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((Short description|Class of artificial neural network))
((Machine learning|Artificial neural network))
[[File:Restricted Boltzmann machine.svg|thumb|Diagram of a restricted Boltzmann machine with three visible units and four hidden units (no bias units).]]
A '''restricted Boltzmann machine''' ('''RBM''') (also called a '''restricted Sherrington–Kirkpatrick model with external field''' or '''restricted stochastic Ising–Lenz–Little model''') is a [[generative model|generative]] [[stochastic neural network|stochastic]] [[artificial neural network]] that can learn a [[probability distribution]] over its set of inputs.<ref>((citation|title=Solvable Model of a Spin-Glass|number=35|year=1975|author1= Sherrington, David|author2=Kirkpatrick, Scott|journal=Physical Review Letters|volume=35|pages=1792–1796|doi=10.1103/PhysRevLett.35.1792|bibcode=1975PhRvL..35.1792S))</ref>

RBMs were initially proposed under the name '''Harmonium''' by [[Paul Smolensky]] in 1986,<ref>((cite book
|last=Smolensky
|first=Paul
|editor1-first=David E.
|editor1-last=Rumelhart
|editor2-first=James L.
|editor2-last=McLelland
|title=Parallel Distributed Processing: Explorations in the Microstructure of Cognition, Volume 1: Foundations
|publisher=MIT Press
|year=1986
|pages=[https://archive.org/details/paralleldistribu00rume/page/194 194–281]
|chapter=Chapter 6: Information Processing in Dynamical Systems: Foundations of Harmony Theory
|chapter-url=https://stanford.edu/~jlmcc/papers/PDP/Volume%201/Chap6_PDP86.pdf
|isbn=0-262-68053-X
|title-link=Connectionism
))</ref> and rose to prominence after [[Geoffrey Hinton]] and collaborators used fast learning algorithms for them in the mid-2000s. RBMs have found applications in [[dimensionality reduction]],<ref>((Cite journal | last1 = Hinton | first1 = G. E. | last2 = Salakhutdinov | first2 = R. R. | title = Reducing the Dimensionality of Data with Neural Networks | doi = 10.1126/science.1127647 | journal = Science | volume = 313 | issue = 5786 | pages = 504–507 | year = 2006 | pmid = 16873662 | url = http://www.cs.toronto.edu/~hinton/science.pdf | bibcode = 2006Sci...313..504H | s2cid = 1658773 | access-date = 2015-12-02 | archive-date = 2015-12-23 | archive-url = https://web.archive.org/web/20151223152006/http://www.cs.toronto.edu/~hinton/science.pdf | url-status = dead ))</ref> [[statistical classification|classification]],<ref>((Cite conference | last1 = Larochelle | first1 = H. | last2 = Bengio | first2 = Y. | doi = 10.1145/1390156.1390224 | title = Classification using discriminative restricted Boltzmann machines | conference = Proceedings of the 25th international conference on Machine learning - ICML '08 | pages = 536 | year = 2008 | isbn = 9781605582054 | url = http://machinelearning.org/archive/icml2008/papers/601.pdf))</ref> [[collaborative filtering]],<ref name="softCF">((Cite conference | doi = 10.1145/1273496.1273596| title = Restricted Boltzmann machines for collaborative filtering| conference = Proceedings of the 24th international conference on Machine learning - ICML '07| pages = 791| year = 2007| last1 = Salakhutdinov | first1 = R. | last2 = Mnih | first2 = A. | last3 = Hinton | first3 = G. | isbn = 9781595937933))</ref> [[feature learning]],<ref name="coates2011">((cite conference
|last1 = Coates
|first1 = Adam
|last2 = Lee
|first2 = Honglak
|last3 = Ng
|first3 = Andrew Y.
|title = An analysis of single-layer networks in unsupervised feature learning
|conference = International Conference on Artificial Intelligence and Statistics (AISTATS)
|year = 2011
|url = http://cs.stanford.edu/~acoates/papers/coatesleeng_aistats_2011.pdf
|access-date = 2014-12-19
|archive-date = 2014-12-20
|archive-url = https://web.archive.org/web/20141220030058/http://cs.stanford.edu/~acoates/papers/coatesleeng_aistats_2011.pdf
|url-status = dead
))</ref> [[topic model]]ling<ref name="softTM">Ruslan Salakhutdinov and Geoffrey Hinton (2010). [http://books.nips.cc/papers/files/nips22/NIPS2009_0817.pdf Replicated softmax: an undirected topic model] ((Webarchive|url=https://web.archive.org/web/20120525063031/http://books.nips.cc/papers/files/nips22/NIPS2009_0817.pdf |date=2012-05-25 )). ''[[Neural Information Processing Systems]]'' '''23'''.</ref> and even [[Many-body problem|many body quantum mechanics]].<ref>((Cite journal|last1=Carleo|first1=Giuseppe|last2=Troyer|first2=Matthias|date=2017-02-10|title=Solving the quantum many-body problem with artificial neural networks|journal=Science|language=en|volume=355|issue=6325|pages=602–606|doi=10.1126/science.aag2302|pmid=28183973|issn=0036-8075|arxiv=1606.02318|bibcode=2017Sci...355..602C|s2cid=206651104))</ref><ref>((Cite journal|last1=Melko|first1=Roger G.|last2=Carleo|first2=Giuseppe|last3=Carrasquilla|first3=Juan|last4=Cirac|first4=J. Ignacio|date=September 2019|title=Restricted Boltzmann machines in quantum physics|journal=Nature Physics|language=en|volume=15|issue=9|pages=887–892|doi=10.1038/s41567-019-0545-1|bibcode=2019NatPh..15..887M|issn=1745-2481|doi-access=|s2cid=256704838 ))</ref> They can be trained in either [[supervised learning|supervised]] or [[unsupervised learning|unsupervised]] ways, depending on the task.

