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{{Short description|Neural network that learns efficient data encoding in an unsupervised manner}}
{{Distinguish|Autocoder|Autocode}}
{{Use dmy dates|date=March 2020|cs1-dates=y}}
{{Machine learning|Artificial neural network}}
An '''autoencoder''' is a type of [[artificial neural network]] used to learn [[Feature learning|efficient codings]] of unlabeled data ([[unsupervised learning]]).<ref name=":12">{{cite journal|doi=10.1002/aic.690370209|title=Nonlinear principal component analysis using autoassociative neural networks|journal=AIChE Journal|volume=37|issue=2|pages=233–243|date=1991|last1=Kramer|first1=Mark A.|bibcode=1991AIChE..37..233K |url= https://www.researchgate.net/profile/Abir_Alobaid/post/To_learn_a_probability_density_function_by_using_neural_network_can_we_first_estimate_density_using_nonparametric_methods_then_train_the_network/attachment/59d6450279197b80779a031e/AS:451263696510979@1484601057779/download/NL+PCA+by+using+ANN.pdf}}</ref><ref name=":13">{{Cite journal |last=Kramer |first=M. A. |date=1992-04-01 |title=Autoassociative neural networks |url=https://dx.doi.org/10.1016/0098-1354%2892%2980051-A |journal=Computers & Chemical Engineering |series=Neutral network applications in chemical engineering |language=en |volume=16 |issue=4 |pages=313–328 |doi=10.1016/0098-1354(92)80051-A |issn=0098-1354}}</ref> An autoencoder learns two functions: an encoding function that transforms the input data, and a decoding function that recreates the input data from the encoded representation. The autoencoder learns an [[Feature learning|efficient representation]] (encoding) for a set of data, typically for [[dimensionality reduction]].

Variants exist, aiming to force the learned representations to assume useful properties.<ref name=":0" /> Examples are regularized autoencoders (''Sparse'', ''Denoising'' and ''Contractive''), which are effective in learning representations for subsequent [[Statistical classification|classification]] tasks,<ref name=":4" /> and [[Variational_autoencoder|''Variational'' autoencoders]], with applications as [[generative model]]s.<ref name=":11">{{cite journal |arxiv=1906.02691|doi=10.1561/2200000056|bibcode=2019arXiv190602691K|title=An Introduction to Variational Autoencoders|date=2019|last1=Welling|first1=Max|last2=Kingma|first2=Diederik P.|journal=Foundations and Trends in Machine Learning|volume=12|issue=4|pages=307–392|s2cid=174802445}}</ref> Autoencoders are applied to many problems, including [[face recognition|facial recognition]],<ref>Hinton GE, Krizhevsky A, Wang SD. [http://www.cs.toronto.edu/~fritz/absps/transauto6.pdf Transforming auto-encoders.] In International Conference on Artificial Neural Networks 2011 Jun 14 (pp. 44-51). Springer, Berlin, Heidelberg.</ref> feature detection,<ref name=":2">{{Cite book|last=Géron|first=Aurélien|title=Hands-On Machine Learning with Scikit-Learn, Keras, and TensorFlow|publisher=O’Reilly Media, Inc.|year=2019|location=Canada|pages=739–740}}</ref> anomaly detection and acquiring the meaning of words.<ref>{{cite journal|doi=10.1016/j.neucom.2008.04.030|title=Modeling word perception using the Elman network|journal=Neurocomputing|volume=71|issue=16–18|pages=3150|date=2008|last1=Liou|first1=Cheng-Yuan|last2=Huang|first2=Jau-Chi|last3=Yang|first3=Wen-Chie|url=http://ntur.lib.ntu.edu.tw//handle/246246/155195 }}</ref><ref>{{cite journal|doi=10.1016/j.neucom.2013.09.055|title=Autoencoder for words|journal=Neurocomputing|volume=139|pages=84–96|date=2014|last1=Liou|first1=Cheng-Yuan|last2=Cheng|first2=Wei-Chen|last3=Liou|first3=Jiun-Wei|last4=Liou|first4=Daw-Ran}}</ref> Autoencoders are also generative models which can randomly generate new data that is similar to the input data (training data).<ref name=":2" />

== Mathematical principles ==

=== Definition ===
An autoencoder is defined by the following components: <blockquote>Two sets: the space of decoded messages <math>\mathcal X</math>; the space of encoded messages <math>\mathcal Z</math>. Almost always, both <math>\mathcal X</math> and <math>\mathcal Z</math> are Euclidean spaces, that is, <math>\mathcal X = \R^m, \mathcal Z = \R^n</math> for some <math>m, n</math>. </blockquote><blockquote>Two parametrized families of functions: the encoder family <math>E_\phi:\mathcal{X} \rightarrow \mathcal{Z}</math>, parametrized by <math>\phi</math>; the decoder family <math>D_\theta:\mathcal{Z} \rightarrow \mathcal{X}</math>, parametrized by <math>\theta</math>.</blockquote>For any <math>x\in \mathcal X</math>, we usually write <math>z = E_\phi(x)</math>, and refer to it as the code, the [[latent variable]], latent representation, latent vector, etc. Conversely, for any <math>z\in \mathcal Z</math>, we usually write <math>x' = D_\theta(z)</math>, and refer to it as the (decoded) message.

Usually, both the encoder and the decoder are defined as [[multilayer perceptron]]s. For example, a one-layer-MLP encoder <math>E_\phi</math> is:

:<math>E_\phi(\mathbf x) = \sigma(Wx+b)</math>

where <math>\sigma</math> is an element-wise [[activation function]] such as a [[sigmoid function]] or a [[rectified linear unit]], <math>W</math> is a matrix called "weight", and <math>b</math> is a vector called "bias".

