Normalizing Flows

Introduced by Rezende et al. in Variational Inference with Normalizing Flows

Normalizing Flows are a method for constructing complex distributions by transforming a probability density through a series of invertible mappings. By repeatedly applying the rule for change of variables, the initial density ‘flows’ through the sequence of invertible mappings. At the end of this sequence we obtain a valid probability distribution and hence this type of flow is referred to as a normalizing flow.

In the case of finite flows, the basic rule for the transformation of densities considers an invertible, smooth mapping $f : \mathbb{R}^{d} \rightarrow \mathbb{R}^{d}$ with inverse $f^{-1} = g$, i.e. the composition $g \cdot f\left(z\right) = z$. If we use this mapping to transform a random variable $z$ with distribution $q\left(z\right)$, the resulting random variable $z' = f\left(z\right)$ has a distribution:

$$ q\left(\mathbf{z}'\right) = q\left(\mathbf{z}\right)\bigl\vert{\text{det}}\frac{\delta{f}^{-1}}{\delta{\mathbf{z'}}}\bigr\vert = q\left(\mathbf{z}\right)\bigl\vert{\text{det}}\frac{\delta{f}}{\delta{\mathbf{z}}}\bigr\vert ^{-1} $$ ? where the last equality can be seen by applying the chain rule (inverse function theorem) and is a property of Jacobians of invertible functions. We can construct arbitrarily complex densities by composing several simple maps and successively applying the above equation. The density $q_{K}\left(\mathbf{z}\right)$ obtained by successively transforming a random variable $z_{0}$ with distribution $q_{0}$ through a chain of $K$ transformations $f_{k}$ is:

$$ z_{K} = f_{K} \cdot \dots \cdot f_{2} \cdot f_{1}\left(z_{0}\right) $$

$$ \ln{q}_{K}\left(z_{K}\right) = \ln{q}_{0}\left(z_{0}\right) − \sum^{K}_{k=1}\ln\vert\det\frac{\delta{f_{k}}}{\delta{\mathbf{z_{k-1}}}}\vert $$ ? The path traversed by the random variables $z_{k} = f_{k}\left(z_{k}−1\right)$ with initial distribution $q_{0}\left(z_{0}\right)$ is called the flow and the path formed by the successive distributions $q_{k}$ is a normalizing flow.

