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{{Short description|Statistical model used in machine learning}}
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A '''flow-based generative model''' is a [[generative model]] used in [[machine learning]] that explicitly models a [[probability distribution]] by leveraging '''normalizing flow''',<ref>{{cite journal |last1=Tabak |first1=Esteban G. |last2=Vanden-Eijnden |first2=Eric |title=Density estimation by dual ascent of the log-likelihood |journal=Communications in Mathematical Sciences |date=2010 |volume=8 |issue=1 |pages=217–233 |doi=10.4310/CMS.2010.v8.n1.a11 |url=https://projecteuclid.org/journals/communications-in-mathematical-sciences/volume-8/issue-1/Density-estimation-by-dual-ascent-of-the-log-likelihood/cms/1266935020.full}}</ref><ref>{{cite journal |last1=Tabak |first1=Esteban G. |last2=Turner |first2=Cristina V. |title=A family of nonparametric density estimation algorithms |journal=Communications on Pure and Applied Mathematics |date=2012 |volume=66 |issue=2 |pages=145–164 |doi=10.1002/cpa.21423 |hdl=11336/8930 |s2cid=17820269 |url=https://onlinelibrary.wiley.com/doi/abs/10.1002/cpa.21423|hdl-access=free }}</ref><ref>{{cite journal |first1=George |last1=Papamakarios |first2=Eric |last2=Nalisnick |first3=Danilo |last3=Jimenez Rezende |first4=Shakir |last4=Mohamed |first5=Balaji |last5=Bakshminarayanan |year=2021 |title=Normalizing flows for probabilistic modeling and inference |journal=Journal of Machine Learning Research |volume=22 |issue=1 |pages=2617–2680 |arxiv=1912.02762 |url=https://dl.acm.org/doi/abs/10.5555/3546258.3546315 }}</ref> which is a statistical method using the [[Probability density function#Function of random variables and change of variables in the probability density function|change-of-variable]] law of probabilities to transform a simple distribution into a complex one.

The direct modeling of likelihood provides many advantages. For example, the negative log-likelihood can be directly computed and minimized as the [[loss function]]. Additionally, novel samples can be generated by sampling from the initial distribution, and applying the flow transformation.

In contrast, many alternative generative modeling methods such as [[Autoencoder#Variational autoencoder (VAE)|variational autoencoder (VAE)]] and [[generative adversarial network]] do not explicitly represent the likelihood function.

== Method ==

[[File:Normalizing-flow.svg|thumb|Scheme for normalizing flows]] 

Let <math>z_0</math> be a (possibly multivariate) [[random variable]] with distribution <math>p_0(z_0)</math>.

For <math>i = 1, ..., K</math>, let <math>z_i = f_i(z_{i-1})</math> be a sequence of random variables transformed from <math>z_0</math>. The functions <math>f_1, ..., f_K</math> should be invertible, i.e. the [[inverse function]] <math>f^{-1}_i</math> exists. The final output <math>z_K</math> models the target distribution.

The log likelihood of <math>z_K</math> is (see [[#Derivation of log likelihood|derivation]]):

: <math>\log p_K(z_K) = \log p_0(z_0) - \sum_{i=1}^{K} \log \left|\det \frac{df_i(z_{i-1})}{dz_{i-1}}\right|</math>

To efficiently compute the log likelihood, the functions <math>f_1, ..., f_K</math> should be 1. easy to invert, and 2. easy to compute the determinant of its Jacobian. In practice, the functions <math>f_1, ..., f_K</math> are modeled using [[Deep learning|deep neural networks]], and are trained to minimize the negative log-likelihood of data samples from the target distribution. These architectures are usually designed such that only the forward pass of the neural network is required in both the inverse and the Jacobian determinant calculations. Examples of such architectures include NICE,<ref name=":1">{{cite arXiv | eprint=1410.8516| last1=Dinh| first1=Laurent| last2=Krueger| first2=David| last3=Bengio| first3=Yoshua| title=NICE: Non-linear Independent Components Estimation| year=2014| class=cs.LG}}</ref> RealNVP,<ref name=":2">{{cite arXiv | eprint=1605.08803| last1=Dinh| first1=Laurent| last2=Sohl-Dickstein| first2=Jascha| last3=Bengio| first3=Samy| title=Density estimation using Real NVP| year=2016| class=cs.LG}}</ref> and Glow.<ref name="glow">{{cite arXiv | eprint=1807.03039| last1=Kingma| first1=Diederik P.| last2=Dhariwal| first2=Prafulla| title=Glow: Generative Flow with Invertible 1x1 Convolutions| year=2018| class=stat.ML}}</ref>

=== Derivation of log likelihood ===

Consider <math>z_1</math> and <math>z_0</math>. Note that <math>z_0 = f^{-1}_1(z_1)</math>.