=== Training an autoencoder ===
An autoencoder, by itself, is simply a tuple of two functions. To judge its ''quality'', we need a ''task''. A task is defined by a reference probability distribution <math>\mu_{ref}</math> over <math>\mathcal X</math>, and a "reconstruction quality" function <math>d: \mathcal X \times \mathcal X \to [0, \infty]</math>, such that <math>d(x, x')</math> measures how much <math>x'</math> differs from <math>x</math>.

With those, we can define the loss function for the autoencoder as<math display="block">L(\theta, \phi) := \mathbb \mathbb E_{x\sim \mu_{ref}}[d(x, D_\theta(E_\phi(x)))]</math>The ''optimal'' autoencoder for the given task <math>(\mu_{ref}, d)</math> is then <math>\arg\min_{\theta, \phi}L(\theta, \phi)</math>. The search for the optimal autoencoder can be accomplished by any mathematical optimization technique, but usually by [[gradient descent]]. This search process is referred to as "training the autoencoder".

In most situations, the reference distribution is just the [[Empirical measure|empirical distribution]] given by a dataset <math>\{x_1, ..., x_N\} \subset \mathcal X</math>, so that<math display="block">\mu_{ref} = \frac{1}{N}\sum_{i=1}^N \delta_{x_i}</math>

where <math>\delta_{x_i}</math> is the [[Dirac measure]], the quality function is just L2 loss: <math>d(x, x') = \|x - x'\|_2^2</math>, and <math>\|\cdot\|_2</math> is the Euclidean norm. Then the problem of searching for the optimal autoencoder is just a [[Least squares|least-squares]] optimization:<math display="block">\min_{\theta, \phi} L(\theta, \phi), \text{where } L(\theta, \phi) = \frac{1}{N}\sum_{i=1}^N \|x_i - D_\theta(E_\phi(x_i))\|_2^2</math>

=== Interpretation ===
[[File:Autoencoder_schema.png|thumb|Schema of a basic autoencoder]]
An autoencoder has two main parts: an encoder that maps the message to a code, and a decoder that reconstructs the message from the code. An optimal autoencoder would perform as close to perfect reconstruction as possible, with "close to perfect" defined by the reconstruction quality function <math>d</math>.

The simplest way to perform the copying task perfectly would be to duplicate the signal. To suppress this behavior, the code space <math>\mathcal Z</math> usually has fewer dimensions than the message space <math>\mathcal{X}</math>.

Such an autoencoder is called ''undercomplete''. It can be interpreted as [[Data compression|compressing]] the message, or [[Dimensionality reduction|reducing its dimensionality]].<ref name=":12" /><ref name=":7" />

At the limit of an ideal undercomplete autoencoder, every possible code <math>z</math> in the code space is used to encode a message <math>x</math> that really appears in the distribution <math>\mu_{ref}</math>, and the decoder is also perfect: <math>D_\theta(E_\phi(x)) = x</math>. This ideal autoencoder can then be used to generate messages indistinguishable from real messages, by feeding its decoder arbitrary code <math>z</math> and obtaining <math>D_\theta(z)</math>, which is a message that really appears in the distribution <math>\mu_{ref}</math>.

If the code space <math>\mathcal Z</math> has dimension larger than (''overcomplete''), or equal to, the message space <math>\mathcal{X}</math>, or the hidden units are given enough capacity, an autoencoder can learn the [[identity function]] and become useless. However, experimental results found that overcomplete autoencoders might still [[feature learning|learn useful features]].<ref name="bengio">{{Cite journal |last1=Bengio |first1=Y. |date=2009 |title=Learning Deep Architectures for AI |url=http://www.iro.umontreal.ca/~lisa/pointeurs/TR1312.pdf |journal=Foundations and Trends in Machine Learning |volume=2 |issue=8 |pages=1795–7 |citeseerx=10.1.1.701.9550 |doi=10.1561/2200000006 |pmid=23946944|s2cid=207178999 }}</ref>

In the ideal setting, the code dimension and the model capacity could be set on the basis of the complexity of the data distribution to be modeled. A standard way to do so is to add modifications to the basic autoencoder, to be detailed below.<ref name=":0" />

== History ==
The autoencoder was first proposed as a nonlinear generalization of [[Principal component analysis|principal components analysis]] (PCA) by Kramer.<ref name=":12" /> The autoencoder has also been called the autoassociator,<ref>{{Cite journal |last1=Japkowicz |first1=Nathalie |author-link=Nathalie Japkowicz |last2=Hanson |first2=Stephen José |author-link2=Stephen José Hanson |last3=Gluck |first3=Mark A. |date=2000-03-01 |title=Nonlinear Autoassociation Is Not Equivalent to PCA |journal=Neural Computation |volume=12 |issue=3 |pages=531–545 |doi=10.1162/089976600300015691 |issn=0899-7667 |pmid=10769321 |s2cid=18490972}}</ref> or Diabolo network.<ref>{{Cite journal |last1=Schwenk |first1=Holger |last2=Bengio |first2=Yoshua |date=1997 |title=Training Methods for Adaptive Boosting of Neural Networks |url=https://proceedings.neurips.cc/paper/1997/hash/9cb67ffb59554ab1dabb65bcb370ddd9-Abstract.html |journal=Advances in Neural Information Processing Systems |publisher=MIT Press |volume=10}}</ref><ref name="bengio" /> Its first applications date to early 1990s.<ref name=":0" /><ref>{{Cite journal |last=Schmidhuber |first=Jürgen |date=January 2015 |title=Deep learning in neural networks: An overview |journal=Neural Networks |volume=61 |pages=85–117 |arxiv=1404.7828 |doi=10.1016/j.neunet.2014.09.003 |pmid=25462637 |s2cid=11715509}}</ref><ref>Hinton, G. E., & Zemel, R. S. (1994). Autoencoders, minimum description length and Helmholtz free energy. In ''Advances in neural information processing systems 6'' (pp. 3-10).</ref> Their most traditional application was [[dimensionality reduction]] or [[feature learning]], but the concept became widely used for learning [[generative model]]s of data.<ref name="VAE">{{cite arXiv |eprint=1312.6114 |class=stat.ML |author1=Diederik P Kingma |first2=Max |last2=Welling |title=Auto-Encoding Variational Bayes |date=2013}}</ref><ref name="gan_faces">Generating Faces with Torch, Boesen A., Larsen L. and Sonderby S.K., 2015 {{url|http://torch.ch/blog/2015/11/13/gan.html}}</ref> Some of the most powerful [[Artificial intelligence|AIs]] in the 2010s involved autoencoders stacked inside [[Deep learning|deep]] neural networks.<ref name="domingos">{{cite book |last1=Domingos |first1=Pedro |title=The Master Algorithm: How the Quest for the Ultimate Learning Machine Will Remake Our World |title-link=The Master Algorithm |date=2015 |publisher=Basic Books |isbn=978-046506192-1 |at="Deeper into the Brain" subsection |chapter=4 |author-link=Pedro Domingos}}</ref>