Source: Variational Inference with Normalizing Flows

Latest Papers

PAPER DATE
Blind Image Restoration with Flow Based Priors
Leonhard HelmingerMichael BernasconiAbdelaziz DjelouahMarkus GrossChristopher Schroers
2020-09-09
Variational Mixture of Normalizing Flows
Guilherme G. P. Freitas PiresMário A. T. Figueiredo
2020-09-01
Same Same But DifferNet: Semi-Supervised Defect Detection with Normalizing Flows
| Marco RudolphBastian WandtBodo Rosenhahn
2020-08-28
Lossy Image Compression with Normalizing Flows
Leonhard HelmingerAbdelaziz DjelouahMarkus GrossChristopher Schroers
2020-08-24
ClimAlign: Unsupervised statistical downscaling of climate variables via normalizing flows
Brian GroenkeLuke MadausClaire Monteleoni
2020-08-11
StyleFlow: Attribute-conditioned Exploration of StyleGAN-Generated Images using Conditional Continuous Normalizing Flows
Rameen AbdalPeihao ZhuNiloy MitraPeter Wonka
2020-08-06
Learning from a Complementary-label Source Domain: Theory and Algorithms
Yiyang ZhangFeng LiuZhen FangBo YuanGuangquan ZhangJie Lu
2020-08-04
Imitative Planning using Conditional Normalizing Flow
Shubhankar AgarwalHarshit SikchiCole GulinoEric Wilkinson
2020-07-31
Clarinet: A One-step Approach Towards Budget-friendly Unsupervised Domain Adaptation
| Yiyang ZhangFeng LiuZhen FangBo YuanGuangquan ZhangJie Lu
2020-07-29
SeismoGlow -- Data augmentation for the class imbalance problem
Ruy Luiz MilidiúLuis Felipe Müller
2020-07-23
Discrete Point Flow Networks for Efficient Point Cloud Generation
| Roman KlokovEdmond BoyerJakob Verbeek
2020-07-20
AdvFlow: Inconspicuous Black-box Adversarial Attacks using Normalizing Flows
| Hadi M. DolatabadiSarah ErfaniChristopher Leckie
2020-07-15
Faster Uncertainty Quantification for Inverse Problems with Conditional Normalizing Flows
Ali SiahkoohiGabrio RizzutiPhilipp A. WitteFelix J. Herrmann
2020-07-15
Projected Latent Markov Chain Monte Carlo: Conditional Inference with Normalizing Flows
Chris CannellaMohammadreza SoltaniVahid Tarokh
2020-07-13
Variational Inference with Continuously-Indexed Normalizing Flows
| Anthony CateriniRob CornishDino SejdinovicArnaud Doucet
2020-07-10
NVAE: A Deep Hierarchical Variational Autoencoder
Arash VahdatJan Kautz
2020-07-08
SurVAE Flows: Surjections to Bridge the Gap between VAEs and Flows
| Didrik NielsenPriyank JainiEmiel HoogeboomOle WintherMax Welling
2020-07-06
Sliced Iterative Generator
| Biwei DaiUros Seljak
2020-07-01
Mixture of Discrete Normalizing Flows for Variational Inference
| Tomasz KuśmierczykArto Klami
2020-06-28
Normalizing Flows Across Dimensions
Edmond CunninghamRenos ZabounidisAbhinav AgrawalIna FiterauDaniel Sheldon
2020-06-23
Riemannian Continuous Normalizing Flows
Emile MathieuMaximilian Nickel
2020-06-18
Advances in Black-Box VI: Normalizing Flows, Importance Weighting, and Optimization
Abhinav AgrawalDaniel SheldonJustin Domke
2020-06-18
Neural Manifold Ordinary Differential Equations
| Aaron LouDerek LimIsay KatsmanLeo HuangQingxuan JiangSer-Nam LimChristopher De Sa
2020-06-18
AI Feynman 2.0: Pareto-optimal symbolic regression exploiting graph modularity
| Silviu-Marian UdrescuAndrew TanJiahai FengOrisvaldo NetoTailin WuMax Tegmark
2020-06-18
Categorical Normalizing Flows via Continuous Transformations
| Phillip LippeEfstratios Gavves
2020-06-17
Flows Succeed Where GANs Fail: Lessons from Low-Dimensional Data
Tianci LiuJeffrey Regier
2020-06-17
MoFlow: An Invertible Flow Model for Generating Molecular Graphs
Chengxi ZangFei Wang
2020-06-17
Understanding and mitigating exploding inverses in invertible neural networks
| Jens BehrmannPaul VicolKuan-Chieh WangRoger GrosseJörn-Henrik Jacobsen
2020-06-16
Posterior Network: Uncertainty Estimation without OOD Samples via Density-Based Pseudo-Counts
Bertrand CharpentierDaniel ZügnerStephan Günnemann
2020-06-16
Density Deconvolution with Normalizing Flows
Tim DockhornJames A. RitchieYaoliang YuIain Murray
2020-06-16
Why Normalizing Flows Fail to Detect Out-of-Distribution Data
Polina KirichenkoPavel IzmailovAndrew Gordon Wilson
2020-06-15
