A list of awesome resources on 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.}
Normalizing flows in the form of Gaussianization in an iterative format. Also shows connections to information theory.
Introduces the additive coupling layer (NICE) and shows how to use it for image generation and inpainting.
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.}
They show how to go beyond mean-field variational inference by using flows to increase the flexibility of the variational family.
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.
Jun 15, 2016 - Improving Variational Inference with Inverse Autoregressive Flow by Diederik Kingma et al.
Mar 6, 2017 - Multiplicative Normalizing Flows for Variational Bayesian Neural Networks by Christos Louizos, Max 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.
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.
Introduces Sylvester normalizing flows which remove the single-unit bottleneck from planar flows for increased flexibility in the variational posterior.
April 3, 2018 - Neural Autoregressive Flows by Chin-Wei Huang, David Krueger, Alexandre Lacoste, Aaron Courville.
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. (Author's Code)
July 3, 2018 - Deep Density Destructors by Inouye & Ravikumar
Normalizing flows but from an iterative perspective. Features a Tree-based density estimator.
Jul 9, 2018 - 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.
Oct 2, 2018 - 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)
Nov 6, 2018 - FloWaveNet : A Generative Flow for Raw Audio by Kim et. al.
A flow-based generative model for raw audo synthesis. (Author's Code)
Apr 9, 2019 - Block Neural Autoregressive Flow - De Cao et. al.
May 17, 2019 - Integer Discrete Flows and Lossless Compression by Hoogeboom et. al.
A normalizing flow to be used for ordinal discrete data. They introduce a flexible transformation layer called integer discrete coupling.
May 30, 2019 - Graph Normalizing Flows by Jenny Liu 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.
Jul 21, 2019 - Noise Regularization for Conditional Density Estimation by Rothfuss et. al.
Normalizing flows for conditional density estimation. This paper proposes noise regularization to reduce overfitting. (Blog | )
Aug 25, 2019 - Normalizing Flows: An Introduction and Review of Current Methods by Kobyzev 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.
Jun 10, 2019 - Neural Spline Flows by Conor Durkan et. al.
Uses monotonic ration splines as a coupling layer. This is currently one of the state of the art.
Dec 5, 2019 - Normalizing Flows for Probabilistic Modeling and Inference by Papamakarios et al.
A thorough and very readable review article by some of the guys at DeepMind involved in the development of flows. Highly recommended.
Jan 15, 2020 - Invertible Generative Modeling using Linear Rational Splines by Dolatabadi et. al.
A successor to the Neural spline flows which features an easy-to-compute inverse.
Jan 17, 2020 - Training Normalizing Flows with the Information Bottleneck for Competitive Generative Classification by Ardizzone et. al.
They introduce a class of conditional normalizing flows with an information bottleneck objective. (Author's Code)
Feb 16, 2020 - Stochastic Normalizing Flows by Hao Wu, Jonas Köhler, Frank Noé.
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).
Feb 21, 2020 - Stochastic Normalizing Flows by Liam Hodgkinson, Chris van der Heide, Fred Roosta, Michael W. Mahoney.
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.
Feb 24, 2020 - Modeling Continuous Stochastic Processes with Dynamic Normalizing Flows by Deng et. al.
They propose a normalizing flow using differential deformation of the Wiener process. Applied to time series. (Tweet)
Feb 27, 2020 - 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. (Author's Code)
Mar 4, 2020 - Gaussianization Flows by Meng 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. (Author's Code)
Mar 31, 2020 - 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 | Author's Code)
April 8, 2020 - 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) (Author's Code)
Jun 3, 2020 - Equivariant Flows: exact likelihood generative learning for symmetric densities by Jonas Köhler, Leon Klein, Frank Noé.
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.
Jun 15, 2020 - Why Normalizing Flows Fail to Detect Out-of-Distribution Data by Kirichenko 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 | Author's Code)
July 6, 2020 - SurVAE Flows: Surjections to Bridge the Gap between VAEs and Flows by Didrik Nielsen 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 mapsx ~ 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 | Author's Code)
July 15, 2020 - AdvFlow: Inconspicuous Black-box Adversarial Attacks using Normalizing Flows by Dolatabadi et. al.
An adversarial attack method on image classifiers that use normalizing flows. (Author's Code)
Sept 21, 2020 - Haar Wavelet based Block Autoregressive Flows for Trajectories by Bhattacharyya et. al.
Introduce a Haar wavelet-based block autoregressive model.
Normalizing flows for inverse problems.
Uses normalizing flows in conjunction with Monte Carlo estimation to have more expressive distributions and better posterior estimation.
Uses normalizing flows for super-resolution.
Uses conditional normalizing flows for inverse problems. (Video)
Introduces Sylvester normalizing flows which remove the single-unit bottleneck from planar flows for increased flexibility in the variational posterior.
Mar 24, 2019 - PixelCNN, Wavenet, Normalizing Flows - Santiago Pascual - UPC Barcelona 2018
Sep 24, 2019 - Graph Normalizing Flows by Jenny Liu (University of Toronto, Vector Institute).
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.
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".
A great 3blue1brown-style video explaining the basics of normalizing flows.
A really thorough explanation of normalizing flows. Also includes some sample code.
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.
Part 1: Distributions and Determinants. Part 2: Modern Normalizing Flows. Lots of great graphics.
Apr 3, 2018 - Normalizing Flows by Adam Kosiorek.
Oct 13, 2018 - Flow-based Deep Generative Models by Lilian Weng.
FrEIAby 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.
nflowsby 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.
Large first-party library that offers RNVP, MAF among other autoregressive models plus a collection of composable bijectors.
NuXby 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.
Another library that has normalizing flows using JAX as the backend. Has some of the SOTA methods.
pytorch-flowsby Ilya Kostrikov.
PyTorch implementations of density estimation algorithms: MAF, RNVP, Glow.
normalizing_flowsby Kamen Bliznashki.
Pytorch implementations of density estimation algorithms: BNAF, Glow, MAF, RealNVP, planar flows.
pytorch_flowsby acids-ircam
A great repo with some basic PyTorch implementations of normalizing flows from scratch.
pytorch-normalizing-flowsby Andrej Karpathy.
A Jupyter notebook with PyTorch implementations of the most commonly used flows: NICE, RNVP, MAF, Glow, NSF.
Example of how to use FFJORD as a continuous normalizing flow (CNF). Based on the PyTorch suite
torchdynwhich offers continuous neural architectures.
Normalizing Flows - Introduction (Part 1)by pyro.ai
A tutorial about how to use the
pyro-ppllibrary (based on PyTorch) to use Normalizing flows. They provide some SOTA methods including NSF and MAF. Parts 2 and 3 coming later.
Features an example of how Normalizing flows can be used to get more robust posteriors from Monte Carlo methods. Uses the
numpyrolibrary which is a PPL with JAX as the backend. The NF implementations include the basic ones like IAF and BNAF.
autograd, so pretty low-level.
normalizing_flows_overview.ipynbby 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.
destructive-deep-learningby 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:
- compute one layer or block layer (e.g. Marginal Gaussianization + PCA rotation)
- 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.
See something that's missing from this list? PRs welcome!
If you're unsure if a paper or resource belongs in this list, feel free to open an issue and start a discussion. This repo is meant to be a community effort. So don't hesitate to voice an opinion.