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In addition to being seen as an [[autoencoder]] neural network architecture, variational autoencoders can also be studied within the mathematical formulation of [[variational Bayesian methods]], connecting a neural encoder network to its decoder through a probabilistic [[latent space]] (for example, as a [[multivariate Gaussian distribution]]) that corresponds to the parameters of a variational distribution.
Thus, the encoder maps each point (such as an image) from a large complex dataset into a distribution within the latent space, rather than to a single point in that space. The decoder has the opposite function, which is to map from the latent space to the input space, again according to a distribution (although in practice, noise rarely added during the decoding stage). By mapping a point to a distribution instead of a single point, the network can avoid overfitting the training data. 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.{{cn|date=June 2024}}
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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