Exponential Tilting of Generative Models: Improving Sample Quality by Training and Sampling from Latent Energy
Zhisheng XiaoQing YanYali Amit
2020-06-15
Ordering Dimensions with Nested Dropout Normalizing Flows
Artur BekasovIain Murray
2020-06-15
HyperFlow: Representing 3D Objects as Surfaces
Przemysław SpurekMaciej ZiębaJacek TaborTomasz Trzciński
2020-06-15
Neural Ordinary Differential Equations on Manifolds
Luca FalorsiPatrick Forré
2020-06-11
Learning normalizing flows from Entropy-Kantorovich potentials
Chris FinlayAugusto GerolinAdam M ObermanAram-Alexandre Pooladian
2020-06-10
Recurrent Flow Networks: A Recurrent Latent Variable Model for Spatio-Temporal Density Modelling
| Daniele GammelliFilipe Rodrigues
2020-06-09
SoftFlow: Probabilistic Framework for Normalizing Flow on Manifolds
Hyeongju KimHyeonseung LeeWoo Hyun KangJoun Yeop LeeNam Soo Kim
2020-06-08
Auto-decoding Graphs
Sohil Atul ShahVladlen Koltun
2020-06-04
Equivariant Flows: exact likelihood generative learning for symmetric densities
Jonas KöhlerLeon KleinFrank Noé
2020-06-03
Graphical Normalizing Flows
Antoine WehenkelGilles Louppe
2020-06-03
The Expressive Power of a Class of Normalizing Flow Models
Zhifeng KongKamalika Chaudhuri
2020-05-31
CLARINET: A RISC-V Based Framework for Posit Arithmetic Empiricism
Riya JainNiraj SharmaFarhad MerchantSachin PatkarRainer Leupers
2020-05-30
OT-Flow: Fast and Accurate Continuous Normalizing Flows via Optimal Transport
| Derek OnkenSamy Wu FungXingjian LiLars Ruthotto
2020-05-29
Discretize-Optimize vs. Optimize-Discretize for Time-Series Regression and Continuous Normalizing Flows
Derek OnkenLars Ruthotto
2020-05-27
Exhaustive Neural Importance Sampling applied to Monte Carlo event generation
Sebastian Pina-OteyFederico SánchezThorsten LuxVicens Gaitan
2020-05-26
A comparison of Vietnamese Statistical Parametric Speech Synthesis Systems
Huy Kinh PhanViet Lam PhungTuan Anh DinhBao Quoc Nguyen
2020-05-26
Style-Controllable Speech-Driven Gesture Synthesis Using Normalising Flows
| Simon AlexandersonGustav Eje HenterTaras KucherenkoJonas Beskow
2020-05-25
Constraining the Reionization History using Bayesian Normalizing Flows
Héctor J. HortúaLuigi MalagoRiccardo Volpi
2020-05-14
Anomaly Detection in Trajectory Data with Normalizing Flows
Madson L. D. DiasCésar Lincoln C. MattosTiciana L. C. da SilvaJosé Antônio F. de MacedoWellington C. P. Silva
2020-04-13
Variational Autoencoders with Normalizing Flow Decoders
Rogan MorrowWei-Chen Chiu
2020-04-12
Scaling Bayesian inference of mixed multinomial logit models to very large datasets
Filipe Rodrigues
2020-04-11
Stochastic Flows and Geometric Optimization on the Orthogonal Group
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2020-03-30
Modeling Contrary-to-Duty with CP-nets
Roberta CalegariAndrea LoreggiaEmiliano LoriniFrancesca RossiGiovanni Sartor
2020-03-23
Gaussianization Flows
Chenlin MengYang SongJiaming SongStefano Ermon
2020-03-04
Implicit Functions in Feature Space for 3D Shape Reconstruction and Completion
| Julian ChibaneThiemo AlldieckGerard Pons-Moll
2020-03-03
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Achille ThinNikita KotelevskiiJean-Stanislas DenainLeo GrinsztajnAlain DurmusMaxim PanovEric Moulines
2020-02-27
Gradient Boosted Normalizing Flows
Robert GiaquintoArindam Banerjee
2020-02-27
Emosaic: Visualizing Affective Content of Text at Varying Granularity
Philipp GeuderMarie Claire LeidingerMartin von LupinMarian DörkTobias Schröder
2020-02-24
Stochastic Normalizing Flows
Liam HodgkinsonChris van der HeideFred RoostaMichael W. Mahoney
2020-02-21
Schoenberg-Rao distances: Entropy-based and geometry-aware statistical Hilbert distances
| Gaëtan HadjeresFrank Nielsen
2020-02-19
Gravitational-wave parameter estimation with autoregressive neural network flows
Stephen R. GreenChristine SimpsonJonathan Gair
2020-02-18
Stochastic Normalizing Flows
Hao WuJonas KöhlerFrank Noé
2020-02-16
Latent Variable Modelling with Hyperbolic Normalizing Flows
| Avishek Joey BoseAriella SmofskyRenjie LiaoPrakash PanangadenWilliam L. Hamilton