As their name implies, RBMs are a variant of [[Boltzmann machine]]s, with the restriction that their [[Artificial neural network#Neuron|neurons]] must form a [[bipartite graph]]: 
a pair of nodes from each of the two groups of units (commonly referred to as the "visible" and "hidden" units respectively) may have a symmetric connection between them; and there are no connections between nodes within a group. By contrast, "unrestricted" Boltzmann machines may have connections between [[Artificial neural network#Hidden units|hidden units]]. This restriction allows for more efficient training [[algorithms]] than are available for the general class of Boltzmann machines, in particular the [[Gradient descent|gradient-based]] '''contrastive divergence''' algorithm.<ref name="oncd">Miguel Á. Carreira-Perpiñán and Geoffrey Hinton (2005). [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.221.8829&rep=rep1&type=pdf#page=42 On contrastive divergence learning]. ''Artificial Intelligence and Statistics''.</ref>

Restricted Boltzmann machines can also be used in [[deep learning]] networks. In particular, [[deep belief network]]s can be formed by "stacking" RBMs and optionally fine-tuning the resulting deep network with [[gradient descent]] and [[backpropagation]].<ref>((Cite journal 
| last1 = Hinton | first1 = G. 
| title = Deep belief networks 
| doi = 10.4249/scholarpedia.5947 
 |doi-access= free| journal = Scholarpedia 
| volume = 4 
| issue = 5 
| pages = 5947 
| year = 2009 
| bibcode = 2009SchpJ...4.5947H))</ref>

==Structure==
The standard type of RBM has binary-valued ([[Boolean algebra|Boolean]]) hidden and visible units, and consists of a [[Matrix (mathematics)|matrix]] of weights <math>W</math> of size <math>m\times n</math>. Each weight element <math>(w_{i,j})</math> of the matrix is associated with the connection between the visible (input) unit <math>v_i</math> and the hidden unit <math>h_j</math>. In addition, there are bias weights (offsets) <math>a_i</math> for <math>v_i</math> and <math>b_j</math> for <math>h_j</math>. Given the weights and biases, the ''energy'' of a configuration (pair of boolean vectors) ((math|(''v'',''h''))) is defined as

