In [[machine learning]], a '''variational autoencoder (VAE)''',<ref name=":0">{{cite arXiv |last1=Kingma |first1=Diederik P. |last2=Welling |first2=Max |title=Auto-Encoding Variational Bayes |date=2014-05-01 |class=stat.ML |eprint=1312.6114}}</ref> is an [[artificial neural network]] architecture introduced by Diederik P. Kingma and [[Max Welling]], belonging to the families of [[graphical model|probabilistic graphical models]] and [[variational Bayesian methods]].
Variational autoencoders are often associated with the [[autoencoder]] model because of its architectural affinity, but with significant differences in the goal and mathematical formulation. Variational autoencoders allow statistical inference problems to be rewritten (such as inferring the value of one [[random variable]] from another random variable) as statistical optimization problems (i.e find the parameter values that minimize some objective function).<ref>{{cite journal |last1=Kramer |first1=Mark A. |title=Nonlinear principal component analysis using autoassociative neural networks |journal=AIChE Journal |date=1991 |volume=37 |issue=2 |pages=233–243 |doi=10.1002/aic.690370209 |url=https://aiche.onlinelibrary.wiley.com/doi/abs/10.1002/aic.690370209 |language=en}}</ref><ref>{{cite journal |last1=Hinton |first1=G. E. |last2=Salakhutdinov |first2=R. R. |title=Reducing the Dimensionality of Data with Neural Networks |journal=Science |date=2006-07-28 |volume=313 |issue=5786 |pages=504–507 |doi=10.1126/science.1127647 |pmid=16873662 |bibcode=2006Sci...313..504H |s2cid=1658773 |url=https://www.science.sciencemag.org/contentdoi/313abs/578610.1126/504science.abstract1127647?casa_token=ZLsQ9vPfFA4AAAAA:3iBJRtRFr9RzkbbGpAJQtghIAndmRGEPVxW%3A3iBJRtRFr9RzkbbGpAJQtghIAndmRGEPVxW-yixDgfiXqWuuaQs8WjDMf-fkzTIe8RKn_J9o1aFozD4 |language=en}}</ref> <ref>{{cite web |title=A Beginner's Guide to Variational Methods: Mean-Field Approximation |url=https://blog.evjang.com/2016/08/variational-bayes.html |website=Eric Jang |language=en |date=2016-07-08}}</ref> They are meant to map the input variable to a multivariate latent distribution. Although this type of model was initially designed for [[unsupervised learning]],<ref>{{cite arXiv |last1=Dilokthanakul |first1=Nat |last2=Mediano |first2=Pedro A. M. |last3=Garnelo |first3=Marta |last4=Lee |first4=Matthew C. H. |last5=Salimbeni |first5=Hugh |last6=Arulkumaran |first6=Kai |last7=Shanahan |first7=Murray |title=Deep Unsupervised Clustering with Gaussian Mixture Variational Autoencoders |date=2017-01-13 |class=cs.LG |eprint=1611.02648}}</ref><ref>{{cite book |last1=Hsu |first1=Wei-Ning |last2=Zhang |first2=Yu |last3=Glass |first3=James |title=2017 IEEE Automatic Speech Recognition and Understanding Workshop (ASRU) |chapter=Unsupervised ___domain adaptation for robust speech recognition via variational autoencoder-based data augmentation |date=December 2017 |pages=16–23 |doi=10.1109/ASRU.2017.8268911 |arxiv=1707.06265 |isbn=978-1-5090-4788-8 |s2cid=22681625 |chapter-url=https://ieeexplore.ieee.org/abstract/document/8268911?casa_token=i8S9DzueB5gAAAAA:SnZUh5mfUYtRpusQLMJxN7eC_-6-qOQs9vpkEcA0Ai_ju-nJH7o1H1DN6nDFdeCY-LgGg3OVKQ}}</ref> its effectiveness has been proven for [[semi-supervised learning]]<ref>{{cite book |last1=Ehsan Abbasnejad |first1=M. |last2=Dick |first2=Anthony |last3=van den Hengel |first3=Anton |title=Infinite Variational Autoencoder for Semi-Supervised Learning |date=2017 |pages=5888–5897 |url=https://openaccess.thecvf.com/content_cvpr_2017/html/Abbasnejad_Infinite_Variational_Autoencoder_CVPR_2017_paper.html}}</ref><ref>{{cite journal |last1=Xu |first1=Weidi |last2=Sun |first2=Haoze |last3=Deng |first3=Chao |last4=Tan |first4=Ying |title=Variational Autoencoder for Semi-Supervised Text Classification |journal=Proceedings of the AAAI Conference on Artificial Intelligence |date=2017-02-12 |volume=31 |issue=1 |url=https://ojs.aaai.org/index.php/AAAI/article/view/10966 |language=en}}</ref> and [[supervised learning]].<ref>{{cite journal |last1=Kameoka |first1=Hirokazu |last2=Li |first2=Li |last3=Inoue |first3=Shota |last4=Makino |first4=Shoji |title=Supervised Determined Source Separation with Multichannel Variational Autoencoder |journal=Neural Computation |date=2019-09-01 |volume=31 |issue=9 |pages=1891–1914 |doi=10.1162/neco_a_01217 |pmid=31335290 |s2cid=198168155 |url=https://direct.mit.edu/neco/article/31/9/1891/8494/Supervised-Determined-Source-Separation-with}}</ref>