By the [[Probability density function#Function of random variables and change of variables in the probability density function|change of variable]] formula, the distribution of <math>z_1</math> is:

: <math>p_1(z_1) = p_0(z_0)\left|\det \frac{df_1^{-1}(z_1)}{dz_1}\right|</math>

Where <math>\det \frac{df_1^{-1}(z_1)}{dz_1}</math> is the [[determinant]] of the [[Jacobian matrix and determinant|Jacobian matrix]] of <math>f^{-1}_1</math>.

By the [[inverse function theorem]]:

: <math>p_1(z_1) = p_0(z_0)\left|\det \left(\frac{df_1(z_0)}{dz_0}\right)^{-1}\right|</math>

By the identity <math>\det(A^{-1}) = \det(A)^{-1}</math> (where <math>A</math> is an [[invertible matrix]]), we have:

: <math>p_1(z_1) = p_0(z_0)\left|\det \frac{df_1(z_0)}{dz_0}\right|^{-1}</math>

The log likelihood is thus:

: <math>\log p_1(z_1) = \log p_0(z_0) - \log \left|\det \frac{df_1(z_0)}{dz_0}\right|</math>

In general, the above applies to any <math>z_i</math> and <math>z_{i-1}</math>. Since <math>\log p_i(z_i)</math> is equal to <math>\log p_{i-1}(z_{i-1})</math> subtracted by a non-recursive term, we can infer by [[Mathematical induction|induction]] that:

: <math>\log p_K(z_K) = \log p_0(z_0) - \sum_{i=1}^{K} \log \left|\det \frac{df_i(z_{i-1})}{dz_{i-1}}\right|</math>

== Training method ==
As is generally done when training a deep learning model, the goal with normalizing flows is to minimize the [[Kullback–Leibler divergence]] between the model's likelihood and the target distribution to be estimated. Denoting <math>p_\theta</math> the model's likelihood and <math>p^*</math> the target distribution to learn, the (forward) KL-divergence is:

: <math>D_{KL}[p^{*}(x)||p_{\theta}(x)] = -\mathbb{E}_{p^{*}(x)}[\log(p_{\theta}(x))] + \mathbb{E}_{p^{*}(x)}[\log(p^{*}(x))]</math>

The second term on the right-hand side of the equation corresponds to the entropy of the target distribution and is independent of the parameter <math>\theta</math> we want the model to learn, which only leaves the expectation of the negative log-likelihood to minimize under the target distribution. This intractable term can be approximated with a Monte-Carlo method by [[importance sampling]]. Indeed, if we have a dataset <math>\{x_{i}\}_{i=1:N}</math> of samples each independently drawn from the target distribution <math>p^*(x)</math>, then this term can be estimated as:

: <math>-\hat{\mathbb{E}}_{p^{*}(x)}[\log(p_{\theta}(x))] = -\frac{1}{N} \sum_{i=0}^{N} \log(p_{\theta}(x_{i})) </math>

Therefore, the learning objective

: <math>\underset{\theta}{\operatorname{arg\,min}}\ D_{KL}[p^{*}(x)||p_{\theta}(x)]</math>
is replaced by 
: <math>\underset{\theta}{\operatorname{arg\,max}}\ \sum_{i=0}^{N} \log(p_{\theta}(x_{i}))</math>

In other words, minimizing the [[Kullback–Leibler divergence]] between the model's likelihood and the target distribution is equivalent to [[Maximum likelihood estimation|maximizing the model likelihood]] under observed samples of the target distribution.<ref>{{Cite journal |last1=Papamakarios |first1=George |last2=Nalisnick |first2=Eric |last3=Rezende |first3=Danilo Jimenez |last4=Shakir |first4=Mohamed |last5=Balaji |first5=Lakshminarayanan |date=March 2021 |title=Normalizing Flows for Probabilistic Modeling and Inference |journal=Journal of Machine Learning Research |url=https://jmlr.org/papers/v22/19-1028.html |volume=22 |issue=57 |pages=1–64 |arxiv=1912.02762}}</ref>