==Variations==

=== Regularized autoencoders ===
Various techniques exist to prevent autoencoders from learning the [[identity function]] and to improve their ability to capture important information and learn richer representations.

====Sparse autoencoder (SAE)====
Inspired by the [[sparse coding]] hypothesis in neuroscience, sparse autoencoders are variants of autoencoders, such that the codes <math>E_\phi(x)</math> for messages tend to be ''sparse codes'', that is, <math>E_\phi(x)</math> is close to zero in most entries. Sparse autoencoders may include more (rather than fewer) hidden units than inputs, but only a small number of the hidden units are allowed to be active at the same time.<ref name="domingos" /> Encouraging sparsity improves performance on classification tasks.<ref name=":5">{{Cite journal |last1=Frey |first1=Brendan |last2=Makhzani |first2=Alireza |date=2013-12-19 |title=k-Sparse Autoencoders |arxiv=1312.5663 |bibcode=2013arXiv1312.5663M}}</ref> [[File:Autoencoder sparso.png|thumb|Simple schema of a single-layer sparse autoencoder. The hidden nodes in bright yellow are activated, while the light yellow ones are inactive. The activation depends on the input.]]
There are two main ways to enforce sparsity. One way is to simply clamp all but the highest-k activations of the latent code to zero. This is the '''k-sparse autoencoder'''.<ref name=":1">{{cite arXiv |eprint=1312.5663 |class=cs.LG |first1=Alireza |last1=Makhzani |first2=Brendan |last2=Frey |title=K-Sparse Autoencoders |date=2013}}</ref>

The k-sparse autoencoder inserts the following "k-sparse function" in the latent layer of a standard autoencoder:<math display="block">f_k(x_1, ..., x_n) = (x_1 b_1, ..., x_n b_n)</math>where <math>b_i = 1</math> if <math>|x_i|</math> ranks in the top k, and 0 otherwise.

Backpropagating through <math>f_k</math> is simple: set gradient to 0 for <math>b_i = 0</math> entries, and keep gradient for <math>b_i=1</math> entries. This is essentially a generalized [[Rectifier (neural networks)|ReLU]] function.<ref name=":1" />

The other way is a [[Relaxation (approximation)|relaxed version]] of the k-sparse autoencoder. Instead of forcing sparsity, we add a '''sparsity regularization loss''', then optimize for<math display="block">\min_{\theta, \phi}L(\theta, \phi) + \lambda L_{sparsity} (\theta, \phi)</math>where <math>\lambda > 0</math> measures how much sparsity we want to enforce.<ref name=":6" />

Let the autoencoder architecture have <math>K</math> layers. To define a sparsity regularization loss, we need a "desired" sparsity <math>\hat \rho_k</math> for each layer, a weight <math>w_k</math> for how much to enforce each sparsity, and a function <math>s: [0, 1]\times [0, 1] \to [0, \infty]</math> to measure how much two sparsities differ.

For each input <math>x</math>, let the actual sparsity of activation in each layer <math>k</math> be<math display="block">\rho_k(x) = \frac 1n \sum_{i=1}^n a_{k, i}(x)</math>where <math>a_{k, i}(x)</math> is the activation in the <math>i</math> -th neuron of the <math>k</math> -th layer upon input <math>x</math>.

The sparsity loss upon input <math>x</math> for one layer is <math>s(\hat\rho_k, \rho_k(x))</math>, and the sparsity regularization loss for the entire autoencoder is the expected weighted sum of sparsity losses:<math display="block">L_{sparsity}(\theta, \phi) = \mathbb \mathbb E_{x\sim\mu_X}\left[\sum_{k\in 1:K} w_k s(\hat\rho_k, \rho_k(x)) \right]</math>Typically, the function <math>s</math> is either the [[Kullback–Leibler divergence|Kullback-Leibler (KL) divergence]], as<ref name=":5" /><ref name=":6">Ng, A. (2011). [https://web.stanford.edu/class/cs294a/sparseAutoencoder_2011new.pdf Sparse autoencoder]. ''CS294A Lecture notes'', ''72''(2011), 1-19.</ref><ref>{{Cite journal|last1=Nair|first1=Vinod|last2=Hinton|first2=Geoffrey E.|date=2009|title=3D Object Recognition with Deep Belief Nets|url=http://dl.acm.org/citation.cfm?id=2984093.2984244|journal=Proceedings of the 22nd International Conference on Neural Information Processing Systems|series=NIPS'09|location=USA|publisher=Curran Associates Inc.|pages=1339–1347|isbn=9781615679119}}</ref><ref>{{Cite journal|last1=Zeng|first1=Nianyin|last2=Zhang|first2=Hong|last3=Song|first3=Baoye|last4=Liu|first4=Weibo|last5=Li|first5=Yurong|last6=Dobaie|first6=Abdullah M.|date=2018-01-17|title=Facial expression recognition via learning deep sparse autoencoders|journal=Neurocomputing|volume=273|pages=643–649|doi=10.1016/j.neucom.2017.08.043|issn=0925-2312}}</ref>

