2855:
142:
20:
5240:
2702:, it is equivalent to maximizing the log-likelihood of the data. Therefore, the training procedure performs gradient ascent on the log-likelihood of the observed data. This is in contrast to the EM algorithm, where the posterior distribution of the hidden nodes must be calculated before the maximization of the expected value of the complete data likelihood during the M-step.
3448:
2869:
Although learning is impractical in general
Boltzmann machines, it can be made quite efficient in a restricted Boltzmann machine (RBM) which does not allow intralayer connections between hidden units and visible units, i.e. there is no connection between visible to visible and hidden to hidden units.
3894:
The explicit analogy drawn with statistical mechanics in the
Boltzmann Machine formulation led to the use of terminology borrowed from physics (e.g., "energy" rather than "harmony"), which became standard in the field. The widespread adoption of this terminology may have been encouraged by the fact
2858:
Graphical representation of a restricted
Boltzmann machine. The four blue units represent hidden units, and the three red units represent visible states. In restricted Boltzmann machines there are only connections (dependencies) between hidden and visible units, and none between units of the same
3135:
3663:
However, the slow speed of DBMs limits their performance and functionality. Because exact maximum likelihood learning is intractable for DBMs, only approximate maximum likelihood learning is possible. Another option is to use mean-field inference to estimate data-dependent expectations and
3668:(MCMC). This approximate inference, which must be done for each test input, is about 25 to 50 times slower than a single bottom-up pass in DBMs. This makes joint optimization impractical for large data sets, and restricts the use of DBMs for tasks such as feature representation.
3157:
1848:
To train the network so that the chance it will converge to a global state according to an external distribution over these states, the weights must be set so that the global states with the highest probabilities get the lowest energies. This is done by training.
3542:
1479:
1387:
3631:
2870:
After training one RBM, the activities of its hidden units can be treated as data for training a higher-level RBM. This method of stacking RBMs makes it possible to train many layers of hidden units efficiently and is one of the most common
2348:). The other is the "negative" phase where the network is allowed to run freely, i.e. only the input nodes have their state determined by external data, but the output nodes are allowed to float. The gradient with respect to a given weight,
2827:
Unfortunately, Boltzmann machines experience a serious practical problem, namely that it seems to stop learning correctly when the machine is scaled up to anything larger than a trivial size. This is due to important effects, specifically:
2835:
connection strengths are more plastic when the connected units have activation probabilities intermediate between zero and one, leading to a so-called variance trap. The net effect is that noise causes the connection strengths to follow a
860:
1828:
The network runs by repeatedly choosing a unit and resetting its state. After running for long enough at a certain temperature, the probability of a global state of the network depends only upon that global state's energy, according to a
345:
2683:, biologically) does not need information about anything other than the two neurons it connects. This is more biologically realistic than the information needed by a connection in many other neural network training algorithms, such as
1649:
1056:
1564:
2819:
Theoretically the
Boltzmann machine is a rather general computational medium. For instance, if trained on photographs, the machine would theoretically model the distribution of photographs, and could use that model to, for example,
4095:
1298:
1202:
3697:, when models are tested on data with both image-text modalities or with single modality. Multimodal deep Boltzmann machines are also able to predict missing modalities given the observed ones with reasonably good precision.
2158:
2961:
1784:
2809:
2491:
2320:
Boltzmann machine training involves two alternating phases. One is the "positive" phase where the visible units' states are clamped to a particular binary state vector sampled from the training set (according to
23:
A graphical representation of an example
Boltzmann machine. Each undirected edge represents dependency. In this example there are 3 hidden units and 4 visible units. This is not a restricted Boltzmann machine.
922:
1833:, and not on the initial state from which the process was started. This means that log-probabilities of global states become linear in their energies. This relationship is true when the machine is "at
3443:{\displaystyle p({\boldsymbol {\nu }})={\frac {1}{Z}}\sum _{h}e^{\sum _{ij}W_{ij}^{(1)}\nu _{i}h_{j}^{(1)}+\sum _{jl}W_{jl}^{(2)}h_{j}^{(1)}h_{l}^{(2)}+\sum _{lm}W_{lm}^{(3)}h_{l}^{(2)}h_{m}^{(3)}},}
4551:
Chen, Richard J.; Lu, Ming Y.; Williamson, Drew F. K.; Chen, Tiffany Y.; Lipkova, Jana; Noor, Zahra; Shaban, Muhammad; Shady, Maha; Williams, Mane; Joo, Bumjin; Mahmood, Faisal (8 August 2022).
2956:
3895:
that its use led to the adoption of a variety of concepts and methods from statistical mechanics. The various proposals to use simulated annealing for inference were apparently independent.
3770:, the visible units (input) are real-valued. The difference is in the hidden layer, where each hidden unit has a binary spike variable and a real-valued slab variable. A spike is a discrete
1837:", meaning that the probability distribution of global states has converged. Running the network beginning from a high temperature, its temperature gradually decreases until reaching a
1857:
The units in the
Boltzmann machine are divided into 'visible' units, V, and 'hidden' units, H. The visible units are those that receive information from the 'environment', i.e. the
3685:
Multimodal deep
Boltzmann machines are successfully used in classification and missing data retrieval. The classification accuracy of multimodal deep Boltzmann machine outperforms
3456:
3660:, they pursue the inference and training procedure in both directions, bottom-up and top-down, which allow the DBM to better unveil the representations of the input structures.
495:
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2017:
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628:
381:
173:
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This learning rule is biologically plausible because the only information needed to change the weights is provided by "local" information. That is, the connection (
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1086:
450:
229:
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the required time order to collect equilibrium statistics grows exponentially with the machine's size, and with the magnitude of the connection strengths
4924:
Hofstadter, Douglas R. (1988). "A Non-Deterministic
Approach to Analogy, Involving the Ising Model of Ferromagnetism". In Caianiello, Eduardo R. (ed.).