A pseudocode for training normalizing flows is as follows:<ref>{{Cite journal |last1=Kobyzev |first1=Ivan |last2=Prince |first2=Simon J.D. |last3=Brubaker |first3=Marcus A. |date=November 2021 |title=Normalizing Flows: An Introduction and Review of Current Methods |url=https://ieeexplore.ieee.org/document/9089305 |journal=IEEE Transactions on Pattern Analysis and Machine Intelligence |volume=43 |issue=11 |pages=3964–3979 |doi=10.1109/TPAMI.2020.2992934 |pmid=32396070 |arxiv=1908.09257 |s2cid=208910764 |issn=1939-3539}}</ref>

* INPUT. dataset <math>x_{1:n}</math>, normalizing flow model <math>f_\theta(\cdot), p_0 </math>.
* SOLVE. <math>\max_\theta \sum_j \ln p_\theta(x_j)</math> by gradient descent
* RETURN. <math>\hat\theta</math>

== Variants ==

=== Planar Flow ===
The earliest example.<ref name=":0">{{cite arXiv | eprint=1505.05770| author1=Danilo Jimenez Rezende| last2=Mohamed| first2=Shakir| title=Variational Inference with Normalizing Flows| year=2015| class=stat.ML}}</ref> Fix some activation function <math>h</math>, and let <math>\theta = (u, w, b)</math> with the appropriate dimensions, then<math display="block">x = f_\theta(z) = z + u h(\langle w, z \rangle + b)</math>The inverse <math>f_\theta^{-1}</math> has no closed-form solution in general.

The Jacobian is <math>|\det (I + h'(\langle w, z \rangle + b) uw^T)| = |1 + h'(\langle w, z \rangle + b) \langle u, w\rangle|</math>.

For it to be invertible everywhere, it must be nonzero everywhere. For example, <math>h = \tanh</math> and <math>\langle u, w \rangle > -1</math> satisfies the requirement.

=== Nonlinear Independent Components Estimation (NICE) ===
Let <math>x, z\in \R^{2n}</math> be even-dimensional, and split them in the middle.<ref name=":1" /> Then the normalizing flow functions are<math display="block">x = \begin{bmatrix}
	  x_1 \\ x_2
	  \end{bmatrix}= f_\theta(z) = \begin{bmatrix}
	  z_1 \\z_2
	  \end{bmatrix} + \begin{bmatrix}
	  0 \\ m_\theta(z_1)
	  \end{bmatrix}</math>where <math>m_\theta</math> is any neural network with weights <math>\theta</math>.

<math>f_\theta^{-1}</math> is just <math>z_1 = x_1, z_2 = x_2 - m_\theta(x_1)</math>, and the Jacobian is just 1, that is, the flow is volume-preserving.

When <math>n=1</math>, this is seen as a curvy shearing along the <math>x_2</math> direction.

=== Real Non-Volume Preserving (Real NVP) ===
The Real Non-Volume Preserving model generalizes NICE model by:<ref name=":2" /><math display="block">x = \begin{bmatrix}
	  x_1 \\ x_2
	  \end{bmatrix}= f_\theta(z) = \begin{bmatrix}
	  z_1 \\ e^{s_\theta(z_1)} \odot z_2
	  \end{bmatrix} + \begin{bmatrix}
	  0 \\ m_\theta(z_1)
	  \end{bmatrix}</math>

Its inverse is <math>z_1 = x_1, z_2 = e^{-s_\theta (x_1)}\odot (x_2 - m_\theta (x_1))</math>, and its Jacobian is <math>\prod^n_{i=1} e^{s_\theta(z_{1, })}</math>. The NICE model is recovered by setting <math>s_\theta = 0</math>.
Since the Real NVP map keeps the first and second halves of the vector <math>x</math> separate, it's usually required to add a permutation <math>(x_1, x_2) \mapsto (x_2, x_1)</math> after every Real NVP layer.