::<math>s(\rho, \hat\rho) = KL(\rho || \hat{\rho}) = \rho \log \frac{\rho}{\hat{\rho}}+(1- \rho)\log \frac{1-\rho}{1-\hat{\rho}}</math>
or the L1 loss, as <math>s(\rho, \hat\rho) = |\rho- \hat\rho|</math>, or the L2 loss, as <math>s(\rho, \hat\rho) = |\rho- \hat\rho|^2</math>.

Alternatively, the sparsity regularization loss may be defined without reference to any "desired sparsity", but simply force as much sparsity as possible. In this case, one can define the sparsity regularization loss as <math display="block">L_{sparsity}(\theta, \phi) = \mathbb \mathbb E_{x\sim\mu_X}\left[
\sum_{k\in 1:K} w_k \|h_k\|
\right]</math>where <math>h_k</math> is the activation vector in the <math>k</math>-th layer of the autoencoder. The norm <math>\|\cdot\|</math> is usually the L1 norm (giving the L1 sparse autoencoder) or the L2 norm (giving the L2 sparse autoencoder).

====Denoising autoencoder (DAE)====

Denoising autoencoders (DAE) try to achieve a ''good'' representation by changing the ''reconstruction criterion''.<ref name=":0" /><ref name=":4" />

A DAE, originally called a "robust autoassociative network",<ref name=":13"/> is trained by intentionally corrupting the inputs of a standard autoencoder during training. A noise process is defined by a probability distribution <math>\mu_T</math> over functions <math>T:\mathcal X \to \mathcal X</math>. That is, the function <math>T</math> takes a message <math>x\in \mathcal X</math>, and corrupts it to a noisy version <math>T(x)</math>. The function <math>T</math> is selected randomly, with a probability distribution <math>\mu_T</math>.

Given a task <math>(\mu_{ref}, d)</math>, the problem of training a DAE is the optimization problem:<math display="block">\min_{\theta, \phi}L(\theta, \phi) = \mathbb \mathbb E_{x\sim \mu_X, T\sim\mu_T}[d(x, (D_\theta\circ E_\phi \circ T)(x))]</math>That is, the optimal DAE should take any noisy message and attempt to recover the original message without noise, thus the name "denoising"''.''

Usually, the noise process <math>T</math> is applied only during training and testing, not during downstream use.

The use of DAE depends on two assumptions:
* There exist representations to the messages that are relatively stable and robust to the type of noise we are likely to encounter;
* The said representations capture structures in the input distribution that are useful for our purposes.<ref name=":4">{{Cite journal|last1=Vincent|first1=Pascal|last2=Larochelle|first2=Hugo|date=2010|title=Stacked Denoising Autoencoders: Learning Useful Representations in a Deep Network with a Local Denoising Criterion|journal=Journal of Machine Learning Research|volume=11|pages=3371–3408}}</ref>

Example noise processes include:

* additive isotropic [[Additive white Gaussian noise|Gaussian noise]], 
* masking noise (a fraction of the input is randomly chosen and set to 0)
* salt-and-pepper noise (a fraction of the input is randomly chosen and randomly set to its minimum or maximum value).<ref name=":4" />

==== Contractive autoencoder (CAE) ====
A contractive autoencoder adds the contractive regularization loss to the standard autoencoder loss:<math display="block">\min_{\theta, \phi}L(\theta, \phi) + \lambda L_{contractive} (\theta, \phi)</math>where <math>\lambda > 0</math> measures how much contractive-ness we want to enforce. The contractive regularization loss itself is defined as the expected [[Frobenius norm]] of the [[Jacobian matrix and determinant|Jacobian matrix]] of the encoder activations with respect to the input:<math display="block">L_{contractive}(\theta, \phi) = \mathbb E_{x\sim \mu_{ref}} \|\nabla_x E_\phi(x) \|_F^2</math>To understand what <math>L_{contractive}</math> measures, note the fact<math display="block">\|E_\phi(x + \delta x) - E_\phi(x)\|_2 \leq \|\nabla_x E_\phi(x) \|_F \|\delta x\|_2</math>for any message <math>x\in \mathcal X</math>, and small variation <math>\delta x</math> in it. Thus, if <math>\|\nabla_x E_\phi(x) \|_F^2</math> is small, it means that a small neighborhood of the message maps to a small neighborhood of its code. This is a desired property, as it means small variation in the message leads to small, perhaps even zero, variation in its code, like how two pictures may look the same even if they are not exactly the same.

The DAE can be understood as an infinitesimal limit of CAE: in the limit of small Gaussian input noise, DAEs make the reconstruction function resist small but finite-sized input perturbations, while CAEs make the extracted features resist infinitesimal input perturbations.