:<math>E(v,h) = -\sum_i a_i v_i - \sum_j b_j h_j -\sum_i \sum_j v_i w_{i,j} h_j</math>

or, in matrix notation,

:<math>E(v,h) = -a^{\mathrm{T)) v - b^{\mathrm{T)) h -v^{\mathrm{T)) W h.</math>

This energy function is analogous to that of a [[Hopfield network]]. As with general Boltzmann machines, the [[joint probability distribution]] for the visible and hidden vectors is defined in terms of the energy function as follows,<ref name="guide">Geoffrey Hinton (2010). ''[http://www.cs.toronto.edu/~hinton/absps/guideTR.pdf A Practical Guide to Training Restricted Boltzmann Machines]''. UTML TR 2010–003, University of Toronto.</ref>

:<math>P(v,h) = \frac{1}{Z} e^{-E(v,h)}</math>

where <math>Z</math> is a [[Partition function (mathematics)|partition function]] defined as the sum of <math>e^{-E(v,h)}</math> over all possible configurations, which can be interpreted as a [[normalizing constant]] to ensure that the probabilities sum to 1. The [[Marginal distribution|marginal probability]] of a visible vector is the sum of <math>P(v,h)</math> over all possible hidden layer configurations,<ref name="guide"/>

:<math>P(v) = \frac{1}{Z} \sum_{\{h\)) e^{-E(v,h)}</math>,

and vice versa. Since the underlying graph structure of the RBM is [[Bipartite graph|bipartite]] (meaning there are no intra-layer connections), the hidden unit activations are [[Conditional independence|mutually independent]] given the visible unit activations. Conversely, the visible unit activations are mutually independent given the hidden unit activations.<ref name="oncd"/> That is, for ''m'' visible units and ''n'' hidden units, the [[conditional probability]] of a configuration of the visible units ((mvar|v)), given a configuration of the hidden units ((mvar|h)), is

:<math>P(v|h) = \prod_{i=1}^m P(v_i|h)</math>.

Conversely, the conditional probability of ((mvar|h)) given ((mvar|v)) is

:<math>P(h|v) = \prod_{j=1}^n P(h_j|v)</math>.

The individual activation probabilities are given by

:<math>P(h_j=1|v) = \sigma \left(b_j + \sum_{i=1}^m w_{i,j} v_i \right)</math> and <math>\,P(v_i=1|h) = \sigma \left(a_i + \sum_{j=1}^n w_{i,j} h_j \right)</math>

where <math>\sigma</math> denotes the [[Logistic function|logistic sigmoid]].

The visible units of Restricted Boltzmann Machine can be [[Multinomial distribution|multinomial]], although the hidden units are [[Bernoulli distribution|Bernoulli]].((clarify|date=April 2021|reason=What is meant by Bernoulli in this context?)) In this case, the logistic function for visible units is replaced by the [[softmax function]]

:<math>P(v_i^k = 1|h) = \frac{\exp(a_i^k + \Sigma_j W_{ij}^k h_j)} {\Sigma_{k'=1}^K \exp(a_i^{k'} + \Sigma_j W_{ij}^{k'} h_j)}</math>

where ''K'' is the number of discrete values that the visible values have. They are applied in topic modeling,<ref name="softTM"/> and [[recommender system]]s.<ref name="softCF"/>

===Relation to other models===
Restricted Boltzmann machines are a special case of [[Boltzmann machine]]s and [[Markov random field]]s.<ref name="cdconvergence">((cite journal | first1 = Ilya | last1 = Sutskever | first2 = Tijmen | last2 = Tieleman | year = 2010 | title = On the convergence properties of contrastive divergence | journal = Proc. 13th Int'l Conf. On AI and Statistics (AISTATS) | url = http://machinelearning.wustl.edu/mlpapers/paper_files/AISTATS2010_SutskeverT10.pdf | url-status = dead | archive-url = https://web.archive.org/web/20150610230811/http://machinelearning.wustl.edu/mlpapers/paper_files/AISTATS2010_SutskeverT10.pdf | archive-date = 2015-06-10 ))</ref><ref name="RBMTutorial">Asja Fischer and Christian Igel. [http://image.diku.dk/igel/paper/TRBMAI.pdf  Training Restricted Boltzmann Machines: An Introduction] ((Webarchive|url=https://web.archive.org/web/20150610230447/http://image.diku.dk/igel/paper/TRBMAI.pdf# |date=2015-06-10 )). Pattern Recognition 47, pp. 25-39, 2014</ref>
Their [[graphical model]] corresponds to that of [[factor analysis]].<ref>((cite journal |author1=María Angélica Cueto |author2=Jason Morton |author3=Bernd Sturmfels |year=2010 |title=Geometry of the restricted Boltzmann machine |journal=Algebraic Methods in Statistics and Probability |volume=516 |publisher=American Mathematical Society |arxiv=0908.4425 |bibcode=2009arXiv0908.4425A ))</ref>