=== Generative Flow (Glow) ===
In generative flow model,<ref name="glow" /> each layer has 3 parts:

* channel-wise affine transform<math display="block">y_{cij} = s_c(x_{cij} + b_c)</math>with Jacobian <math>\prod_c s_c^{HW}</math>.
* invertible 1x1 convolution<math display="block">z_{cij} = \sum_{c'} K_{cc'} y_{cij}</math>with Jacobian <math>\det(K)^{HW}</math>. Here <math>K</math> is any invertible matrix.
* Real NVP, with Jacobian as described in Real NVP.

The idea of using the invertible 1x1 convolution is to permute all layers in general, instead of merely permuting the first and second half, as in Real NVP.

=== Masked autoregressive flow (MAF) ===
An autoregressive model of a distribution on <math>\R^n</math> is defined as the following stochastic process:<ref>{{Cite journal |last1=Papamakarios |first1=George |last2=Pavlakou |first2=Theo |last3=Murray |first3=Iain |date=2017 |title=Masked Autoregressive Flow for Density Estimation |url=https://proceedings.neurips.cc/paper/2017/hash/6c1da886822c67822bcf3679d04369fa-Abstract.html |journal=Advances in Neural Information Processing Systems |publisher=Curran Associates, Inc. |volume=30|arxiv=1705.07057 }}</ref>

<math display="block">\begin{align}
		  x_1 \sim& N(\mu_1, \sigma_1^2)\\
		  x_2 \sim& N(\mu_2(x_1), \sigma_2(x_1)^2)\\
		  &\cdots \\
		  x_n \sim& N(\mu_n(x_{1:n-1}), \sigma_n(x_{1:n-1})^2)\\
\end{align}</math>where <math>\mu_i: \R^{i-1} \to \R</math> and <math>\sigma_i: \R^{i-1} \to (0, \infty)</math> are fixed functions that define the autoregressive model.

By the [[reparameterization trick]], the autoregressive model is generalized to a normalizing flow:<math display="block">\begin{align}
		  x_1 =& \mu_1 + \sigma_1 z_1\\
		  x_2 =& \mu_2(x_1) + \sigma_2(x_1) z_2\\
		  &\cdots \\
		  x_n =& \mu_n(x_{1:n-1}) + \sigma_n(x_{1:n-1}) z_n\\
\end{align}</math>The autoregressive model is recovered by setting <math>z \sim N(0, I_{n})</math>.

The forward mapping is slow (because it's sequential), but the backward mapping is fast (because it's parallel).

The Jacobian matrix is lower-diagonal, so the Jacobian is <math>\sigma_1 \sigma_2(x_1)\cdots \sigma_n(x_{1:n-1})</math>.

Reversing the two maps <math>f_\theta</math> and <math>f_\theta^{-1}</math> of MAF results in Inverse Autoregressive Flow (IAF), which has fast forward mapping and slow backward mapping.<ref>{{Cite journal |last1=Kingma |first1=Durk P |last2=Salimans |first2=Tim |last3=Jozefowicz |first3=Rafal |last4=Chen |first4=Xi |last5=Sutskever |first5=Ilya |last6=Welling |first6=Max |date=2016 |title=Improved Variational Inference with Inverse Autoregressive Flow |url=https://proceedings.neurips.cc/paper/2016/hash/ddeebdeefdb7e7e7a697e1c3e3d8ef54-Abstract.html |journal=Advances in Neural Information Processing Systems |publisher=Curran Associates, Inc. |volume=29|arxiv=1606.04934 }}</ref>

=== Continuous Normalizing Flow (CNF) ===

Instead of constructing flow by function composition, another approach is to formulate the flow as a continuous-time dynamic.<ref name="ffjord">{{cite arXiv | eprint=1810.01367| last1=Grathwohl| first1=Will| last2=Chen| first2=Ricky T. Q.| last3=Bettencourt| first3=Jesse| last4=Sutskever| first4=Ilya| last5=Duvenaud| first5=David| title=FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models| year=2018| class=cs.LG}}</ref><ref>{{Cite arXiv |last1=Lipman |first1=Yaron |last2=Chen |first2=Ricky T. Q. |last3=Ben-Hamu |first3=Heli |last4=Nickel |first4=Maximilian |last5=Le |first5=Matt |date=2022-10-01 |title=Flow Matching for Generative Modeling |class=cs.LG | eprint=2210.02747}}</ref> Let <math>z_0</math> be the latent variable with distribution <math>p(z_0)</math>. Map this latent variable to data space with the following flow function:

: <math>x = F(z_0) = z_T = z_0 + \int_0^T f(z_t, t) dt</math>

where <math>f</math> is an arbitrary function and can be modeled with e.g. neural networks.