3130:{\displaystyle {\boldsymbol {h}}^{(1)}\in \{0,1\}^{F_{1}},{\boldsymbol {h}}^{(2)}\in \{0,1\}^{F_{2}},\ldots ,{\boldsymbol {h}}^{(L)}\in \{0,1\}^{F_{L}}}
2049:
3656:, using limited, labeled data to fine-tune the representations built using a large set of unlabeled sensory input data. However, unlike DBNs and deep
4387:
1707:
145:
A graphical representation of a
Boltzmann machine with a few weights labeled. Each undirected edge represents dependency and is weighted with weight
2711:
1841:
at a lower temperature. It then may converge to a distribution where the energy level fluctuates around the global minimum. This process is called
5132:
Kothari P (2020): https://www.forbes.com/sites/tomtaulli/2020/02/02/coronavirus-can-ai-artificial-intelligence-make-a-difference/?sh=1eca51e55817
2384:
4529:
5256:
5261:
5194:
4155:
4463:
4992:
871:
90:, but if the connectivity is properly constrained, the learning can be made efficient enough to be useful for practical problems.
3706:
3866:
The original contribution in applying such energy-based models in cognitive science appeared in papers by Hinton and
Sejnowski.
4823:. IEEE Conference on Computer Vision and Pattern Recognition (CVPR). Washington, D.C.: IEEE Computer Society. pp. 448–453.
2691:
5551:
5187:
4273:
2893:
3798:
175:. In this example there are 3 hidden units (blue) and 4 visible units (white). This is not a restricted Boltzmann machine.
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2914:
4795:
4664:
4933:
4613:
4438:
4700:
4618:
2699:
2020:
5455:
3869:
The seminal publication by John Hopfield connected physics and statistical mechanics, mentioning spin glasses.
3720:
3657:
3537:{\displaystyle {\boldsymbol {h}}=\{{\boldsymbol {h}}^{(1)},{\boldsymbol {h}}^{(2)},{\boldsymbol {h}}^{(3)}\}}
1112:. We then rearrange terms and consider that the probabilities of the unit being on and off must sum to one:
82:. Boltzmann machines with unconstrained connectivity have not been proven useful for practical problems in
5224:
5085:
5026:
3927:
3790:
3786:
3767:
3759:
3748:
3741:
3138:
2864:
4402:
3641:), while lower layers form a directed generative model. In a DBM all layers are symmetric and undirected.
4163:
3690:
2877:
An extension to the restricted Boltzmann machine allows using real valued data rather than binary data.
1474:{\displaystyle -{\frac {\Delta E_{i}}{T}}=\ln \left({\frac {1-p_{\text{i=on}}}{p_{\text{i=on}}}}\right)}
5332:
5248:
5000:
Parallel Distributed Processing: Explorations in the Microstructure of Cognition. Volume 1: Foundations
1382:{\displaystyle {\frac {\Delta E_{i}}{T}}=\ln \left({\frac {p_{\text{i=on}}}{1-p_{\text{i=on}}}}\right)}
455:
5556:
3844:
3678:
1657:
180:
48:
4358:
2203:
is a function of the weights, since they determine the energy of a state, and the energy determines
5508:
5491:
5099:
5040:
3698:
3665:
3626:{\displaystyle \theta =\{{\boldsymbol {W}}^{(1)},{\boldsymbol {W}}^{(2)},{\boldsymbol {W}}^{(3)}\}}
703:
569:
5355:
5345:
5271:
4337:
3954:
Learning rule that uses conditional "local" information can be derived from the reversed form of
3915:
935:
that the energy of a state is proportional to the negative log probability of that state) gives:
522:
75:
4493:
Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics
4257:
4249:
2547:
2502:
855:{\displaystyle \Delta E_{i}=\sum _{j>i}w_{ij}\,s_{j}+\sum _{j<i}w_{ji}\,s_{j}+\theta _{i}}
5402:
5094:
5035:
4521:
3686:
1830:
932:
94:
2623:
2206:
1986:
1950:
1911:
1864:
340:{\displaystyle E=-\left(\sum _{i<j}w_{ij}\,s_{i}\,s_{j}+\sum _{i}\theta _{i}\,s_{i}\right)}
5287:
5210:
4435:
Proceedings of the Twelfth International Conference on Artificial Intelligence and Statistics
3648:, DBMs can learn complex and abstract internal representations of the input in tasks such as
1905:
1861:
is a set of binary vectors over the set V. The distribution over the training set is denoted
98:
5138:
4359:"Context-Dependent Pre-trained Deep Neural Networks for Large Vocabulary Speech Recognition"
4182:
2351:
2266:
603:
356:
148:
5430:
4847:
4760:
4212:
4124:
3633:
are the model parameters, representing visible-hidden and hidden-hidden interactions. In a
2324:
1644:{\displaystyle \exp \left(-{\frac {\Delta E_{i}}{T}}\right)={\frac {1}{p_{\text{i=on}}}}-1}
1064:
1051:{\displaystyle \Delta E_{i}=-k_{B}\,T\ln(p_{\text{i=off}})-(-k_{B}\,T\ln(p_{\text{i=on}}))}
428:
184:
126:
1559:{\displaystyle -{\frac {\Delta E_{i}}{T}}=\ln \left({\frac {1}{p_{\text{i=on}}}}-1\right)}
8:
5496:
5475:
5169:
3937:
3899:
3775:
3694:
3645:
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2617:
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1834:
64:
56:
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633:
74:
nature of their training algorithm (being trained by Hebb's rule), and because of their
5523:
5435:
5412:
5393:
5373:
5337:
5120:
5061:
4836:"Neural networks and physical systems with emergent collective computational abilities"
4751:
Sherrington, David; Kirkpatrick, Scott (1975-12-29). "Solvable Model of a Spin-Glass".
