Awesome Normalizing Flows

A list of awesome resources for understanding and applying normalizing flows (NF): a relatively simple yet powerful new tool in statistics for constructing expressive probability distributions from simple base distributions using a chain (flow) of trainable smooth bijective transformations (diffeomorphisms).

^{Figure inspired by Lilian Weng. Created in TikZ. View source.}

📝 Publications
🛠️ Applications
📺 Videos
📦 Packages
🧑‍💻 Code
🌐 Blog Posts
🎉 Open to Suggestions!

📝 Publications

2022-01-14 - E(n) Equivariant Normalizing Flows by Satorras, Hoogeboom et al.

Introduces equivariant graph neural networks into the normalizing flow framework which combine to give invertible equivariant functions. Demonstrates their flow beats prior equivariant models and allows sampling of molecular configurations with positions, atom types and charges.
2021-07-03 - CInC Flow: Characterizable Invertible 3x3 Convolution by Nagar, Dufraisse et al.

These authors sought to improve the emerging convolutions as they were expensive. So they investigated the conditions for when 3x3 convolutions are invertible under which conditions (e.g. padding) and saw successful speedups. Furthermore, they developed a more expressive, invertible Quad coupling layer. [Code]
2021-04-14 - Orthogonalizing Convolutional Layers with the Cayley Transform by Trockman, Kolter

The authors parameterized the multichannel convolution to be orthogonal via the Cayley transform (skew-symmetric convolutions in the Fourier domain). This enables the inverse to be computed efficiently. [Code]
2021-04-14 - Improving Normalizing Flows via Better Orthogonal Parameterizations by Goliński, Lezcano-Casado et al.

The authors parameterize the 1x1 convolution via the exponential map and the Cayley map. They demonstrate an improved optimization for the Sylvester normalizing flows.
2020-09-21 - Haar Wavelet based Block Autoregressive Flows for Trajectories by Bhattacharyya, Straehle et al.

Introduce a Haar wavelet-based block autoregressive model.
2020-07-15 - AdvFlow: Inconspicuous Black-box Adversarial Attacks using Normalizing Flows by Dolatabadi, Erfani et al.

An adversarial attack method on image classifiers that use normalizing flows. [Code]
2020-07-06 - SurVAE Flows: Surjections to Bridge the Gap between VAEs and Flows by Nielsen, Jaini et al.

They present a generalized framework that encompasses both Flows (deterministic maps) and VAEs (stochastic maps). By seeing deterministic maps x = f(z) as limiting cases of stochastic maps x ~ p(x|z), the ELBO is reinterpreted as a change of variables formula for the stochastic maps. Moreover, they present a few examples of surjective layers using stochastic maps, which can be composed together with flow layers. [Video] [Code]
2020-06-15 - Why Normalizing Flows Fail to Detect Out-of-Distribution Data by Kirichenko, Izmailov et al.

This study how traditional normalizing flow models can suffer from out-of-distribution data. They offer a solution to combat this issue by modifying the coupling layers. [Tweet] [Code]
2020-06-03 - Equivariant Flows: exact likelihood generative learning for symmetric densities by Köhler, Klein et al.

Shows that distributions generated by equivariant NFs faithfully reproduce symmetries in the underlying density. Proposes building blocks for flows which preserve typical symmetries in physical/chemical many-body systems. Shows that symmetry-preserving flows can provide better generalization and sampling efficiency.
2020-06-02 - The Convolution Exponential and Generalized Sylvester Flows by Hoogeboom, Satorras et al.

These authors introduce the exponential convolution to add the spatial dependencies in linear layers as an improvement of the 1x1 convolutions. It uses matrix exponentials to create cheap and invertible layers. They also use this new architecture to create convolutional Sylvester flows and graph convolutional exponentials. [Code]
2020-05-11 - iUNets: Fully invertible U-Nets with Learnable Upand Downsampling by Etmann, Ke et al.