==== Minimal description length autoencoder ====
<ref>{{Cite journal |last1=Hinton |first1=Geoffrey E |last2=Zemel |first2=Richard |date=1993 |title=Autoencoders, Minimum Description Length and Helmholtz Free Energy |url=https://proceedings.neurips.cc/paper/1993/hash/9e3cfc48eccf81a0d57663e129aef3cb-Abstract.html |journal=Advances in Neural Information Processing Systems |publisher=Morgan-Kaufmann |volume=6}}</ref>
{{Empty section|date=March 2024}}

=== Concrete autoencoder ===
The concrete autoencoder is designed for discrete feature selection.<ref>{{cite arXiv|last1=Abid|first1=Abubakar|last2=Balin|first2=Muhammad Fatih|last3=Zou|first3=James|date=2019-01-27|title=Concrete Autoencoders for Differentiable Feature Selection and Reconstruction|eprint=1901.09346|class=cs.LG}}</ref> A concrete autoencoder forces the latent space to consist only of a user-specified number of features. The concrete autoencoder uses a continuous [[Relaxation (approximation)|relaxation]] of the [[categorical distribution]] to allow gradients to pass through the feature selector layer, which makes it possible to use standard [[backpropagation]] to learn an optimal subset of input features that minimize reconstruction loss.

===Variational autoencoder (VAE)===
{{Main|Variational autoencoder}}

[[Variational autoencoder]]s (VAEs) belong to the families of [[variational Bayesian methods]]. Despite the architectural similarities with basic autoencoders, VAEs are architected with different goals and have a different mathematical formulation. The latent space is, in this case, composed of a mixture of distributions instead of fixed vectors.

Given an input dataset <math>x</math> characterized by an unknown probability function <math>P(x)</math> and a multivariate latent encoding vector <math>z</math>, the objective is to model the data as a distribution <math>p_\theta(x)</math>, with <math>\theta</math> defined as the set of the network parameters so that <math>p_\theta(x) = \int_{z}p_\theta(x,z)dz </math>.

==Advantages of depth==
[[File:Autoencoder_structure.png|350x350px|Schematic structure of an autoencoder with 3 fully connected hidden layers. The code (z, or h for reference in the text) is the most internal layer.|thumb]]
Autoencoders are often trained with a single-layer encoder and a single-layer decoder, but using many-layered (deep) encoders and decoders offers many advantages.<ref name=":0" />

* Depth can exponentially reduce the computational cost of representing some functions.
* Depth can exponentially decrease the amount of training data needed to learn some functions.
* Experimentally, deep autoencoders yield better compression compared to shallow or linear autoencoders.<ref name=":7" />

=== Training ===
[[Geoffrey Hinton]] developed the [[deep belief network]] technique for training many-layered deep autoencoders. His method involves treating each neighboring set of two layers as a [[restricted Boltzmann machine]] so that pretraining approximates a good solution, then using backpropagation to fine-tune the results.<ref name=":7">{{cite journal|last1=Hinton|first1=G. E.|last2=Salakhutdinov|first2=R.R.|title=Reducing the Dimensionality of Data with Neural Networks|journal=Science|date=28 July 2006|volume=313|issue=5786|pages=504–507|doi=10.1126/science.1127647|pmid=16873662|bibcode=2006Sci...313..504H|s2cid=1658773}}</ref>

Researchers have debated whether joint training (i.e. training the whole architecture together with a single global reconstruction objective to optimize) would be better for deep auto-encoders.<ref name=":9">{{cite arXiv |eprint=1405.1380|last1=Zhou|first1=Yingbo|last2=Arpit|first2=Devansh|last3=Nwogu|first3=Ifeoma|last4=Govindaraju|first4=Venu|title=Is Joint Training Better for Deep Auto-Encoders?|class=stat.ML|date=2014}}</ref> A 2015 study showed that joint training learns better data models along with more representative features for classification as compared to the layerwise method.<ref name=":9" /> However, their experiments showed that the success of joint training depends heavily on the regularization strategies adopted.<ref name=":9" /><ref>R. Salakhutdinov and G. E. Hinton, “Deep Boltzmann machines,” in AISTATS, 2009, pp. 448–455.</ref>

== Applications ==
The two main applications of autoencoders are dimensionality reduction and information retrieval,<ref name=":0">{{Cite book|url=http://www.deeplearningbook.org|title=Deep Learning|last1=Goodfellow|first1=Ian|last2=Bengio|first2=Yoshua|last3=Courville|first3=Aaron|publisher=MIT Press|date=2016|isbn=978-0262035613}}</ref> but modern variations have been applied to other tasks.

=== Dimensionality reduction ===
[[File:PCA vs Linear Autoencoder.png|thumb|Plot of the first two Principal Components (left) and a two-dimension hidden layer of a Linear Autoencoder (Right) applied to the [[Fashion MNIST]] dataset.<ref name=":10">{{Cite web|url=https://github.com/zalandoresearch/fashion-mnist|title=Fashion MNIST|website=[[GitHub]]|date=2019-07-12}}</ref> The two models being both linear learn to span the same subspace. The projection of the data points is indeed identical, apart from rotation of the subspace. While PCA selects a specific orientation up to reflections in the general case, the cost function of a simple autoencoder is invariant to rotations of the latent space.]][[Dimensionality reduction]] was one of the first [[deep learning]] applications.<ref name=":0" />

For Hinton's 2006 study,<ref name=":7" /> he pretrained a multi-layer autoencoder with a stack of [[Restricted Boltzmann machine|RBMs]] and then used their weights to initialize a deep autoencoder with gradually smaller hidden layers until hitting a bottleneck of 30 neurons. The resulting 30 dimensions of the code yielded a smaller reconstruction error compared to the first 30 components of a principal component analysis (PCA), and learned a representation that was qualitatively easier to interpret, clearly separating data clusters.<ref name=":0" /><ref name=":7" />