4600:
4302:
3957:
3881:
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3653:
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406:
386:
198:
122:
4878:
4835:
4724:
Mitchell, T; Beauchamp, J (1988). "Bayesian Variable Selection in Linear Regression".
3823:
In more general mathematical setting, the Boltzmann distribution is also known as the
5425:
5398:
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5299:
5292:
5266:
5112:
5077:
5065:
5053:
4939:
4929:
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4865:
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4230:
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3779:
1817:
1293:{\displaystyle {\frac {\Delta E_{i}}{T}}=\ln(p_{\text{i=on}})-\ln(1-p_{\text{i=on}})}
927:
Substituting the energy of each state with its relative probability according to the
102:
60:
3637:
only the top two layers form a restricted Boltzmann machine (which is an undirected
2019:
produced by the machine. The similarity of the two distributions is measured by the
1197:{\displaystyle {\frac {\Delta E_{i}}{T}}=\ln(p_{\text{i=on}})-\ln(p_{\text{i=off}})}
5420:
5124:
5104:
5045:
5018:
5003:
4962:
4873:
4855:
4768:
4737:
4733:
4582:
4564:
4312:
4261:
4220:
4172:
4132:
4090:{\displaystyle G'=\sum _{v}{P^{-}(v)\ln \left({\frac {P^{-}(v)}{P^{+}(v)}}\right)}}
3949:
3836:
3832:
3818:
3771:
3716:
2904:
2695:
2240:
1790:
928:
216:
118:
83:
79:
40:
5470:
2153:{\displaystyle G=\sum _{v}{P^{+}(v)\ln \left({\frac {P^{+}(v)}{P^{-}(v)}}\right)}}
5465:
5460:
5388:
5383:
5350:
5277:
5073:
5014:
4988:
4984:
4291:"On the Anatomy of MCMC-Based Maximum Likelihood Learning of Energy-Based Models"
4265:
4115:
Sherrington, David; Kirkpatrick, Scott (1975), "Solvable Model of a Spin-Glass",
3911:
3794:
3763:
3755:
3752:
3638:
2908:
2901:
2897:
2684:
110:
106:
5108:
4772:
4569:
4553:"Pan-cancer integrative histology-genomic analysis via multimodal deep learning"
4136:
5513:
5229:
5049:
4840:
Proceedings of the National Academy of Sciences of the United States of America
4177:
4153:
3888:
3873:
2854:
1900:
The distribution over global states converges as the Boltzmann machine reaches
1779:{\displaystyle p_{\text{i=on}}={\frac {1}{1+\exp(-{\frac {\Delta E_{i}}{T}})}}}
5174:
4966:
4331:
4225:
4200:
2804:{\displaystyle {\frac {\partial {G}}{\partial {\theta _{i}}}}=-{\frac {1}{R}}}
5545:
5501:
4943:
4910:
4869:
4780:
4614:"New AI technology integrates multiple data types to predict cancer outcomes"
4578:
4317:
4234:
3840:
3824:
3814:
2871:
2610:
188:
70:
Boltzmann machines are theoretically intriguing because of the locality and
5518:
5116:
5057:
4903:
The Copycat Project: An Experiment in Nondeterminism and Creative Analogies
4860:
4800:. 5th Annual Congress of the Cognitive Science Society. Rochester, New York
4596:
1858:
130:
4887:
3887:
Similar ideas (with a change of sign in the energy function) are found in
5327:
5322:
3943:
3903:
3860:
3733:
2837:
2676:
when the network is free-running is given by the Boltzmann distribution.
2486:{\displaystyle {\frac {\partial {G}}{\partial {w_{ij}}}}=-{\frac {1}{R}}}
1813:
220:
52:
5179:
5317:
4957:
Liou, C.-Y.; Lin, S.-L. (1989). "The other variant Boltzmann machine".
3856:
3828:
2821:
1820:
found in probability expressions in variants of the Boltzmann machine.
192:
114:
44:
4154:
Ackley, David H.; Hinton, Geoffrey E.; Sejnowski, Terrence J. (1985).
2874:
strategies. As each new layer is added the generative model improves.
5378:
2880:
One example of a practical RBM application is in speech recognition.
2587:
are both on when the machine is at equilibrium on the negative phase.
2542:
are both on when the machine is at equilibrium on the positive phase.
141:
87:
19:
4697:
Proceedings of the 28th International Conference on Machine Learning
4114:
2705:
Training the biases is similar, but uses only single node activity:
680:
The difference in the global energy that results from a single unit
133:
as energy are used as a starting point to define the learning task.
5282:
5019:"Training Products of Experts by Minimizing Contrastive Divergence"
4307:
3907:
3843:
the Boltzmann distribution is used in the sampling distribution of
3737:
865:
This can be expressed as the difference of energies of two states:
5447:
5146:
2680:
71:
3701:
brings a more interesting and powerful model for multimodality.
3793:
provides extra modeling capacity using additional terms in the
3710:
3702:
4522:"Harvard boffins build multimodal AI system to predict cancer"
4295:
Proceedings of the AAAI Conference on Artificial Intelligence
917:{\displaystyle \Delta E_{i}=E_{\text{i=off}}-E_{\text{i=on}}}
4288:
2888:
A deep Boltzmann machine (DBM) is a type of binary pairwise
4250:"Fast Teaching of Boltzmann Machines with Local Inhibition"
4688:
Courville, Aaron; Bergstra, James; Bengio, Yoshua (2011).
4648:
Courville, Aaron; Bergstra, James; Bengio, Yoshua (2011).
1092:
and is absorbed into the artificial notion of temperature
4819:
Hinton, Geoffrey E.; Sejnowski, Terrence J. (June 1983).
2859:
type (no hidden-hidden, nor visible-visible connections).