The authors extend the classical UNet to be fully invertible by enabling invertible, orthogonal upsampling and downsampling layers. It is rather efficient so it should be able to enable stable training of deeper and larger networks.
2020-04-08 - Normalizing Flows with Multi-Scale Autoregressive Priors by Mahajan, Bhattacharyya et al.

Improves the representational power of flow-based models by introducing channel-wise dependencies in their latent space through multi-scale autoregressive priors (mAR). [Code]
2020-03-31 - Flows for simultaneous manifold learning and density estimation by Brehmer, Cranmer

Normalizing flows that learn the data manifold and probability density function on that manifold. [Tweet] [Code]
2020-03-04 - Gaussianization Flows by Meng, Song et al.

Uses a repeated composition of trainable kernel layers and orthogonal transformations. Very competitive versus some of the SOTA like Real-NVP, Glow and FFJORD. [Code]
2020-02-27 - Gradient Boosted Normalizing Flows by Giaquinto, Banerjee

Augment traditional normalizing flows with gradient boosting. They show that training multiple models can achieve good results and it's not necessary to have more complex distributions. [Code]
2020-02-24 - Modeling Continuous Stochastic Processes with Dynamic Normalizing Flows by Deng, Chang et al.

They propose a normalizing flow using differential deformation of the Wiener process. Applied to time series. [Tweet]
2020-02-21 - Stochastic Normalizing Flows by Hodgkinson, Heide et al.

Name clash for a very different technique from the above SNF: an extension of continuous normalizing flows using stochastic differential equations (SDE). Treats Brownian motion in the SDE as a latent variable and approximates it by a flow. Aims to enable efficient training of neural SDEs which can be used for constructing efficient Markov chains.
2020-02-16 - Stochastic Normalizing Flows by Wu, Köhler et al.

Introduces SNF, an arbitrary sequence of deterministic invertible functions (the flow) and stochastic processes such as MCMC or Langevin Dynamics. The aim is to increase expressiveness of the chosen deterministic invertible function, while the trainable flow improves sampling efficiency over pure MCMC [Tweet).]
2020-01-17 - Training Normalizing Flows with the Information Bottleneck for Competitive Generative Classification by Ardizzone, Mackowiak et al.

They introduce a class of conditional normalizing flows with an information bottleneck objective. [Code]
2020-01-15 - Invertible Generative Modeling using Linear Rational Splines by Dolatabadi, Erfani et al.

A successor to the Neural spline flows which features an easy-to-compute inverse.
2019-12-05 - Normalizing Flows for Probabilistic Modeling and Inference by Papamakarios, Nalisnick et al.

A thorough and very readable review article by some of the guys at DeepMind involved in the development of flows. Highly recommended.
2019-08-25 - Normalizing Flows: An Introduction and Review of Current Methods by Kobyzev, Prince et al.

Another very thorough and very readable review article going through the basics of NFs as well as some of the state-of-the-art. Also highly recommended.
2019-07-21 - Noise Regularization for Conditional Density Estimation by Rothfuss, Ferreira et al.

Normalizing flows for conditional density estimation. This paper proposes noise regularization to reduce overfitting. Blog
2019-07-18 - MintNet: Building Invertible Neural Networks with Masked Convolutions by Song, Meng et al.

The authors create an autoregressive-like coupling layer via masked convolutions which is fast and efficient to evaluate. [Code]
2019-07-18 - Densely connected normalizing flows by Grcić, Grubišić et al.

The authors created a nested coupling structure to add more expressivity to standard coupling layers. They also utilize slicing/factorization for dimensionality reduction and Nystromer for the coupling layer conditioning network. They achieved SOTA results for normalizing flow models. [Code]
2019-06-15 - Invertible Convolutional Flow by Karami, Schuurmans et al.

The authors introduce convolutional layers that are circular and symmetric. The layer is invertible and cheap to evaluate. They also showcase how one can design non-linear elementwise bijectors that induce special properties via constraining the loss function. [Code]
2019-06-15 - Invertible Convolutional Networks by Finzi, Izmailov et al.