==Training algorithm==
Restricted Boltzmann machines are trained to maximize the product of probabilities assigned to some training set <math>V</math> (a matrix, each row of which is treated as a visible vector <math>v</math>),

:<math>\arg\max_W \prod_{v \in V} P(v)</math>

or equivalently, to maximize the [[Expected value|expected]] [[log probability]] of a training sample <math>v</math> selected randomly from <math>V</math>:<ref name="cdconvergence"/><ref name="RBMTutorial"/>

:<math>\arg\max_W \mathbb{E} \left[ \log P(v)\right]</math>

The algorithm most often used to train RBMs, that is, to optimize the weight matrix <math>W</math>, is the contrastive divergence (CD) algorithm due to [[Geoffrey Hinton|Hinton]], originally developed to train PoE ([[product of experts]]) models.<ref>Geoffrey Hinton (1999). [http://www.gatsby.ucl.ac.uk/publications/papers/06-1999.pdf Products of Experts]. ''ICANN 1999''.</ref><ref>((Cite journal | last1 = Hinton | first1 = G. E. | title = Training Products of Experts by Minimizing Contrastive Divergence | doi = 10.1162/089976602760128018 | journal = Neural Computation | volume = 14 | issue = 8 | pages = 1771–1800 | year = 2002 | pmid =  12180402| s2cid = 207596505 | url = http://www.cs.toronto.edu/~hinton/absps/tr00-004.pdf))</ref>
The algorithm performs [[Gibbs sampling]] and is used inside a [[gradient descent]] procedure (similar to the way backpropagation is used inside such a procedure when training feedforward neural nets) to compute weight update.

The basic, single-step contrastive divergence (CD-1) procedure for a single sample can be summarized as follows:

# Take a training sample ((mvar|v)), compute the probabilities of the hidden units and sample a hidden activation vector ((mvar|h)) from this probability distribution.
# Compute the [[outer product]] of ((mvar|v)) and ((mvar|h)) and call this the ''positive gradient''.
# From ((mvar|h)), sample a reconstruction ((mvar|v')) of the visible units, then resample the hidden activations ((mvar|h')) from this. (Gibbs sampling step)
# Compute the [[outer product]] of ((mvar|v')) and ((mvar|h')) and call this the ''negative gradient''.
# Let the update to the weight matrix <math>W</math> be the positive gradient minus the negative gradient, times some learning rate: <math>\Delta W = \epsilon (vh^\mathsf{T} - v'h'^\mathsf{T})</math>.
# Update the biases ((mvar|a)) and ((mvar|b)) analogously: <math>\Delta a = \epsilon (v - v')</math>, <math>\Delta b = \epsilon (h - h')</math>.