The inverse function is then naturally:<ref name="ffjord" />

: <math>z_0 = F^{-1}(x) = z_T + \int_T^0 f(z_t, t) dt = z_T - \int_0^T f(z_t,t) dt </math>

And the log-likelihood of <math>x</math> can be found as:<ref name="ffjord" />

: <math>\log(p(x)) = \log(p(z_0)) - \int_0^T \text{Tr}\left[\frac{\partial f}{\partial z_t} \right] dt</math>

Since the trace depends only on the diagonal of the Jacobian <math>\partial_{z_t} f</math>, this allows "free-form" Jacobian.<ref>{{cite arXiv |last1=Grathwohl |first1=Will |last2=Chen |first2=Ricky T. Q. |last3=Bettencourt |first3=Jesse |last4=Sutskever |first4=Ilya |last5=Duvenaud |first5=David |date=2018-10-22 |title=FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models |class=cs.LG |eprint=1810.01367 }}</ref> Here, "free-form" means that there is no restriction on the Jacobian's form. It is contrasted with previous discrete models of normalizing flow, where the Jacobian is carefully designed to be only upper- or lower-diagonal, so that the Jacobian can be evaluated efficiently.

The trace can be estimated by "Hutchinson's trick":<ref name="Finlay 3154–3164">{{Cite journal |last1=Finlay |first1=Chris |last2=Jacobsen |first2=Joern-Henrik |last3=Nurbekyan |first3=Levon |last4=Oberman |first4=Adam |date=2020-11-21 |title=How to Train Your Neural ODE: the World of Jacobian and Kinetic Regularization |url=https://proceedings.mlr.press/v119/finlay20a.html |journal=International Conference on Machine Learning |language=en |publisher=PMLR |pages=3154–3164|arxiv=2002.02798 }}</ref><ref>{{Cite journal |last=Hutchinson |first=M.F. |date=January 1989 |title=A Stochastic Estimator of the Trace of the Influence Matrix for Laplacian Smoothing Splines |url=http://www.tandfonline.com/doi/abs/10.1080/03610918908812806 |journal=Communications in Statistics - Simulation and Computation |language=en |volume=18 |issue=3 |pages=1059–1076 |doi=10.1080/03610918908812806 |issn=0361-0918}}</ref><blockquote>Given any matrix <math>W\in \R^{n\times n}</math>, and any random <math>u\in \R^n</math> with <math>E[uu^T] = I</math>, we have <math>E[u^T W u] = tr(W)</math>. (Proof: expand the expectation directly.)</blockquote>Usually, the random vector is sampled from <math>N(0, I)</math> (normal distribution) or <math>\{\pm n^{-1/2}\}^n</math> ([[Rademacher distribution|Radamacher distribution]]).

When <math>f</math> is implemented as a neural network, [[neural ODE]] methods<ref>{{cite conference |last1=Chen |first1=Ricky T. Q. |last2=Rubanova |first2=Yulia |last3=Bettencourt |first3=Jesse |last4=Duvenaud |first4=David K. |year=2018 |editor1-last=Bengio |editor1-first=S. |editor2-last=Wallach |editor2-first=H. |editor3-last=Larochelle |editor3-first=H. |editor4-last=Grauman |editor4-first=K. |editor5-last=Cesa-Bianchi |editor5-first=N. |editor6-last=Garnett |editor6-first=R. |title=Neural Ordinary Differential Equations |url=https://proceedings.neurips.cc/paper_files/paper/2018/file/69386f6bb1dfed68692a24c8686939b9-Paper.pdf |conference= |publisher=Curran Associates, Inc. |volume=31 |arxiv=1806.07366 |book-title=Advances in Neural Information Processing Systems}}</ref> would be needed. Indeed, CNF was first proposed in the same paper that proposed neural ODE.