The authors showcase how one can turn standard convolutional neural network layers to be invertible via Fourier transformations. They also introduce better activations which might be better suited to normalizing flows, e.g. SneakyRELU
2019-06-10 - Neural Spline Flows by Durkan, Bekasov et al.

Uses monotonic ration splines as a coupling layer. This is currently one of the state of the art.
2019-05-30 - Graph Normalizing Flows by Liu, Kumar et al.

A new, reversible graph network for prediction and generation. They perform similarly to message passing neural networks on supervised tasks, but at significantly reduced memory use, allowing them to scale to larger graphs. Combined with a novel graph auto-encoder for unsupervised learning, graph normalizing flows are a generative model for graph structures.
2019-05-24 - Fast Flow Reconstruction via Robust Invertible n x n Convolution by Truong, Luu et al.

The authors seek to overcome the limitation of 1x1 convolutions and proposes invertible nxn convolutions via a clever convolutional affine function.
2019-05-17 - Integer Discrete Flows and Lossless Compression by Hoogeboom, Peters et al.

A normalizing flow to be used for ordinal discrete data. They introduce a flexible transformation layer called integer discrete coupling.
2019-04-09 - Block Neural Autoregressive Flow) by Cao, Titov et al.

Introduces (B-NAF), a more efficient probability density approximator. Claims to be competitive with other flows across datasets while using orders of magnitude fewer parameters.
2019-02-19 - MaCow: Masked Convolutional Generative Flow by Ma, Kong et al.

The authors introduce a masked convolutional generative flow (MaCow) layer using a small kernel to capture local connectivity. They showed some improvement over the GLOW model while being fast and stable.
2019-01-30 - Emerging Convolutions for Generative Normalizing Flows by Hoogeboom, Berg et al.

The authors introduce autoregressive-like convolutional layers that operate on the channel and spatial axes. This improved upon the performance of image datasets compared to the standard 1x1 Convolutions. The trade-off is that the inverse operator is quite expensive however the authors provide a fast C++ implementation. [Code]
2018-11-06 - FloWaveNet : A Generative Flow for Raw Audio by Kim, Lee et al.

A flow-based generative model for raw audo synthesis. [Code]
2018-10-02 - FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models by Grathwohl, Chen et al.

Uses Neural ODEs as a solver to produce continuous-time normalizing flows (CNF).
2018-07-09 - Glow: Generative Flow with Invertible 1x1 Convolutions by Kingma, Dhariwal

They show that flows using invertible 1x1 convolution achieve high likelihood on standard generative benchmarks and can efficiently synthesize realistic-looking, large images.
2018-07-03 - Deep Density Destructors by Inouye, Ravikumar

Normalizing flows but from an iterative perspective. Features a Tree-based density estimator.
2018-04-03 - Neural Autoregressive Flows by Huang, Krueger et al.

Unifies and generalize autoregressive and normalizing flow approaches, replacing the (conditionally) affine univariate transformations of MAF/IAF with a more general class of invertible univariate transformations expressed as monotonic neural networks. Also demonstrates that the proposed neural autoregressive flows (NAF) are universal approximators for continuous probability distributions. [Code]
2018-03-15 - Sylvester Normalizing Flow for Variational Inference by Berg, Hasenclever et al.

Introduces Sylvester normalizing flows which remove the single-unit bottleneck from planar flows for increased flexibility in the variational posterior.
2017-11-17 - Convolutional Normalizing Flows by Zheng, Yang et al.

Introduces normalizing flows that take advantage of convolutions (based on convolution over the dimensions of random input vector) to improve the posterior in the variational inference framework. This also reduced the number of parameters due to the convolutions.
2017-05-19 - Masked Autoregressive Flow for Density Estimation by Papamakarios, Pavlakou et al.