Representing dimensions can improve performance on tasks such as classification.<ref name=":0" /> Indeed, the hallmark of dimensionality reduction is to place semantically related examples near each other.<ref name=":3">{{Cite journal|last1=Salakhutdinov|first1=Ruslan|last2=Hinton|first2=Geoffrey|date=2009-07-01|title=Semantic hashing|journal=International Journal of Approximate Reasoning|series=Special Section on Graphical Models and Information Retrieval|volume=50|issue=7|pages=969–978|doi=10.1016/j.ijar.2008.11.006|issn=0888-613X|doi-access=free}}</ref>

==== Principal component analysis ====
[[File:Reconstruction autoencoders vs PCA.png|thumb|Reconstruction of 28x28pixel images by an Autoencoder with a code size of two (two-units hidden layer) and the reconstruction from the first two Principal Components of PCA. Images come from the [[Fashion MNIST dataset]].<ref name=":10" />]]
If linear activations are used, or only a single sigmoid hidden layer, then the optimal solution to an autoencoder is strongly related to [[principal component analysis]] (PCA).<ref>{{Cite journal|last1=Bourlard|first1=H.|last2=Kamp|first2=Y.|date=1988|title=Auto-association by multilayer perceptrons and singular value decomposition|journal=Biological Cybernetics|volume=59|issue=4–5|pages=291–294|doi=10.1007/BF00332918|pmid=3196773|s2cid=206775335|url=http://infoscience.epfl.ch/record/82601}}</ref><ref>{{cite book|title=Proceedings of the 5th ACM Conference on Bioinformatics, Computational Biology, and Health Informatics - BCB '14|last1=Chicco|first1=Davide|last2=Sadowski|first2=Peter|last3=Baldi|first3=Pierre|date=2014|isbn=9781450328944|pages=533|chapter=Deep autoencoder neural networks for gene ontology annotation predictions|doi=10.1145/2649387.2649442|hdl=11311/964622|s2cid=207217210|url=http://dl.acm.org/citation.cfm?id=2649442}}</ref> The weights of an autoencoder with a single hidden layer of size <math>p</math> (where <math>p</math> is less than the size of the input) span the same vector subspace as the one spanned by the first <math>p</math> principal components, and the output of the autoencoder is an orthogonal projection onto this subspace. The autoencoder weights are not equal to the principal components, and are generally not orthogonal, yet the principal components may be recovered from them using the [[singular value decomposition]].<ref>{{cite arXiv|last1=Plaut|first1=E|title=From Principal Subspaces to Principal Components with Linear Autoencoders|eprint=1804.10253|date=2018|class=stat.ML}}</ref>

However, the potential of autoencoders resides in their non-linearity, allowing the model to learn more powerful generalizations compared to PCA, and to reconstruct the input with significantly lower information loss.<ref name=":7" />

=== Information retrieval and Search engine optimization===
[[Information retrieval]] benefits particularly from [[dimensionality reduction]] in that search can become more efficient in certain kinds of low dimensional spaces. Autoencoders were indeed applied to semantic hashing, proposed by [[Russ Salakhutdinov|Salakhutdinov]] and Hinton in 2007.<ref name=":3" /> By training the algorithm to produce a low-dimensional binary code, all database entries could be stored in a [[hash table]] mapping binary code vectors to entries. This table would then support information retrieval by returning all entries with the same binary code as the query, or slightly less similar entries by flipping some bits from the query encoding.

The encoder-decoder architecture, often used in natural language processing and neural networks, can be scientifically applied in the field of SEO (Search Engine Optimization) in various ways:

# '''Text Processing''': By using an autoencoder, it's possible to compress the text of web pages into a more compact vector representation. This can help reduce page loading times and improve indexing by search engines.
# 
# '''Noise Reduction''': Autoencoders can be used to remove noise from the textual data of web pages. This can lead to a better understanding of the content by search engines, thereby enhancing ranking in search engine result pages.
# 
# '''Meta Tag and Snippet Generation''': Autoencoders can be trained to automatically generate meta tags, snippets, and descriptions for web pages using the page content. This can optimize the presentation in search results, increasing the Click-Through Rate (CTR).
# 
# '''Content Clustering''': Using an autoencoder, web pages with similar content can be automatically grouped together. This can help organize the website logically and improve navigation, potentially positively affecting user experience and search engine rankings.
# 
# '''Generation of Related Content''': An autoencoder can be employed to generate content related to what is already present on the site. This can enhance the website's attractiveness to search engines and provide users with additional relevant information.
# 
# '''Keyword Detection''': Autoencoders can be trained to identify keywords and important concepts within the content of web pages. This can assist in optimizing keyword usage for better indexing.
# 
# '''Semantic Search''': By using autoencoder techniques, semantic representation models of content can be created. These models can be used to enhance search engines' understanding of the themes covered in web pages.

In essence, the encoder-decoder architecture or autoencoders can be leveraged in SEO to optimize web page content, improve their indexing, and enhance their appeal to both search engines and users.