There are two main deficiencies of CNF, one is that a continuous flow must be a [[homeomorphism]], thus preserve orientation and [[ambient isotopy]] (for example, it's impossible to flip a left-hand to a right-hand by continuous deforming of space, and it's impossible to [[Sphere eversion|turn a sphere inside out]], or undo a knot), and the other is that the learned flow <math>f</math> might be ill-behaved, due to degeneracy (that is, there are an infinite number of possible <math>f</math> that all solve the same problem).

By adding extra dimensions, the CNF gains enough freedom to reverse orientation and go beyond ambient isotopy (just like how one can pick up a polygon from a desk and flip it around in 3-space, or unknot a knot in 4-space), yielding the "augmented neural ODE".<ref>{{Cite journal |last1=Dupont |first1=Emilien |last2=Doucet |first2=Arnaud |last3=Teh |first3=Yee Whye |date=2019 |title=Augmented Neural ODEs |url=https://proceedings.neurips.cc/paper/2019/hash/21be9a4bd4f81549a9d1d241981cec3c-Abstract.html |journal=Advances in Neural Information Processing Systems |publisher=Curran Associates, Inc. |volume=32}}</ref>

Any homeomorphism of <math>\R^n</math> can be approximated by a neural ODE operating on <math>\R^{2n+1}</math>, proved by combining [[Whitney embedding theorem]] for manifolds and the [[universal approximation theorem]] for neural networks.<ref>{{cite arXiv |last1=Zhang |first1=Han |last2=Gao |first2=Xi |last3=Unterman |first3=Jacob |last4=Arodz |first4=Tom |date=2019-07-30 |title=Approximation Capabilities of Neural ODEs and Invertible Residual Networks |class=cs.LG |language=en |eprint=1907.12998}}</ref>

To regularize the flow <math>f</math>, one can impose regularization losses. The paper <ref name="Finlay 3154–3164"/> proposed the following regularization loss based on [[Optimal transport|optimal transport theory]]:<math display="block">\lambda_{K} \int_{0}^{T}\left\|f(z_t, t)\right\|^{2} dt
+\lambda_{J} \int_{0}^{T}\left\|\nabla_z f(z_t, t)\right\|_F^{2} dt
</math>where <math>\lambda_K, \lambda_J >0
</math> are hyperparameters. The first term punishes the model for oscillating the flow field over time, and the second term punishes it for oscillating the flow field over space. Both terms together guide the model into a flow that is smooth (not "bumpy") over space and time.

== Downsides ==
Despite normalizing flows success in estimating high-dimensional densities, some downsides still exist in their designs. First of all, their latent space where input data is projected onto is not a lower-dimensional space and therefore, flow-based models do not allow for compression of data by default and require a lot of computation. However, it is still possible to perform image compression with them.<ref name="Lossy Image Compression with Normal">{{cite arXiv | eprint=2008.10486| last1=Helminger| first1=Leonhard| last2=Djelouah| first2=Abdelaziz| last3=Gross| first3=Markus| last4=Schroers| first4=Christopher| title=Lossy Image Compression with Normalizing Flows| year=2020| class=cs.CV}}</ref>

Flow-based models are also notorious for failing in estimating the likelihood of out-of-distribution samples (i.e.: samples that were not drawn from the same distribution as the training set).<ref>{{cite arXiv | eprint=1810.09136v3| last1=Nalisnick| first1=Eric| last2=Matsukawa| first2=Teh| last3=Zhao| first3=Yee Whye| last4=Song| first4=Zhao| title=Do Deep Generative Models Know What They Don't Know?| year=2018| class=stat.ML}}</ref> Some hypotheses were formulated to explain this phenomenon, among which the [[Typical Set|typical set]] hypothesis,<ref>{{cite arXiv | eprint=1906.02994| last1=Nalisnick| first1=Eric| last2=Matsukawa| first2=Teh| last3=Zhao| first3=Yee Whye| last4=Song| first4=Zhao| title=Detecting Out-of-Distribution Inputs to Deep Generative Models Using Typicality| year=2019| class=stat.ML}}</ref> estimation issues when training models,<ref>{{Cite journal |last1=Zhang |first1=Lily |last2=Goldstein |first2=Mark |last3=Ranganath |first3=Rajesh |date=2021 |title=Understanding Failures in Out-of-Distribution Detection with Deep Generative Models |journal=Proceedings of Machine Learning Research |url=https://proceedings.mlr.press/v139/zhang21g.html |volume=139 |pages=12427–12436|pmid=35860036 |pmc=9295254 }}</ref> or fundamental issues due to the entropy of the data distributions.<ref>{{Cite arXiv|last1=Caterini |first1=Anthony L. |last2=Loaiza-Ganem |first2=Gabriel |date=2022 |title=Entropic Issues in Likelihood-Based OOD Detection |pages=21–26|class=stat.ML |eprint=2109.10794 }}</ref>