Introduces MAF, a stack of autoregressive models forming a normalizing flow suitable for fast density estimation but slow at sampling. Analogous to Inverse Autoregressive Flow (IAF) except the forward and inverse passes are exchanged. Generalization of RNVP.
2017-03-06 - Multiplicative Normalizing Flows for Variational Bayesian Neural Networks by Louizos, Welling

They introduce a new type of variational Bayesian neural network that uses flows to generate auxiliary random variables which boost the flexibility of the variational family by multiplying the means of a fully-factorized Gaussian posterior over network parameters. This turns the usual diagonal covariance Gaussian into something that allows for multimodality and non-linear dependencies between network parameters.
2016-06-15 - Improving Variational Inference with Inverse Autoregressive Flow by Kingma, Salimans et al.

Introduces inverse autoregressive flow (IAF), a new type of flow which scales well to high-dimensional latent spaces.
2016-05-27 - Density estimation using Real NVP by Dinh, Sohl-Dickstein et al.

They introduce the affine coupling layer (RNVP), a major improvement in terms of flexibility over the additive coupling layer (NICE) with unit Jacobian while keeping a single-pass forward and inverse transformation for fast sampling and density estimation, respectively.
2015-05-21 - Variational Inference with Normalizing Flows by Rezende, Mohamed

They show how to go beyond mean-field variational inference by using flows to increase the flexibility of the variational family.
2015-02-12 - Masked Autoencoder for Distribution Estimation by Germain, Gregor et al.

Introduces MADE, a feed-forward network that uses carefully constructed binary masks on its weights to control the precise flow of information through the network. The masks ensure that each output unit receives signals only from input units that come before it in some arbitrary order. Yet all outputs can be computed in a single pass.
A popular and efficient method to bestow flows with autoregressivity is to construct them from MADE nets.

^{Figure created in TikZ. View source.}
2014-10-30 - Non-linear Independent Components Estimation by Dinh, Krueger et al.

Introduces the additive coupling layer (NICE) and shows how to use it for image generation and inpainting.
2011-04-01 - Iterative Gaussianization: from ICA to Random Rotations by Laparra, Camps-Valls et al.

Normalizing flows in the form of Gaussianization in an iterative format. Also shows connections to information theory.

🛠️ Applications

2020-12-06 - Normalizing Kalman Filters for Multivariate Time Series Analysis by Bézenac, Rangapuram et al.

Augments state space models with normalizing flows and thereby mitigates imprecisions stemming from idealized assumptions. Aimed at forecasting real-world data and handling varying levels of missing data. (Also available at Amazon Science.)
2020-11-02 - On the Sentence Embeddings from Pre-trained Language Models by Li, Zhou et al.

Proposes to use flows to transform anisotropic sentence embedding distributions from BERT to a smooth and isotropic Gaussian, learned through unsupervised objective. Demonstrates performance gains over SOTA sentence embeddings on semantic textual similarity tasks. Code available at https://github.com/bohanli/BERT-flow.
2020-10-13 - Targeted free energy estimation via learned mappings by Wirnsberger, Ballard et al.

Normalizing flows used to estimate free energy differences.
2020-07-15 - Faster Uncertainty Quantification for Inverse Problems with Conditional Normalizing Flows by Siahkoohi, Rizzuti et al.

Uses conditional normalizing flows for inverse problems. [Video]
2020-06-25 - SRFlow: Learning the Super-Resolution Space with Normalizing Flow by Lugmayr, Danelljan et al.

Uses normalizing flows for super-resolution.
2019-03-09 - NeuTra-lizing Bad Geometry in Hamiltonian Monte Carlo Using Neural Transport by Hoffman, Sountsov et al.

Uses normalizing flows in conjunction with Monte Carlo estimation to have more expressive distributions and better posterior estimation.
2018-08-14 - Analyzing Inverse Problems with Invertible Neural Networks by Ardizzone, Kruse et al.

Normalizing flows for inverse problems.
2018-04-09 - Latent Space Policies for Hierarchical Reinforcement Learning by Haarnoja, Hartikainen et al.