5139:"Restricted Boltzmann Machines: Introduction and Review"
3664:
approximate the expected sufficient statistics by using
5170:
Scholarpedia article by Hinton about Boltzmann machines
4687:
4647:
4483:
3898:
Ising models became considered to be a special case of
730:, assuming a symmetric matrix of weights, is given by:
215:
in a Boltzmann machine is identical in form to that of
4750:
4690:"Unsupervised Models of Images by Spike-and-Slab RBMs"
4357:
Yu, Dong; Dahl, George; Acero, Alex; Deng, Li (2011).
2907:. It is a network of symmetrically coupled stochastic
4794:
Hinton, Geoffery; Sejnowski, Terrence J. (May 1983).
3984:
3960:
3550:
3459:
3160:
2964:
2917:
2714:
2690:
The training of a Boltzmann machine does not use the
2662:
2626:
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2505:
2387:
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2327:
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2269:
2249:
2209:
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2052:
2029:
1989:
1953:
1914:
1867:
1798:
1710:
1687:
1660:
1573:
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1396:
1307:
1211:
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686:
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525:
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389:
359:
232:
201:
187:) defined for the overall network. Its units produce
151:
4550:
3671:
3137:. No connection links units of the same layer (like
4425:
4385:
3872:The idea of applying the Ising model with annealed
2951:{\displaystyle {\boldsymbol {\nu }}\in \{0,1\}^{D}}
1947:Our goal is to approximate the "real" distribution
117:in cognitive sciences communities, particularly in
4991:(1986). D. E. Rumelhart; J. L. McClelland (eds.).
4388:"A better way to pretrain deep Boltzmann machines"
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2526:
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2309:
2285:
2255:
2231:
2195:
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2035:
2011:
1975:
1936:
1889:
1816:of the system. This relation is the source of the
1804:
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4961:. Washington, D.C., USA: IEEE. pp. 449–454.
4959:International Joint Conference on Neural Networks
4900:
4723:
3797:. One of these terms enables the model to form a
2163:where the sum is over all the possible states of
105:. They were heavily popularized and promoted by
33:Sherrington–Kirkpatrick model with external field
5543:
5078:"A fast learning algorithm for deep belief nets"
4484:Larochelle, Hugo; Salakhutdinov, Ruslan (2010).
4426:Hinton, Geoffrey; Salakhutdinov, Ruslan (2009).
4386:Hinton, Geoffrey; Salakhutdinov, Ruslan (2012).
78:and the resemblance of their dynamics to simple
4993:"Learning and Relearning in Boltzmann Machines"
4983:
4833:
4818:
4793:
4726:Journal of the American Statistical Association
4650:"A Spike and Slab Restricted Boltzmann Machine"
4486:"Efficient Learning of Deep Boltzmann Machines"
4437:. Vol. 3. pp. 448–455. Archived from
4428:"Efficient Learning of Deep Boltzmann Machines"
4356:
2849:
2239:, as promised by the Boltzmann distribution. A
183:, is a network of units with a total "energy" (
4956:
3859:model of Sherrington–Kirkpatrick's stochastic
3778:over continuous domain; their mixture forms a
3751:), which models continuous-valued inputs with
3142:
5195:
4156:"A Learning Algorithm for Boltzmann Machines"
4699:. Vol. 10. pp. 1–8. Archived from
3762:and its variants, a spike-and-slab RBM is a
3620:
3557:
3531:
3468:
3111:
3098:
3052:
3039:
2999:
2986:
2939:
2926:
484:
472:
4461:
16:Type of stochastic recurrent neural network
5202:
5188:
4923:
3713:models that revolutionized multimodality.
2616:This result follows from the fact that at
596:is the activation threshold for the unit.)
5209:
5098:
5039:
4928:. Teaneck, New Jersey: World Scientific.
4877:
4859:
4586:
4568:
4316:
4306:
4289:Nijkamp, E.; Hill, M. E; Han, T. (2020),
4224:
4176:
3805:the slab variables given an observation.
2883:
1019:
974:
828:
785:
675:
321:
287:
276:
5136:
4905:. Defense Technical Information Center.
4657:JMLR: Workshop and Conference Proceeding
4464:"Scaling Learning Algorithms towards AI"
3719:– at least one system under development
2853:
1904:. We denote this distribution, after we
700:equaling 0 (off) versus 1 (on), written
383:is the connection strength between unit
140:
18:
4381:
4379:
4254:International Neural Network Conference
3902:, which find widespread application in
3604:
3583:
3562:
3515:
3494:
3473:
3461:
3168:
3079:
3020:
2967:
2919:
191:results. Boltzmann machine weights are
5544:
5013:
4681:
4627:from the original on 20 September 2022
4532:from the original on 20 September 2022
4247:
4198:
3727:
2911:. It comprises a set of visible units
630:are represented as a symmetric matrix
47:model with an external field, i.e., a
5183:
4717:
4641:
4477:
4455:
3715:Multimodal deep learning is used for
3145:, the probability assigned to vector
4462:Bengio, Yoshua; LeCun, Yann (2007).
4419:
4376:
4333:Recent Developments in Deep Learning
4248:Osborn, Thomas R. (1 January 1990).
4149:
4147:
4145:
3855:The Boltzmann machine is based on a
1823:
59:technique applied in the context of
13:
4977:
4901:Hofstadter, D. R. (January 1984).
4495:. pp. 693–700. Archived from
4199:Hinton, Geoffrey E. (2007-05-24).
2728:
2718:
2401:
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1591:
1495:
1403:
1311:
1215:
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875:
740:
707:
14:
5568:
5175:Talk at Google by Geoffrey Hinton
5163:
4797:Analyzing Cooperative Computation
4611:Teaching hospital press release:
4519:
4256:. Springer Netherlands. pp.