2020-02-15
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2020-01-29
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2020-01-18
Training Normalizing Flows with the Information Bottleneck for Competitive Generative Classification
| Lynton ArdizzoneRadek MackowiakCarsten RotherUllrich Köthe
2020-01-17
SqueezeWave: Extremely Lightweight Vocoders for On-device Speech Synthesis
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i-flow: High-dimensional Integration and Sampling with Normalizing Flows
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2020-01-15
Disentanglement by Nonlinear ICA with General Incompressible-flow Networks (GIN)
Peter SorrensonCarsten RotherUllrich Köthe
2020-01-14
Relational State-Space Model for Stochastic Multi-Object Systems
Fan YangLing ChenFan ZhouYusong GaoWei Cao
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Self-Supervised Learning of Generative Spin-Glasses with Normalizing Flows
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InfoCNF: Efficient Conditional Continuous Normalizing Flow Using Adaptive Solvers
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Normalizing flows for deep anomaly detection
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Normalizing Flows: An Introduction and Review of Current Methods
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A Linear Systems Theory of Normalizing Flows
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Tails of Lipschitz Triangular Flows
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2019-07-10
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2019-07-09
Copula & Marginal Flows: Disentangling the Marginal from its Joint
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2019-07-07
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2018-02-17
Persistence Fisher Kernel: A Riemannian Manifold Kernel for Persistence Diagrams
| Tam LeMakoto Yamada
2018-02-10
Reparametrization of COM-Poisson Regression Models with Applications in the Analysis of Experimental Data
Eduardo E. Ribeiro JrWalmes M. ZevianiWagner H. BonatClarice G. B. DemétrioJohn Hinde
2018-01-29
Evaluation of generative networks through their data augmentation capacity
Timothée LesortFlorian BordesJean-Francois GoudouDavid Filliat
2018-01-01
A Spectral Approach to Generalization and Optimization in Neural Networks
Farzan FarniaJesse ZhangDavid Tse
2018-01-01
CNNs are Globally Optimal Given Multi-Layer Support
Chen HuangChen KongSimon Lucey
2017-12-07
On Frank-Wolfe and Equilibrium Computation
Jacob D. AbernethyJun-Kun Wang
2017-12-01
State Space LSTM Models with Particle MCMC Inference
Xun ZhengManzil ZaheerAmr AhmedYuan WangEric P XingAlexander J Smola
2017-11-30
Implicit Regularization in Nonconvex Statistical Estimation: Gradient Descent Converges Linearly for Phase Retrieval, Matrix Completion, and Blind Deconvolution
Cong MaKaizheng WangYuejie ChiYuxin Chen
2017-11-28
Proximal Alternating Direction Network: A Globally Converged Deep Unrolling Framework
Risheng LiuXin FanShichao ChengXiangyu WangZhongxuan Luo
2017-11-21
Wasserstein Auto-Encoders
| Ilya TolstikhinOlivier BousquetSylvain GellyBernhard Schoelkopf
2017-11-05
Analysis of planar ornament patterns via motif asymmetry assumption and local connections
Venera AdanovaSibel Tari
2017-10-12
Fast and Strong Convergence of Online Learning Algorithms
Zheng-Chu GuoLei Shi
2017-10-10
Generative Adversarial Mapping Networks
Jianbo GuoGuangxiang ZhuJian Li
2017-09-28
Order-Preserving Abstractive Summarization for Spoken Content Based on Connectionist Temporal Classification
Bo-Ru LuFrank ShyuYun-Nung ChenHung-Yi LeeLin-shan Lee
2017-09-16
Continuous-Time Flows for Efficient Inference and Density Estimation
Changyou ChenChunyuan LiLiqun ChenWenlin WangYunchen PuLawrence Carin
2017-09-04
Deep vs. Diverse Architectures for Classification Problems
Colleen M. Farrelly
2017-08-21
Geometric Enclosing Networks
Trung LeHung VuTu Dinh NguyenDinh Phung
2017-08-16
Follow the Moving Leader in Deep Learning
Shuai ZhengJames T. Kwok
2017-08-01