=== Anomaly detection ===
Another application for autoencoders is [[anomaly detection]].<ref name=":13" /><ref>{{Cite book |last1=Morales-Forero |first1=A. |last2=Bassetto |first2=S. |title=2019 IEEE International Conference on Industrial Engineering and Engineering Management (IEEM) |chapter=Case Study: A Semi-Supervised Methodology for Anomaly Detection and Diagnosis |date=December 2019 |chapter-url=https://ieeexplore.ieee.org/document/8978509 |location=Macao, Macao |publisher=IEEE |pages=1031–1037 |doi=10.1109/IEEM44572.2019.8978509 |isbn=978-1-7281-3804-6|s2cid=211027131 }}</ref><ref>{{Cite book |last1=Sakurada |first1=Mayu |last2=Yairi |first2=Takehisa |title=Proceedings of the MLSDA 2014 2nd Workshop on Machine Learning for Sensory Data Analysis |chapter=Anomaly Detection Using Autoencoders with Nonlinear Dimensionality Reduction |date=December 2014 |chapter-url=http://dl.acm.org/citation.cfm?doid=2689746.2689747 |language=en |location=Gold Coast, Australia QLD, Australia |publisher=ACM Press |pages=4–11 |doi=10.1145/2689746.2689747 |isbn=978-1-4503-3159-3|s2cid=14613395 }}</ref><ref name=":8">An, J., & Cho, S. (2015). [http://dm.snu.ac.kr/static/docs/TR/SNUDM-TR-2015-03.pdf Variational Autoencoder based Anomaly Detection using Reconstruction Probability]. ''Special Lecture on IE'', ''2'', 1-18.</ref><ref>{{Cite book |last1=Zhou |first1=Chong |last2=Paffenroth |first2=Randy C. |title=Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining |chapter=Anomaly Detection with Robust Deep Autoencoders |date=2017-08-04 |chapter-url=https://dl.acm.org/doi/10.1145/3097983.3098052 |language=en |location=Halifax NS Canada |publisher=ACM |pages=665–674 |doi=10.1145/3097983.3098052 |isbn=978-1-4503-4887-4|s2cid=207557733 }}</ref><ref>{{Cite journal|doi=10.1016/j.patrec.2017.07.016|title=A study of deep convolutional auto-encoders for anomaly detection in videos|year=2018|last1=Ribeiro|first1=Manassés|last2=Lazzaretti|first2=André Eugênio|last3=Lopes|first3=Heitor Silvério|journal=Pattern Recognition Letters|volume=105|pages=13–22|bibcode=2018PaReL.105...13R}}</ref> By learning to replicate the most salient features in the training data under some of the constraints described previously, the model is encouraged to learn to precisely reproduce the most frequently observed characteristics. When facing anomalies, the model should worsen its reconstruction performance. In most cases, only data with normal instances are used to train the autoencoder; in others, the frequency of anomalies is small compared to the observation set so that its contribution to the learned representation could be ignored. After training, the autoencoder will accurately reconstruct "normal" data, while failing to do so with unfamiliar anomalous data.<ref name=":8" /> Reconstruction error (the error between the original data and its low dimensional reconstruction) is used as an anomaly score to detect anomalies.<ref name=":8" />

Recent literature has however shown that certain autoencoding models can, counterintuitively, be very good at reconstructing anomalous examples and consequently not able to reliably perform anomaly detection.<ref>{{cite arXiv|last1=Nalisnick|first1=Eric|last2=Matsukawa|first2=Akihiro|last3=Teh|first3=Yee Whye|last4=Gorur|first4=Dilan|last5=Lakshminarayanan|first5=Balaji|date=2019-02-24|title=Do Deep Generative Models Know What They Don't Know?|class=stat.ML|eprint=1810.09136}}</ref><ref>{{Cite journal|last1=Xiao|first1=Zhisheng|last2=Yan|first2=Qing|last3=Amit|first3=Yali|date=2020|title=Likelihood Regret: An Out-of-Distribution Detection Score For Variational Auto-encoder|url=https://proceedings.neurips.cc/paper/2020/hash/eddea82ad2755b24c4e168c5fc2ebd40-Abstract.html|journal=Advances in Neural Information Processing Systems|language=en|volume=33|arxiv=2003.02977}}</ref>

=== Image processing ===
The characteristics of autoencoders are useful in image processing.

One example can be found in lossy [[image compression]], where autoencoders outperformed other approaches and proved competitive against [[JPEG 2000]].<ref>{{cite arXiv |eprint=1703.00395|last1=Theis|first1=Lucas|last2=Shi|first2=Wenzhe|last3=Cunningham|first3=Andrew|last4=Huszár|first4=Ferenc|title=Lossy Image Compression with Compressive Autoencoders|class=stat.ML|date=2017}}</ref><ref>{{cite book |last1=Balle |first1=J |last2=Laparra |first2=V |last3=Simoncelli |first3=EP |chapter=End-to-end optimized image compression |title=International Conference on Learning Representations |date=April 2017 |arxiv=1611.01704}}</ref>

Another useful application of autoencoders in image preprocessing is [[image denoising]].<ref>Cho, K. (2013, February). Simple sparsification improves sparse denoising autoencoders in denoising highly corrupted images. In ''International Conference on Machine Learning'' (pp. 432-440).</ref><ref>{{cite arXiv |eprint=1301.3468|last1=Cho|first1=Kyunghyun|title=Boltzmann Machines and Denoising Autoencoders for Image Denoising|class=stat.ML|date=2013}}</ref><ref>{{Cite journal|doi = 10.1137/040616024|title = A Review of Image Denoising Algorithms, with a New One |url=https://hal.archives-ouvertes.fr/hal-00271141 |year = 2005|last1 = Buades|first1 = A.|last2 = Coll|first2 = B.|last3 = Morel|first3 = J. M.|journal = Multiscale Modeling & Simulation|volume = 4|issue = 2|pages = 490–530|s2cid = 218466166 }}</ref>