4142:
3808:
3679:Multimodal learning § Application
3672:Multimodal deep Boltzmann machine
3544:are the set of hidden units, and
5238:
5076:; Osindero, S.; Teh, Y. (2006).
3740:RBMs, led to the spike-and-slab
3732:The need for deep learning with
3677:This section is an excerpt from
566:in the global energy function. (
490:{\displaystyle s_{i}\in \{0,1\}}
4950:
4917:
4894:
4827:
4812:
4787:
4744:
4544:
4513:
4340:from the original on 2021-12-22
3847:such as the Boltzmann machine.
1674:{\displaystyle p_{\text{i=on}}}
672:with zeros along the diagonal.
4926:Physics of cognitive processes
4738:10.1080/01621459.1988.10478694
4350:
4324:
4282:
4241:
4192:
4108:
4076:
4070:
4055:
4049:
4023:
4017:
3615:
3609:
3594:
3588:
3573:
3567:
3526:
3520:
3505:
3499:
3484:
3478:
3430:
3424:
3409:
3403:
3388:
3382:
3348:
3342:
3327:
3321:
3306:
3300:
3266:
3260:
3235:
3229:
3172:
3164:
3090:
3084:
3031:
3025:
2978:
2972:
2840:until the activities saturate.
2798:
2762:
2643:
2637:
2579:is the probability that units
2534:is the probability that units
2480:
2438:
2226:
2220:
2139:
2133:
2118:
2112:
2086:
2080:
2006:
2000:
1970:
1964:
1931:
1925:
1884:
1878:
1770:
1742:
1287:
1268:
1256:
1243:
1191:
1178:
1166:
1153:
1045:
1042:
1029:
1003:
997:
984:
659:
643:
1:
4101:
3723:such different types of data.
3658:convolutional neural networks
1908:it over the hidden units, as
181:Sherrington–Kirkpatrick model
63:. It is also classified as a
49:Sherrington–Kirkpatrick model
5552:Neural network architectures
5225:Principle of maximum entropy
4821:Optimal Perceptual Inference
4619:Brigham and Women's Hospital
4266:10.1007/978-94-009-0643-3_76
3928:Restricted Boltzmann machine
2865:Restricted Boltzmann machine
2850:Restricted Boltzmann machine
2378:, is given by the equation:
2317:with respect to the weight.
723:{\displaystyle \Delta E_{i}}
589:{\displaystyle -\theta _{i}}
179:A Boltzmann machine, like a
136:
7:
5109:10.1162/neco.2006.18.7.1527
4773:10.1103/physrevlett.35.1792
4570:10.1016/j.ccell.2022.07.004
4137:10.1103/PhysRevLett.35.1792
3921:
3774:at zero, while a slab is a
3691:latent Dirichlet allocation
2958:and layers of hidden units
2814:
2694:, which is heavily used in
2021:Kullback–Leibler divergence
1852:
1681:, the probability that the
539:{\displaystyle \theta _{i}}
10:
5573:
5249:Statistical thermodynamics
5050:10.1162/089976602760128018
4178:10.1207/s15516709cog0901_7
3850:
3845:stochastic neural networks
3812:
3801:of the spike variables by
3676:
2900:) with multiple layers of
2862:
2572:{\displaystyle p_{ij}^{-}}
2527:{\displaystyle p_{ij}^{+}}
5484:
5446:
5411:
5366:
5308:
5247:
5236:
5217:
5002:: 282–317. Archived from
4967:10.1109/IJCNN.1989.118618
4663:: 233–241. Archived from
4226:10.4249/scholarpedia.1668
101:, which is used in their
93:They are named after the
5509:Condensed matter physics
5492:Statistical field theory
5137:Montufar, Guido (2018).
4834:Hopfield, J. J. (1982).
4318:10.1609/aaai.v34i04.5973
3799:conditional distribution
3699:Self-supervised learning
3666:Markov chain Monte Carlo
2844:
2649:{\displaystyle P^{-}(s)}
2263:changes a given weight,
2232:{\displaystyle P^{-}(v)}
2012:{\displaystyle P^{-}(V)}
1976:{\displaystyle P^{+}(V)}
1937:{\displaystyle P^{-}(V)}
1890:{\displaystyle P^{+}(V)}
5367:Mathematical approaches
5356:Lennard-Jones potential
5272:thermodynamic potential
4753:Physical Review Letters
4117:Physical Review Letters
3916:artificial intelligence
3687:support vector machines
51:, that is a stochastic
5403:conformal field theory
4861:10.1073/pnas.79.8.2554
4471:Université de Montréal
4091:
3968:
3627:
3538:
3444:
3131:
2952:
2884:Deep Boltzmann machine
2860:
2824:a partial photograph.
2805:
2670:
2650:
2603:
2573:
2528:
2487:
2372:
2371:{\displaystyle w_{ij}}
2342:
2311:
2293:, by subtracting the
2287:
2286:{\displaystyle w_{ij}}
2257:
2233:
2197:
2177:
2154:
2037:
2013:
1977:
1938:
1891:
1831:Boltzmann distribution
1812:is referred to as the
1806:
1780:
1701:-th unit is on gives:
1695:
1675:
1645:
1560:
1475:
1383:
1294:
1198:
1106:
1082:
1052:
933:Boltzmann distribution
918:
856:
724:
694:
676:Unit state probability
666:
624:
623:{\displaystyle w_{ij}}
590:
560:
540:
511:
491:
446:
417:
397:
377:
376:{\displaystyle w_{ij}}
341:
209:
176:
169:
168:{\displaystyle w_{ij}}
95:Boltzmann distribution
37:stochastic Ising model
24:
5318:Ferromagnetism models
5211:Statistical mechanics
4621:via medicalxpress.com
4401:: 1–9. Archived from
4092:
3969:
3891:'s "Harmony Theory".