A Practical Guide to Training RBMs written by Hinton can be found on his homepage.<ref name="guide"/>

==Stacked Restricted Boltzmann Machine==
((Technical|section|date=August 2023))
((More citations needed section|date=August 2023))
((See also|Deep belief network))

* The difference between the Stacked Restricted Boltzmann Machines and RBM is that RBM has lateral connections within a layer that are prohibited to make analysis tractable. On the other hand, the Stacked Boltzmann consists of a combination of an unsupervised three-layer network with symmetric weights and a supervised fine-tuned top layer for recognizing three classes. 
* The usage of Stacked Boltzmann is to [[Natural-language understanding|understand Natural languages]], [[Document retrieval|retrieve documents]], image generation, and classification. These functions are trained with unsupervised pre-training and/or supervised fine-tuning. Unlike the undirected symmetric top layer, with a two-way unsymmetric layer for connection for RBM. The restricted Boltzmann's connection is three-layers with asymmetric weights, and two networks are combined into one.
* Stacked Boltzmann does share similarities with RBM, the neuron for Stacked Boltzmann is a stochastic binary Hopfield neuron, which is the same as the Restricted Boltzmann Machine. The energy from both Restricted Boltzmann and RBM is given by Gibb's probability measure: <math>E = -\frac12\sum_{i,j}{w_{ij}{s_i}{s_j))+\sum_i{\theta_i}{s_i}</math>. The training process of Restricted Boltzmann is similar to RBM. Restricted Boltzmann train one layer at a time and approximate equilibrium state with a 3-segment pass, not performing back propagation. Restricted Boltzmann uses both supervised and unsupervised on different RBM for pre-training for classification and recognition. The training uses contrastive divergence with Gibbs sampling: Δw<sub>ij</sub> = e*(p<sub>ij</sub> - p'<sub>ij</sub>)
* The restricted Boltzmann's strength is it performs a non-linear transformation so it's easy to expand, and can give a hierarchical layer of features. The Weakness is that it has complicated calculations of integer and real-valued neurons. It does not follow the gradient of any function, so the approximation of Contrastive divergence to maximum likelihood is improvised.<ref name="guide"/>

== Literature ==
* ((Citation|last1=Fischer|first1=Asja|title=An Introduction to Restricted Boltzmann Machines|date=2012|work=Progress in Pattern Recognition, Image Analysis, Computer Vision, and Applications|pages=14–36|place=Berlin, Heidelberg|publisher=Springer Berlin Heidelberg|last2=Igel|first2=Christian|series=Lecture Notes in Computer Science |volume=7441 |doi=10.1007/978-3-642-33275-3_2 |isbn=978-3-642-33274-6 |doi-access=free))

==See also==
* [[Autoencoder]]
* [[Helmholtz machine]]

==References==
((Reflist|30em))

==External links==
* [http://blog.echen.me/2011/07/18/introduction-to-restricted-boltzmann-machines/ Introduction to Restricted Boltzmann Machines]. Edwin Chen's blog, July 18, 2011.
* ((Cite web |url=https://deeplearning4j.org/restrictedboltzmannmachine.html |title=A Beginner's Guide to Restricted Boltzmann Machines |access-date=November 15, 2018 |archive-url=https://web.archive.org/web/20170211042953/https://deeplearning4j.org/restrictedboltzmannmachine.html |archive-date=February 11, 2017 |url-status=bot: unknown |df=mdy-all )). [[Deeplearning4j]] Documentation
* ((Cite web |url=http://deeplearning4j.org/understandingRBMs.html |title=Understanding RBMs |access-date=December 29, 2014 |archive-url=https://web.archive.org/web/20160920122139/http://deeplearning4j.org/understandingRBMs.html |archive-date=September 20, 2016 |url-status=dead |df=mdy-all )). Deeplearning4j Documentation
* [[Python (programming language)|Python]] [https://github.com/AmazaspShumik/sklearn-bayes/blob/master/skbayes/decomposition_models/rbm.py implementation] of Bernoulli RBM and [https://github.com/AmazaspShumik/sklearn-bayes/blob/master/ipython_notebooks_tutorials/decomposition_models/rbm_demo.ipynb tutorial] 
* [https://github.com/swirepe/SimpleRBM SimpleRBM] is a very small RBM code (24kB) useful for you to learn about how RBMs learn and work.
* [[Julia (programming language)|Julia]] implementation of Restricted Boltzmann machines: https://github.com/cossio/RestrictedBoltzmannMachines.jl

[[Category:Neural network architectures]]
[[Category:Stochastic models]]
[[Category:Supervised learning]]
[[Category:Unsupervised learning]]

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