One of the most interesting properties of normalizing flows is the [[Inverse function|invertibility]] of their learned [[Bijection|bijective]] map. This property is given by constraints in the design of the models (cf.: RealNVP, Glow) which guarantee theoretical invertibility. The integrity of the inverse is important in order to ensure the applicability of the [[Probability density function#Function of random variables and change of variables in the probability density function|change-of-variable theorem]], the computation of the [[Jacobian matrix and determinant|Jacobian]] of the map as well as sampling with the model. However, in practice this invertibility is violated and the inverse map explodes because of numerical imprecision.<ref>{{cite arXiv | eprint=2006.09347| last1=Behrmann| first1=Jens| last2=Vicol| first2=Paul| last3=Wang| first3=Kuan-Chieh| last4=Grosse| first4=Roger| last5=Jacobsen| first5=Jörn-Henrik| title=Understanding and Mitigating Exploding Inverses in Invertible Neural Networks| year=2020| class=cs.LG}}</ref>

== Applications ==

Flow-based generative models have been applied on a variety of modeling tasks, including:
* Audio generation<ref>{{cite arXiv | eprint=1912.01219| last1=Ping| first1=Wei| last2=Peng| first2=Kainan| last3=Gorur| first3=Dilan| last4=Lakshminarayanan| first4=Balaji| title=WaveFlow: A Compact Flow-based Model for Raw Audio| year=2019| class=cs.SD}}</ref>
* Image generation<ref name="glow" />
* Molecular graph generation<ref>{{cite arXiv | eprint=2001.09382| last1=Shi| first1=Chence| last2=Xu| first2=Minkai| last3=Zhu| first3=Zhaocheng| last4=Zhang| first4=Weinan| last5=Zhang| first5=Ming| last6=Tang| first6=Jian| title=GraphAF: A Flow-based Autoregressive Model for Molecular Graph Generation| year=2020| class=cs.LG}}</ref>
* Point-cloud modeling<ref>{{cite arXiv | eprint=1906.12320| last1=Yang| first1=Guandao| last2=Huang| first2=Xun| last3=Hao| first3=Zekun| last4=Liu| first4=Ming-Yu| last5=Belongie| first5=Serge| last6=Hariharan| first6=Bharath| title=PointFlow: 3D Point Cloud Generation with Continuous Normalizing Flows| year=2019| class=cs.CV}}</ref>
* Video generation<ref>{{cite arXiv | eprint=1903.01434| last1=Kumar| first1=Manoj| last2=Babaeizadeh| first2=Mohammad| last3=Erhan| first3=Dumitru| last4=Finn| first4=Chelsea| last5=Levine| first5=Sergey| last6=Dinh| first6=Laurent| last7=Kingma| first7=Durk| title=VideoFlow: A Conditional Flow-Based Model for Stochastic Video Generation| year=2019| class=cs.CV}}</ref>
* Lossy image compression<ref name="Lossy Image Compression with Normal"/>
* Anomaly detection<ref>{{cite arXiv | eprint=2008.12577| last1=Rudolph| first1=Marco| last2=Wandt| first2=Bastian| last3=Rosenhahn| first3=Bodo| title=Same Same But DifferNet: Semi-Supervised Defect Detection with Normalizing Flows| year=2021| class=cs.CV}}</ref>

== References ==
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{{reflist}}

== External links ==
* [https://lilianweng.github.io/lil-log/2018/10/13/flow-based-deep-generative-models.html Flow-based Deep Generative Models]
* [https://deepgenerativemodels.github.io/notes/flow/ Normalizing flow models]

[[Category:Machine learning]]
[[Category:Statistical models]]
[[Category:Probabilistic models]]