3628:
3539:
3445:
3132:
2953:
2857:
2806:
2671:
2651:
2604:
2574:
2529:
2488:
2373:
2343:
2341:{\displaystyle P^{+}}
2312:
2288:
2258:
2234:
2198:
2178:
2155:
2038:
2014:
1978:
1939:
1892:
1807:
1781:
1696:
1676:
1646:
1561:
1476:
1384:
1295:
1199:
1107:
1083:
1081:{\displaystyle k_{B}}
1053:
919:
857:
725:
695:
667:
625:
591:
561:
541:
512:
492:
447:
445:{\displaystyle s_{i}}
418:
398:
378:
342:
210:
170:
144:
99:statistical mechanics
22:
3982:
3958:
3900:Markov random fields
3548:
3457:
3158:
2962:
2915:
2712:
2698:. By minimizing the
2660:
2656:of any global state
2624:
2593:
2548:
2503:
2385:
2352:
2325:
2301:
2267:
2247:
2207:
2187:
2167:
2050:
2027:
1987:
1951:
1912:
1865:
1796:
1708:
1685:
1658:
1571:
1486:
1394:
1305:
1209:
1119:
1096:
1065:
942:
872:
737:
704:
684:
634:
604:
570:
550:
546:is the bias of unit
523:
501:
456:
429:
407:
387:
357:
230:
199:
195:. The global energy
149:
5497:elementary particle
5262:partition functions
4852:1982PNAS...79.2554H
4765:1975PhRvL..35.1792S
4217:2007SchpJ...2.1668H
4201:"Boltzmann machine"
4129:1975PhRvL..35.1792S
3938:Markov random field
3758:. Similar to basic
3728:Spike-and-slab RBMs
3695:deep belief network
3434:
3413:
3392:
3352:
3331:
3310:
3270:
3239:
2890:Markov random field
2797:
2779:
2618:thermal equilibrium
2568:
2523:
2479:
2458:
1902:thermal equilibrium
1843:simulated annealing
1839:thermal equilibrium
1835:thermal equilibrium
931:(the property of a
123:energy-based models
65:Markov random field
57:statistical physics
5524:information theory
5431:correlation length
5426:Critical exponents
5413:Critical phenomena
5394:stochastic process
5374:Boltzmann equation
5267:equations of state
5086:Neural Computation
5027:Neural Computation
4732:(404): 1023–1032.
4395:Advances in Neural
4366:Microsoft Research
4087:
4005:
3964:
3946:(Lenz–Ising model)
3878:Douglas Hofstadter
3785:An extension of ss
3654:speech recognition
3623:
3534:
3440:
3414:
3393:
3369:
3368:
3332:
3311:
3287:
3286:
3250:
3216:
3215:
3197:
3127:
2948:
2861:
2801:
2783:
2765:
2666:
2646:
2599:
2569:
2551:
2524:
2506:
2483:
2462:
2441:
2368:
2338:
2307:
2295:partial derivative
2283:
2253:
2229:
2193:
2173:
2150:
2068:
2033:
2009:
1973:
1934:
1887:
1802:
1776:
1691:
1671:
1641:
1556:
1471:
1379:
1290:
1194:
1102:
1090:Boltzmann constant
1078:
1048:
914:
852:
814:
771:
720:
690:
665:{\displaystyle W=}
662:
620:
600:Often the weights
586:
556:
536:
507:
487:
442:
413:
393:
373:
337:
310:
262:
205:
177:
165:
80:physical processes
25:
5539:
5538:
5529:Boltzmann machine
5399:mean-field theory
5300:Maxwell relations
4759:(26): 1792–1796.
4563:(8): 865–878.e6.
4520:Quach, Katyanna.
4336:, 22 March 2010,
4301:(34): 5272–5280,
4275:978-0-7923-0831-7
4164:Cognitive Science
4123:(35): 1792–1796,
4080:
3996:
3967:{\displaystyle G}
3933:Helmholtz machine
3803:marginalizing out
3356:
3274:
3203:
3188:
3186:
2760:
2744:
2669:{\displaystyle s}
2602:{\displaystyle R}
2436:
2420:
2310:{\displaystyle G}
2256:{\displaystyle G}
2196:{\displaystyle G}
2176:{\displaystyle V}
2143:
2059:
2036:{\displaystyle G}
1824:Equilibrium state
1818:logistic function
1805:{\displaystyle T}
1774:
1768:
1718:
1694:{\displaystyle i}
1668:
1633:
1630:
1608:
1543:
1540:
1512:
1465:
1462:
1451:
1420:
1373:
1369:
1352:
1328:
1284:
1253:
1232:
1188:
1163:
1142:
1105:{\displaystyle T}
1039:
994:
911:
898:
799:
756:
693:{\displaystyle i}
559:{\displaystyle i}
510:{\displaystyle i}
416:{\displaystyle i}
396:{\displaystyle j}
301:
247:
217:Hopfield networks
208:{\displaystyle E}
125:" (EBM), because
103:sampling function
61:cognitive science
29:Boltzmann machine
5564:
5557:Ludwig Boltzmann
5421:Phase transition
5242:
5241:
5204:
5197:
5190:
5181:
5180:
5159:
5157:
5155:
5143:
5128:
5102:
5093:(7): 1527–1554.
5082:
5069:
5043:
5034:(8): 1771–1800.
5023:
5010:
5008:
4997:
4989:Sejnowski, T. J.
