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In [[machine learning]], a '''variational autoencoder''' ('''VAE''') is an [[artificial neural network]] architecture introduced by Diederik P. Kingma and [[Max Welling]]. It is part of the families of [[graphical model|probabilistic graphical models]] and [[variational Bayesian methods]].<ref>{{cite book |first1=Lucas |last1=Pinheiro Cinelli |first2=Matheus |last2=Araújo Marins |first3=Eduardo Antônio |last3=Barros da Silva |first4=Sérgio |last4=Lima Netto |display-authors=1 |title=Variational Methods for Machine Learning with Applications to Deep Networks |___location= |publisher=Springer |year=2021 |pages=111–149 |chapter=Variational Autoencoder |isbn=978-3-030-70681-4 |chapter-url=https://books.google.com/books?id=N5EtEAAAQBAJ&pg=PA111 |doi=10.1007/978-3-030-70679-1_5 |s2cid=240802776 }}</ref>
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 are probabilistic generative models that require neural networks as only a part of their overall structure. The neural network components are typically referred to as the encoder and decoder for the first and second component respectively. The first neural network maps the input variable to a [[latent space]] that corresponds to the parameters of a variational distribution. In this way, the encoder can produce multiple different samples that all come from the same distribution. The decoder has the opposite function, which is to map from the latent space to the input space, in order to produce or generate data points. Both networks are typically trained together with the usage of the [[#reparameterization|reparameterization trick]], although the variance of the noise model can be learned separately.
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}}</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 |doi=10.1609/aaai.v31i1.10966 |s2cid=2060721 |url=https://ojs.aaai.org/index.php/AAAI/article/view/10966 |language=en|doi-access=free }}</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>
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