Autoencoders found use in more demanding contexts such as [[medical imaging]] where they have been used for [[image denoising]]<ref>{{Cite book|last=Gondara|first=Lovedeep|title=2016 IEEE 16th International Conference on Data Mining Workshops (ICDMW) |chapter=Medical Image Denoising Using Convolutional Denoising Autoencoders |date=December 2016|location=Barcelona, Spain|publisher=IEEE|pages=241–246|doi=10.1109/ICDMW.2016.0041|isbn=9781509059102|arxiv=1608.04667|bibcode=2016arXiv160804667G|s2cid=14354973}}</ref> as well as [[super-resolution]].<ref>{{Cite journal|last1=Zeng|first1=Kun|last2=Yu|first2=Jun|last3=Wang|first3=Ruxin|last4=Li|first4=Cuihua|last5=Tao|first5=Dacheng|s2cid=20787612|date=January 2017|title=Coupled Deep Autoencoder for Single Image Super-Resolution|journal=IEEE Transactions on Cybernetics|volume=47|issue=1|pages=27–37|doi=10.1109/TCYB.2015.2501373|pmid=26625442|issn=2168-2267}}</ref><ref>{{cite book |last1=Tzu-Hsi |first1=Song |last2=Sanchez |first2=Victor |last3=Hesham |first3=EIDaly |last4=Nasir M. |first4=Rajpoot |title=2017 IEEE 14th International Symposium on Biomedical Imaging (ISBI 2017) |chapter=Hybrid deep autoencoder with Curvature Gaussian for detection of various types of cells in bone marrow trephine biopsy images |date=2017 |pages=1040–1043 |doi=10.1109/ISBI.2017.7950694 |isbn=978-1-5090-1172-8 |s2cid=7433130 }}</ref> In image-assisted diagnosis, experiments have applied autoencoders for [[breast cancer]] detection<ref>{{cite journal |last1=Xu |first1=Jun |last2=Xiang |first2=Lei |last3=Liu |first3=Qingshan |last4=Gilmore |first4=Hannah |last5=Wu |first5=Jianzhong |last6=Tang |first6=Jinghai |last7=Madabhushi |first7=Anant |title=Stacked Sparse Autoencoder (SSAE) for Nuclei Detection on Breast Cancer Histopathology Images |journal=IEEE Transactions on Medical Imaging |date=January 2016 |volume=35 |issue=1 |pages=119–130 |doi=10.1109/TMI.2015.2458702 |pmid=26208307 |pmc=4729702 }}</ref> and for modelling the relation between the cognitive decline of [[Alzheimer's disease]] and the latent features of an autoencoder trained with [[MRI]].<ref>{{cite journal |last1=Martinez-Murcia |first1=Francisco J. |last2=Ortiz |first2=Andres |last3=Gorriz |first3=Juan M. |last4=Ramirez |first4=Javier |last5=Castillo-Barnes |first5=Diego |s2cid=195187846 |title=Studying the Manifold Structure of Alzheimer's Disease: A Deep Learning Approach Using Convolutional Autoencoders |journal=IEEE Journal of Biomedical and Health Informatics |volume=24 |issue=1 |pages=17–26 |doi=10.1109/JBHI.2019.2914970 |pmid=31217131 |date=2020 |doi-access=free |hdl=10630/28806 |hdl-access=free }}</ref>

=== Drug discovery ===
In 2019 molecules generated with variational autoencoders were validated experimentally in mice.<ref>{{cite journal |last1=Zhavoronkov |first1=Alex|s2cid=201716327|date=2019|title=Deep learning enables rapid identification of potent DDR1 kinase inhibitors |journal=Nature Biotechnology |volume=37|issue=9|pages=1038–1040|doi=10.1038/s41587-019-0224-x |pmid=31477924}}</ref><ref>{{cite magazine |last1=Gregory |first1=Barber |title=A Molecule Designed By AI Exhibits 'Druglike' Qualities |url=https://www.wired.com/story/molecule-designed-ai-exhibits-druglike-qualities/ |magazine=Wired}}</ref>

=== Popularity prediction ===
Recently, a stacked autoencoder framework produced promising results in predicting popularity of social media posts,<ref>{{cite book |doi=10.1109/CSCITA.2017.8066548|chapter=Predicting the popularity of instagram posts for a lifestyle magazine using deep learning|title=2017 2nd IEEE International Conference on Communication Systems, Computing and IT Applications (CSCITA)|pages=174–177|date=2017|last1=De|first1=Shaunak|last2=Maity|first2=Abhishek|last3=Goel|first3=Vritti|last4=Shitole|first4=Sanjay|last5=Bhattacharya|first5=Avik|s2cid=35350962|isbn=978-1-5090-4381-1}}</ref> which is helpful for online advertising strategies.

=== Machine translation ===
Autoencoders have been applied to [[machine translation]], which is usually referred to as [[neural machine translation]] (NMT).<ref>{{cite arXiv |eprint=1409.1259|last1=Cho|first1=Kyunghyun|author2=Bart van Merrienboer|last3=Bahdanau|first3=Dzmitry|last4=Bengio|first4=Yoshua|title=On the Properties of Neural Machine Translation: Encoder-Decoder Approaches|class=cs.CL|date=2014}}</ref><ref>{{cite arXiv |eprint=1409.3215|last1=Sutskever|first1=Ilya|last2=Vinyals|first2=Oriol|last3=Le|first3=Quoc V.|title=Sequence to Sequence Learning with Neural Networks|class=cs.CL|date=2014}}</ref> Unlike traditional autoencoders, the output does not match the input - it is in another language. In NMT, texts are treated as sequences to be encoded into the learning procedure, while on the decoder side sequences in the target language(s) are generated. [[Language]]-specific autoencoders incorporate further [[linguistic]] features into the learning procedure, such as Chinese decomposition features.<ref>{{cite arXiv |eprint=1805.01565|last1=Han|first1=Lifeng|last2=Kuang|first2=Shaohui|title=Incorporating Chinese Radicals into Neural Machine Translation: Deeper Than Character Level|class=cs.CL|date=2018}}</ref> Machine translation is rarely still done with autoencoders, due to the availability of more effective [[Transformer (machine learning model)|transformer]] networks.

==See also==
* [[Representation learning]]
* [[Sparse dictionary learning]]
* [[Deep learning]]

==References==
{{Reflist|30em}}

[[Category:Neural network architectures]]
[[Category:Unsupervised learning]]
[[Category:Dimension reduction]]

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