Uses normalizing flows, specifically RealNVPs, as policies for reinforcement learning and also applies them for the hierarchical reinforcement learning setting.

📺 Videos

2021-01-16 - Normalizing Flows - Motivations, The Big Idea & Essential Foundations by Kapil Sachdeva

A comprehensive tutorial on flows explaining the challenges addressed by this class of algorithm. Provides intuition on how to address those challenges, and explains the underlying mathematics using a simple step by step approach.
2020-12-07 - Normalizing Flows by Marc Deisenroth

Part of a NeurIPS 2020 tutorial series titled "There and Back Again: A Tale of Slopes and Expectations". Link to full series.
2020-11-23 - Introduction to Normalizing Flows by Marcus Brubaker

A great introduction to normalizing flows by one of the creators of Stan presented at ECCV 2020. The tutorial also provides an excellent review of various practical implementations.
2020-02-06 - Flow Models by Pieter Abbeel

A really thorough explanation of normalizing flows. Also includes some sample code.
2019-12-06 - What are normalizing flows? by Ari Seff

A great 3blue1brown-style video explaining the basics of normalizing flows.
2019-10-09 - A primer on normalizing flows by Laurent Dinh

The first author on both the NICE and RNVP papers and one of the first in this field gives an introductory talk at "Machine Learning for Physics and the Physics Learning of, 2019".
2019-09-24 - Graph Normalizing Flows by Jenny Liu

Introduces a new graph generating model for use e.g. in drug discovery, where training on molecules that are known to bind/dissolve/etc. may help to generate novel, similarly effective molecules.
2018-10-04 - Sylvester Normalizing Flow for Variational Inference by Rianne van den Berg

Introduces Sylvester normalizing flows which remove the single-unit bottleneck from planar flows for increased flexibility in the variational posterior.

📦 Packages

PyTorch Packages

2020-12-07 - flowtorch by Facebook / Meta

FlowTorch Docs is a PyTorch library for learning and sampling from complex probability distributions using a class of methods called Normalizing Flows.
2020-02-09 - nflows by Bayesiains

A suite of most of the SOTA methods using PyTorch. From an ML group in Edinburgh. They created the current SOTA spline flows. Almost as complete as you'll find from a single repo.
2018-09-07 - FrEIA by VLL Heidelberg

The Framework for Easily Invertible Architectures (FrEIA) is based on RNVP flows. Easy to setup, it allows to define complex Invertible Neural Networks (INNs) from simple invertible building blocks.

TensorFlow Packages

2018-06-22 - TensorFlow Probability by Google

Large first-party library that offers RNVP, MAF among other autoregressive models plus a collection of composable bijectors.

JAX Packages

2021-06-17 - pzflow by John Franklin Crenshaw

A package that focuses on probabilistic modeling of tabular data, with a focus on sampling and posterior calculation.
2021-04-12 - Distrax by DeepMind

Distrax is a lightweight library of probability distributions and bijectors. It acts as a JAX-native re-implementation of a subset of TensorFlow Probability (TFP), with some new features and emphasis on extensibility.
2020-03-23 - jax-flows by Chris Waites

Another library that has normalizing flows using JAX as the backend. Has some of the SOTA methods.
2020-03-09 - NuX by Information Fusion Labs (UMass)

A library that offers normalizing flows using JAX as the backend. Has some SOTA methods. They also feature a surjective flow via quantization.

Julia Packages

2020-02-07 - InvertibleNetworks.jl by SLIM

A Flux compatible library implementing invertible neural networks and normalizing flows using memory-efficient backpropagation. Uses manually implemented gradients to take advantage of the invertibility of building blocks, which allows for scaling to large-scale problem sizes.

