlsanity.com/paper/2207.02862/) (2022)
- [Density Ratio Estimation via Infinitesimal Classification](https://mlsanity.com/paper/2111.11010/) (2021)
- [From Denoising Diffusions to Denoising Markov Models](https://mlsanity.com/paper/2211.03595/) (2022)
- [normflows: A PyTorch Package for Normalizing Flows](https://mlsanity.com/paper/2302.12014/) (2023)
- [Causal Component Analysis](https://mlsanity.com/paper/2305.17225/) (2023)
- [Beyond ELBOs: A Large-Scale Evaluation of Variational Methods for Sampling](https://mlsanity.com/paper/2406.07423/) (2024)
- [Recent Advances of Continual Learning in Computer Vision: An Overview](https://mlsanity.com/paper/2109.11369/) (2021)
- [Smooth Normalizing Flows](https://mlsanity.com/paper/2110.00351/) (2021)
- [Causal normalizing flows: from theory to practice](https://mlsanity.com/paper/2306.05415/) (2023)
- [Rectangular Flows for Manifold Learning](https://mlsanity.com/paper/2106.01413/) (2021)
- [Sequential Neural Score Estimation: Likelihood-Free Inference with Conditional Score Based Diffusion Models](https://mlsanity.com/paper/2210.04872/) (2022)
- [CaloFlow: Fast and Accurate Generation of Calorimeter Showers with Normalizing Flows](https://mlsanity.com/paper/2106.05285/) (2021)
- [Learning Disentangled Representations in the Imaging Domain](https://mlsanity.com/paper/2108.12043/) (2021)
- [Robust Neural Posterior Estimation and Statistical Model Criticism](https://mlsanity.com/paper/2210.06564/) (2022)

## Abstract (arXiv)

A normalizing flow models a complex probability density as an invertible transformation of a simple base density. Flows based on either coupling or autoregressive transforms both offer exact density evaluation and sampling, but rely on the parameterization of an easily invertible elementwise transformation, whose choice determines the flexibility of these models. Building upon recent work, we propose a fully-differentiable module based on monotonic rational-quadratic splines, which enhances the flexibility of both coupling and autoregressive transforms while retaining analytic invertibility. We demonstrate that neural spline flows improve density estimation, variational inference, and generative modeling of images.