4971:
4970:
4954:
4948:
4947:
4921:
4915:
4914:
4898:
4892:
4891:
4881:
4863:
4831:
4825:
4824:
4816:
4810:
4809:
4807:
4805:
4791:
4785:
4784:
4748:
4742:
4741:
4721:
4715:
4714:
4712:
4711:
4705:
4694:
4685:
4679:
4678:
4676:
4675:
4669:
4654:
4645:
4639:
4636:
4634:
4632:
4608:
4590:
4572:
4548:
4542:
4541:
4539:
4537:
4517:
4511:
4510:
4508:
4507:
4501:
4490:
4481:
4475:
4474:
4468:
4459:
4453:
4452:
4450:
4449:
4443:
4432:
4423:
4417:
4416:
4414:
4413:
4407:
4392:
4383:
4374:
4373:
4363:
4354:
4348:
4347:
4346:
4345:
4328:
4322:
4321:
4320:
4310:
4286:
4280:
4279:
4245:
4239:
4238:
4228:
4196:
4190:
4189:
4188:on 18 July 2011.
4187:
4181:. Archived from
4180:
4160:
4151:
4140:
4139:
4112:
4096:
4094:
4093:
4088:
4086:
4085:
4081:
4079:
4069:
4068:
4058:
4048:
4047:
4037:
4016:
4015:
4004:
3992:
3973:
3971:
3970:
3965:
3950:Hopfield network
3837:log-linear model
3833:machine learning
3819:Log-linear model
3772:probability mass
3756:latent variables
3717:cancer screening
3632:
3630:
3629:
3624:
3619:
3618:
3607:
3598:
3597:
3586:
3577:
3576:
3565:
3543:
3541:
3540:
3535:
3530:
3529:
3518:
3509:
3508:
3497:
3488:
3487:
3476:
3464:
3449:
3447:
3446:
3441:
3436:
3435:
3433:
3422:
3412:
3401:
3391:
3380:
3367:
3351:
3340:
3330:
3319:
3309:
3298:
3285:
3269:
3258:
3249:
3248:
3238:
3227:
3214:
3196:
3187:
3179:
3171:
3150:
3144:
3136:
3134:
3133:
3128:
3126:
3125:
3124:
3123:
3094:
3093:
3082:
3067:
3066:
3065:
3064:
3035:
3034:
3023:
3014:
3013:
3012:
3011:
2982:
2981:
2970:
2957:
2955:
2954:
2949:
2947:
2946:
2922:
2905:random variables
2810:
2808:
2807:
2802:
2796:
2791:
2778:
2773:
2761:
2753:
2745:
2743:
2742:
2741:
2740:
2726:
2725:
2716:
2696:machine learning
2675:
2673:
2672:
2667:
2655:
2653:
2652:
2647:
2636:
2635:
2620:the probability
2608:
2606:
2605:
2600:
2578:
2576:
2575:
2570:
2567:
2562:
2533:
2531:
2530:
2525:
2522:
2517:
2492:
2490:
2489:
2484:
2478:
2473:
2457:
2452:
2437:
2429:
2421:
2419:
2418:
2417:
2416:
2399:
2398:
2389:
2377:
2375:
2374:
2369:
2367:
2366:
2347:
2345:
2344:
2339:
2337:
2336:
2316:
2314:
2313:
2308:
2292:
2290:
2289:
2284:
2282:
2281:
2262:
2260:
2259:
2254:
2241:gradient descent
2238:
2236:
2235:
2230:
2219:
2218:
2202:
2200:
2199:
2194:
2182:
2180:
2179:
2174:
2159:
2157:
2156:
2151:
2149:
2148:
2144:
2142:
2132:
2131:
2121:
2111:
2110:
2100:
2079:
2078:
2067:
2042:
2040:
2039:
2034:
2018:
2016:
2015:
2010:
1999:
1998:
1982:
1980:
1979:
1974:
1963:
1962:
1943:
1941:
1940:
1935:
1924:
1923:
1896:
1894:
1893:
1888:
1877:
1876:
1811:
1809:
1808:
1803:
1785:
1783:
1782:
1777:
1775:
1773:
1769:
1764:
1763:
1762:
1749:
1725:
1720:
1719:
1716:
1700:
1698:
1697:
1692:
1680:
1678:
1677:
1672:
1670:
1669:
1666:
1650:
1648:
1647:
1642:
1634:
1632:
1631:
1628:
1619:
1614:
1610:
1609:
1604:
1603:
1602:
1589:
1565:
1563:
1562:
1557:
1555:
1551:
1544:
1542:
1541:
1538:
1529:
1513:
1508:
1507:
1506:
1493:
1480:
1478:
1477:
1472:
1470:
1466:
1464:
1463:
1460:
1454:
1453:
1452:
1449:
1436:
1421:
1416:
1415:
1414:
1401:
1388:
1386:
1385:
1380:
1378:
1374:
1372:
1371:
1370:
1367:
1354:
1353:
1350:
1344:
1329:
1324:
1323:
1322:
1309:
1299:
1297:
1296:
1291:
1286:
1285:
1282:
1255:
1254:
1251:
1233:
1228:
1227:
1226:
1213:
1203:
1201:
1200:
1195:
1190:
1189:
1186:
1165:
1164:
1161:
1143:
1138:
1137:
1136:
1123:
1111:
1109:
1108:
1103:
1087:
1085:
1084:
1079:
1077:
1076:
1057:
1055:
1054:
1049:
1041:
1040:
1037:
1018:
1017:
996:
995:
992:
973:
972:
957:
956:
929:Boltzmann factor
923:
921:
920:
915:
913:
912:
909:
900:
899:
896:
887:
886:
861:
859:
858:
853:
851:
850:
838:
837:
827:
826:
813:
795:
794:
784:
783:
770:
752:
751:
729:
727:
726:
721:
719:
718:
699:
697:
696:
691:
671:
669:
668:
663:
658:
657:
629:
627:
626:
621:
619:
618:
595:
593:
592:
587:
585:
584:
565:
563:
562:
557:
545:
543:
542:
537:
535:
534:
516:
514:
513:
508:
496:
494:
493:
488:
468:
467:
451:
449:
448:
443:
441:
440:
422:
420:
419:
414:
402:
400:
399:
394:
382:
380:
379:
374:
372:
371:
346:
344:
343:
338:
336:
332:
331:
330:
320:
319:
309:
297:
296:
286:
285:
275:
274:
261:
214:
212:
211:
206:
174:
172:
171:
166:
164:
163:
119:machine learning
84:machine learning
43:is a stochastic
41:Ludwig Boltzmann
5572:
5571:
5567:
5566:
5565:
5563:
5562:
5561:
5542:
5541:
5540:
5535:
5480:
5442:
5407:
5389:BBGKY hierarchy
5384:Vlasov equation
5362:
5351:depletion force
5344:Particles with
5304:
5243:
5239:
5234:
5213:
5208:
5166:
5153:
5151:
5141:
5080:
5021:
5006:
4995:
4980:
4978:Further reading
4975:
4974:
4955:
4951:
4936:
4922:
4918:
4899:
4895:
4846:(8). : 2554–8.