Restricted Eigenvalue from Stable Rank with Applications to Sparse Linear Regression
Shiva Prasad KasiviswanathanMark Rudelson
2017-07-25
Big Data Regression Using Tree Based Segmentation
Rajiv SambasivanSourish Das
2017-07-24
Adversarial Examples, Uncertainty, and Transfer Testing Robustness in Gaussian Process Hybrid Deep Networks
John BradshawAlexander G. de G. MatthewsZoubin Ghahramani
2017-07-08
Maintaining cooperation in complex social dilemmas using deep reinforcement learning
Adam LererAlexander Peysakhovich
2017-07-04
Convergence Analysis of Two-layer Neural Networks with ReLU Activation
Yuanzhi LiYang Yuan
2017-05-28
MUTAN: Multimodal Tucker Fusion for Visual Question Answering
| Hedi Ben-younesRémi CadeneMatthieu CordNicolas Thome
2017-05-18
Generalized Ideals and Co-Granular Rough Sets
A Mani
2017-04-18
Identification of Multiword Expressions for Latvian and Lithuanian: Hybrid Approach
MJustina ravickait{\.e}Tomas Krilavi{\v{c}}ius
2017-04-01
On architectural choices in deep learning: From network structure to gradient convergence and parameter estimation
Vamsi K IthapuSathya N RaviVikas Singh
2017-02-28
Wages of wins: could an amateur make money from match outcome predictions?
Albrecht Zimmermann
2017-02-17
Bayesian System Identification based on Hierarchical Sparse Bayesian Learning and Gibbs Sampling with Application to Structural Damage Assessment
Yong HuangJames L. BeckHui Li
2017-01-13
Factor-Adjusted Regularized Model Selection
Jianqing FanYuan KeKaizheng Wang
2016-12-27
Saliency Driven Image Manipulation
| Roey MechrezEli ShechtmanLihi Zelnik-Manor
2016-12-07
Improved Variational Inference with Inverse Autoregressive Flow
| Durk P. KingmaTim SalimansRafal JozefowiczXi ChenIlya SutskeverMax Welling
2016-12-01
Vectors or Graphs? On Differences of Representations for Distributional Semantic Models
Chris Biemann
2016-12-01
Normalizing Flows on Riemannian Manifolds
Mevlana C. GemiciDanilo RezendeShakir Mohamed
2016-11-07
Optimizing Codes for Source Separation in Color Image Demosaicing and Compressive Video Recovery
Alankar KotwalAjit Rajwade
2016-09-07
Hybrid Jacobian and Gauss-Seidel proximal block coordinate update methods for linearly constrained convex programming
Yangyang Xu
2016-08-13
Improving Variational Inference with Inverse Autoregressive Flow
| Diederik P. KingmaTim SalimansRafal JozefowiczXi ChenIlya SutskeverMax Welling
2016-06-15
Density estimation using Real NVP
| Laurent DinhJascha Sohl-DicksteinSamy Bengio
2016-05-27
Interaction pursuit in high-dimensional multi-response regression via distance correlation
Yinfei KongDaoji LiYingying FanJinchi Lv
2016-05-11
The constrained Dantzig selector with enhanced consistency
Yinfei KongZemin ZhengJinchi Lv
2016-05-11
Makeup like a superstar: Deep Localized Makeup Transfer Network
Si LiuXinyu OuRuihe QianWei WangXiaochun Cao
2016-04-25
A strengthening of rational closure in DLs: reasoning about multiple aspects
Valentina Gliozzi
2016-04-01
Identity Mappings in Deep Residual Networks
| Kaiming HeXiangyu ZhangShaoqing RenJian Sun
2016-03-16
Local-likelihood transformation kernel density estimation for positive random variables
Gery GeenensCraig Wang
2016-02-15
Universal Dependency Analysis
Hoang-Vu NguyenJilles Vreeken
2015-10-28
Scalable Bayesian Non-Negative Tensor Factorization for Massive Count Data
Changwei HuPiyush RaiChangyou ChenMatthew HardingLawrence Carin
2015-08-18
A MAP approach for $\ell_q$-norm regularized sparse parameter estimation using the EM algorithm
Rodrigo CarvajalJuan C. AgüeroBoris I. GodoyDimitrios Katselis
2015-08-05
An SVM-like Approach for Expectile Regression
Muhammad FarooqIngo Steinwart
2015-07-14
Supervised Learning of Semantics-Preserving Hash via Deep Convolutional Neural Networks
| Huei-Fang YangKevin LinChu-Song Chen
2015-07-01
On Distributive Subalgebras of Qualitative Spatial and Temporal Calculi
Zhiguo LongSanjiang Li
2015-06-01
Variational Inference with Normalizing Flows
| Danilo Jimenez RezendeShakir Mohamed
2015-05-21

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