🧑‍💻 Code

PyTorch Code

2021-08-21 - NICE: Non-linear Independent Components Estimation by Maxime Vandegar

PyTorch implementation that reproduces results from the paper NICE in about 100 lines of code.
2020-07-19 - Normalizing Flows - Introduction (Part 1) by pyro.ai

A tutorial about how to use the pyro-ppl library (based on PyTorch) to use Normalizing flows. They provide some SOTA methods including NSF and MAF. Parts 2 and 3 coming later.
2020-07-03 - Density Estimation with Neural ODEs and Density Estimation with FFJORDs by torchdyn

Example of how to use FFJORD as a continuous normalizing flow (CNF). Based on the PyTorch suite torchdyn which offers continuous neural architectures.
2019-12-09 - pytorch-normalizing-flows by Andrej Karpathy

A Jupyter notebook with PyTorch implementations of the most commonly used flows: NICE, RNVP, MAF, Glow, NSF.
2019-02-06 - pytorch_flows by acids-ircam

A great repo with some basic PyTorch implementations of normalizing flows from scratch.
2018-12-30 - normalizing_flows by Kamen Bliznashki

Pytorch implementations of density estimation algorithms: BNAF, Glow, MAF, RealNVP, planar flows.
2018-09-01 - pytorch-flows by Ilya Kostrikov

PyTorch implementations of density estimation algorithms: MAF, RNVP, Glow.

TensorFlow Code

2020-11-02 - Variational Inference using Normalizing Flows (VINF) by Pierre Segonne

This repository provides a hands-on TensorFlow implementation of Normalizing Flows as presented in the paper introducing the concept (D. Rezende & S. Mohamed).
2019-07-19 - BERT-flow by Bohan Li

TensorFlow implementation of "On the Sentence Embeddings from Pre-trained Language Models" (EMNLP 2020).

JAX Code

2020-06-12 - Neural Transport by numpyro

Features an example of how Normalizing flows can be used to get more robust posteriors from Monte Carlo methods. Uses the numpyro library which is a PPL with JAX as the backend. The NF implementations include the basic ones like IAF and BNAF.

Others Code

2018-06-11 - destructive-deep-learning by David Inouye
Code base for the paper Deep Density Destructors by Inouye & Ravikumar (2018). An entire suite of iterative methods including tree-based as well as Gaussianization methods which are similar to normalizing flows except they converge iteratively instead of fully parametrized. That is, they still use bijective transforms, compute the Jacobian, check the likelihood and you can still sample and get probability density estimates. The only difference is you repeat the following two steps until convergence:
1. compute one layer or block layer (e.g. Marginal Gaussianization + PCA rotation)
2. check for convergence (e.g log-likelihood using the change-of-variables formula)
Table 1 in the paper has a good comparison with traditional NFs.
2017-07-11 - Normalizing Flows Overview by PyMC3

A very helpful notebook showcasing how to work with flows in practice and comparing it to PyMC3's NUTS-based HMC kernel. Based on Theano.
2017-03-21 - NormFlows by Andy Miller

Simple didactic example using autograd, so pretty low-level.

🌐 Blog Posts

2018-10-21 - Change of Variables for Normalizing Flows by Neal Jean

Short and simple explanation of change of variables theorem i.t.o. probability mass conservation.
2018-10-13 - Flow-based Deep Generative Models by Lilian Weng

Covers change of variables, NICE, RNVP, MADE, Glow, MAF, IAF, WaveNet, PixelRNN.
2018-04-03 - Normalizing Flows by Adam Kosiorek

Introduction to flows covering change of variables, planar flow, radial flow, RNVP and autoregressive flows like MAF, IAF and Parallel WaveNet.
2018-01-17 - Normalizing Flows Tutorial by Eric Jang

Part 1: Distributions and Determinants. Part 2: Modern Normalizing Flows. Lots of great graphics.

🎉 Open to Suggestions

See something that's missing from this list? PRs welcome!

Papers should be peer-reviewed and published in a journal. If you're unsure if a paper or resource belongs in this list, feel free to open an issue or start a discussion. This repo is meant to be a community effort. Don't hesitate to voice an opinion.

Georgehappy1 / awesome-normalizing-flows