4832:
4828:
4817:
4813:
4803:
4801:
4792:
4788:
4749:
4745:
4722:
4718:
4709:
4707:
4703:
4692:
4686:
4682:
4673:
4671:
4667:
4652:
4646:
4642:
4630:
4628:
4612:
4549:
4545:
4535:
4533:
4518:
4514:
4505:
4503:
4499:
4488:
4482:
4478:
4466:
4460:
4456:
4447:
4445:
4441:
4430:
4424:
4420:
4411:
4409:
4405:
4390:
4384:
4377:
4361:
4355:
4351:
4343:
4341:
4330:
4329:
4325:
4287:
4283:
4276:
4246:
4242:
4197:
4193:
4185:
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3959:
3956:
3955:
3924:
3912:computer vision
3853:
3835:it is called a
3821:
3813:Main articles:
3811:
3795:energy function
3764:bipartite graph
3730:
3725:
3724:
3682:
3674:
3639:graphical model
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2898:graphical model
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2685:backpropagation
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2243:algorithm over
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121:, as part of "
111:Terry Sejnowski
107:Geoffrey Hinton
39:), named after
17:
12:
11:
5:
5570:
5560:
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5514:Complex system
5511:
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5232:
5230:ergodic theory
5227:
5221:
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5207:
5206:
5199:
5192:
5184:
5178:
5177:
5172:
5165:
5164:External links
5162:
5161:
5160:
5134:
5129:
5100:10.1.1.76.1541
5070:
5041:10.1.1.35.8613
5011:
5009:on 2010-07-05.
4979:
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4171:(1): 147–169.
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3947:
3941:
3935:
3930:
3923:
3920:
3889:Paul Smolensky
3876:is present in
3874:Gibbs sampling
3852:
3849:
3810:
3809:In Mathematics
3807:
3766:, while like G
3736:inputs, as in
3729:
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2896:probabilistic
2885:
2882:
2863:Main article:
2851:
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452:is the state,
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5530:
5527:
5525:
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5520:
5517:
5516:
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5510:
5507:
5503:
5502:superfluidity
5500:
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5472:
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5140:
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5118:
5114:
5110:
5106:
5101:
5096:
5092:
5088:
5087:
5079:
5075:
5074:Hinton, G. E.
5071:
5067:
5063:
5059:
5055:
5051:
5047:
5042:
5037:
5033:
5029:
5028:
5020:
5016:
5015:Hinton, G. E.
5012:
5005:
5001:
4994:
4990:
4986:
4985:Hinton, G. E.
4982:
4981:
4968:
4964:
4960:
4953:
4945:
4941:
4937:
4935:9971-5-0255-0
4931:
4927:
4920:
4912:
4908:
4904:
4897:
4889:
4885:
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4875:
4871:
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4774:
4770:
4766:
4762:
4758:
4754:
4747:
4739:
4735:
4731:
4727:
4720:
4706:on 2016-03-04
4702:
4698:
4691:
4684:
4670:on 2016-03-04
4666:
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4580:
4576:
4571:
4566:
4562:
4558:
4554:
4547:
4531:
4527:
4523:
4516:
4502:on 2017-08-14
4498:
4494:
4487:
4480:
4472:
4465:
4458:
4444:on 2015-11-06
4440:
4436:
4429:
4422:
4408:on 2017-08-13
4404:
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3864:
3862:
3858:
3848:
3846:
3842:
3841:deep learning
3838:
3834:
3830:
3826:
3825:Gibbs measure
3820:
3816:
3815:Gibbs measure
3806:
3804:
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5150:(Preprint)
4710:2019-08-25
4674:2019-08-25
4506:2017-08-18
4448:2017-08-18
4412:2017-08-18
4344:2020-02-17
4308:1903.12370
4102:References
3857:spin-glass
3829:statistics
3721:integrates
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1235:=
1230:T
1224:i
1220:E
1192:)
1183:p
1179:(
1167:)
1158:p
1154:(
1145:=
1140:T
1134:i
1130:E
1100:T
1074:B
1070:k
1046:)
1043:)
1034:p
1030:(
1021:T
1015:B
1011:k
1004:(
998:)
989:p
985:(
976:T
970:B
966:k
959:=
954:i
950:E
906:E
893:E
889:=
884:i
880:E
848:i
840:+
835:j
831:s
824:i
821:j
817:w
811:i
805:j
797:+
792:j
788:s
781:j
778:i
774:w
768:i
762:j
754:=
749:i
745:E
716:i
712:E
688:i
660:]
655:j
652:i
648:w
644:[
641:=
638:W
616:j
613:i
609:w
582:i
554:i
532:i
517:.
505:i
485:}
482:1
479:,
476:0
473:{
465:i
461:s
438:i
434:s
423:.
411:i
391:j
369:j
366:i
362:w
334:)
328:i
324:s
317:i
307:i
299:+
294:j
290:s
283:i
279:s
272:j
269:i
265:w
259:j
253:i
244:(
237:=
234:E
203:E
161:j
158:i
154:w
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