Autoencoder
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An autoencoder, autoassociator or Diabolo network^{[1]}^{:19} is an artificial neural network used for learning efficient codings.^{[2]}^{[3]} The aim of an autoencoder is to learn a representation (encoding) for a set of data, typically for the purpose of dimensionality reduction. Recently, the autoencoder concept has become more widely used for learning generative models of data.^{[4]}^{[5]}
Contents
Structure
Architecturally, the simplest form of an autoencoder is a feedforward, nonrecurrent neural net which is very similar to the multilayer perceptron (MLP), with an input layer, an output layer and one or more hidden layers connecting them. The differences between autoencoders and MLPs, though, are that in an autoencoder, the output layer has the same number of nodes as the input layer, and that, instead of being trained to predict the target value given inputs , autoencoders are trained to reconstruct their own inputs . Therefore, autoencoders are unsupervised learning models.
An autoencoder always consists of two parts, the encoder and the decoder, which can be defined as transitions and , such that:
In the simplest case, where there is one hidden layer, an autoencoder takes the input and maps it onto :
This is usually referred to as code or latent variables (latent representation). Here, is an elementwise activation function such as a sigmoid function or a rectified linear unit. After that, is mapped onto the reconstruction of the same shape as :
Autoencoders are also trained to minimise reconstruction errors (such as squared errors):
If the feature space has less dimensionality than the input space , then the feature vector can be regarded as a compressed representation of the input . If the hidden layers are larger than the input layer, an autoencoder can potentially learn the identity function and become useless. However, experimental results have shown that autoencoders might still learn useful features in these cases.^{[1]}^{:19}
Different Implementations
Various techniques exist to prevent autoencoders from learning the identity function and to improve their ability to capture important information and learn richer representations.
Denoising Autoencoder
Denoising autoencoders take a partially corrupted input whilst training to recover the original undistorted input. This technique has been introduced with a specific approach to good representation.^{[6]} A good representation is one that can be obtained robustly from a corrupted input and that will be useful for recovering the corresponding clean input. This definition contains the following implicit assumptions:
 The higher level representations are relatively stable and robust to the corruption of the input;
 It is necessary to extract features that are useful for representation of the input distribution.
To train an autoencoder to denoise data, it is necessary to perform preliminary stochastic mapping in order to corrupt the data and use as input for a normal autoencoder, with the only exception being that the loss should be still computed for the initial input instead of .
Variational Autoencoder (VAE)
Variational autoencoder models inherit autoencoder architecture, but make strong assumptions concerning the distribution of latent variables. They use variational approach for latent representation learning, which results in an additional loss component and specific training algorithm called Stochastic Gradient Variational Bayes (SGVB).^{[4]} It assumes that the data is generated by a directed graphical model and that the encoder is learning an approximation to the posterior distribution where and denote the parameters of the encoder (recognition model) and decoder (generative model). The objective of the variational autoencoder in this case has the following form:
Here, stands for the Kullback–Leibler divergence of the approximate posterior from the prior, and the second term is an expected negative reconstruction error. The prior over the latent variables is set to be the centred isotropic multivariate Gaussian .
Relationship with Other Methods
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).^{[7]}
Training
The training algorithm for an autoencoder can be summarized as
 For each input x,
 Do a feedforward pass to compute activations at all hidden layers, then at the output layer to obtain an output
 Measure the deviation of from the input (typically using squared error),
 Backpropagate the error through the net and perform weight updates.
An autoencoder is often trained using one of the many variants of backpropagation (such as conjugate gradient method, steepest descent, etc.). Though these are often reasonably effective, there are fundamental problems with the use of backpropagation to train networks with many hidden layers. Once errors are backpropagated to the first few layers, they become minuscule and insignificant. This means that the network will almost always learn to reconstruct the average of all the training data.^{[citation needed]} Though more advanced backpropagation methods (such as the conjugate gradient method) can solve this problem to a certain extent, they still result in a very slow learning process and poor solutions. This problem can be remedied by using initial weights that approximate the final solution. The process of finding these initial weights is often referred to as pretraining.
Geoffrey Hinton developed a pretraining technique for training manylayered "deep" autoencoders. This method involves treating each neighboring set of two layers as a restricted Boltzmann machine so that the pretraining approximates a good solution, then using a backpropagation technique to finetune the results.^{[8]}
References
 ↑ ^{1.0} ^{1.1} Template:Cite doi/10.1561.2F2200000006
 ↑ Modeling word perception using the Elman network, Liou, C.Y., Huang, J.C. and Yang, W.C., Neurocomputing, Volume 71, 3150–3157 (2008), doi:10.1016/j.neucom.2008.04.030
 ↑ Autoencoder for Words, Liou, C.Y., Cheng, C.W., Liou, J.W., and Liou, D.R., Neurocomputing, Volume 139, 84–96 (2014), doi:10.1016/j.neucom.2013.09.055
 ↑ ^{4.0} ^{4.1} AutoEncoding Variational Bayes, Kingma, D.P. and Welling, M., ArXiv eprints, 2013 arxiv
.org /abs /1312 .6114  ↑ Generating Faces with Torch, Boesen A., Larsen L. and Sonderby S.K., 2015 torch
.ch /blog /2015 /11 /13 /gan .html  ↑ Vincent, Pascal; Larochelle, Hugo; Lajoie, Isabelle; Bengio, Yoshua; Manzagol, PierreAntoine (2010). "Stacked Denoising Autoencoders: Learning Useful Representations in a Deep Network with a Local Denoising Criterion". The Journal of Machine Learning Research. 11: 3371–3408.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ Bourlard, H.; Kamp, Y. (1988). "Autoassociation by multilayer perceptrons and singular value decomposition". Biological Cybernetics. 59 (4–5): 291–294. doi:10.1007/BF00332918. PMID 3196773.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 ↑ Reducing the Dimensionality of Data with Neural Networks (Science, 28 July 2006, Hinton & Salakhutdinov)