Klaus-Robert Müller (born 1964 in Karlsruhe, West Germany) is a German computer scientist and physicist, most noted for his work in machine learning and brain–computer interfaces. == Career == Klaus-Robert Müller received his Diplom in mathematical physics and PhD in theoretical computer science from the University of Karlsruhe. Following his Ph.D. he went to Berlin as a postdoctoral fellow at GMD (German National Research Center for Computer Science) Berlin (now part of Fraunhofer Institute for Open Communication Systems), where he started building up the Intelligent Data Analysis (IDA) group. From 1994 to 1995 he was a research fellow at Shun'ichi Amari's lab at the University of Tokyo. 1999 Müller became an associate professor for neuroinformatics at the University of Potsdam, transitioning to the full professorship for Neural Networks and Time Series Analysis in 2003. Since 2006 he holds the chair for Machine Learning at Technische Universität Berlin. Since 2012 he holds a distinguished professorship at Korea University in Seoul. He co-founded and is co-director of the Berlin Big Data Center (BBDC) of TU Berlin. As of 2017, 29 former doctoral or postdoctoral researchers of Klaus-Robert Müller have become full professors themselves. Bernhard Schölkopf and Alexander J. Smola were supervised by him as members of his research group. Since 2020 he is director of the Berlin Institute for the Foundations of Learning and Data (BIFOLD), a German National AI Competence Center, and director of the European Laboratory for Learning and Intelligent Systems (ELLIS) unit Berlin. In 2020/2021 he spent his sabbatical at Google Brain as a principal scientist. == Research == Müller has contributed extensively to several major interests of machine learning, including support vector machines (SVMs) and kernel methods, and artificial neural networks. He pioneered applying new methods of pattern recognition in domains like brain–computer interfaces, using them for patients with Locked-in syndrome. He is one of the leading computer scientists affiliated with Germany. His current research interests include: Statistical learning theory (Support Vector Machines, Deep Neural Networks, Boosting) Learning of non-stationarity data Fusion of structured heterogeneous multi-modal data, co-adaptation Applications: MEG, EEG, NIRS, ECoG, EMG, Brain Computer Interfaces, computational neuroscience, computer vision, genomic data analysis, computational chemistry and atomistic simulations, digital pathology == Honours and awards == Klaus-Robert Müller was elected a fellow of the German National Academy of Sciences Leopoldina in 2012. In 2017 he was elected member of the Berlin-Brandenburg Academy of Sciences and Humanities and also external scientific member of the Max Planck Society. In 2021 he was elected member of the German Academy of Science and Engineering. His work was honoured with several awards, including: 2026 Gottfried Wilhelm Leibniz Prize 2025 IEEE Neural Network Pioneer Award 2024 Feynman Prize in Nanotechnology 2023 Hector Fellow 2025, 2024, 2023, 2022, 2021, 2020, and 2019 Clarivate Highly Cited Researcher 2017 Vodafone Innovations Award 2017 2014 Science Prize of Berlin 2014 by the Governing Mayor of Berlin 2014 European Research Council Panel Consolidator Grants 2009 Best Paper award by IEEE Engineering in Medicine and Biology Society EMBS 2006 SEL-ALCATEL Research Prize for Technical Communication 1999 Olympus Award for Pattern Recognition == Books == with Holzinger, Andreas; et al., eds. (2022). xxAI – Beyond Explainable Artificial Intelligence. Lecture Notes in Computer Science. Vol. 13200. Springer Cham. doi:10.1007/978-3-031-04083-2. ISBN 978-3-031-04082-5. with Schütt, Kristof T.; et al., eds. (2020). Machine Learning Meets Quantum Physics. Lecture Notes in Physics. Vol. 968. Springer Cham. doi:10.1007/978-3-030-40245-7. ISBN 978-3-030-40244-0. S2CID 242406994. with Samek, Wojciech; et al., eds. (2019). Explainable AI: Interpreting, Explaining and Visualizing Deep Learning. Lecture Notes in Computer Science. Vol. 11700. Springer Cham. doi:10.1007/978-3-030-28954-6. ISBN 978-3-030-28953-9. with Montavon, Grégoire; et al., eds. (2012). Neural Networks: Tricks of the Trade. Lecture Notes in Computer Science. Vol. 7700 (2nd ed.). Springer Berlin, Heidelberg. doi:10.1007/978-3-642-35289-8. ISBN 978-3-642-35288-1. S2CID 39578794.
Lessac Technologies
Lessac Technologies, Inc. (LTI) is an American firm which develops voice synthesis software, licenses technology and sells synthesized novels as MP3 files. The firm currently has seven patents granted and three more pending for its automated methods of converting digital text into human-sounding speech, more accurately recognizing human speech and outputting the text representing the words and phrases of said speech, along with recognizing the speaker's emotional state. The LTI technology is partly based on the work of the late Arthur Lessac, a Professor of Theater at the State University of New York and the creator of Lessac Kinesensic Training, and LTI has licensed exclusive rights to exploit Arthur Lessac's copyrighted works in the fields of speech synthesis and speech recognition. Based on the view that music is speech and speech is music, Lessac's work and books focused on body and speech energies and how they go together. Arthur Lessac's textual annotation system, which was originally developed to assist actors, singers, and orators in marking up scripts to prepare for performance, is adapted in LTI's speech synthesis system as the basic representation of the speech to be synthesized (Lessemes), in contrast to many other systems which use a phonetic representation. LTI's software has two major components: (1) a linguistic front-end that converts plain text to a sequence of prosodic and phonosensory graphic symbols (Lessemes) based on Arthur Lessac's annotation system, which specify the speech units to be synthesized; (2) a signal-processing back-end that takes the Lessemes as acoustic data and produces human-sounding synthesized speech as output, using unit selection and concatenation. LTI's text-to-speech system came in second in the world-wide Blizzard Challenge 2011 and 2012. The first-place team in 2011 also employed LTI's "front-end" technology, but with its own back-end. The Blizzard Challenge, conducted by the Language Technologies Institute of Carnegie Mellon University, was devised as a way to evaluate speech synthesis techniques by having different research groups build voices from the same voice-actor recordings, and comparing the results through listening tests. LTI was founded in 2000 by H. Donald Wilson (chairman), a lawyer, LexisNexis entrepreneur and business associate of Arthur Lessac; and Gary A. Marple (chief inventor), after Marple suggested that Arthur Lessac's kinesensic voice training might be applicable to computational linguistics. After Wilson's death in 2006, his nephew John Reichenbach became the firm's CEO.
Log-linear model
A log-linear model is a mathematical model that takes the form of a function whose logarithm equals a linear combination of the parameters of the model, which makes it possible to apply (possibly multivariate) linear regression. That is, it has the general form exp ( c + ∑ i w i f i ( X ) ) {\displaystyle \exp \left(c+\sum _{i}w_{i}f_{i}(X)\right)} , in which the fi(X) are quantities that are functions of the variable X, in general a vector of values, while c and the wi stand for the model parameters. The term may specifically be used for: A log-linear plot or graph, which is a type of semi-log plot. Poisson regression for contingency tables, a type of generalized linear model. The specific applications of log-linear models are where the output quantity lies in the range 0 to ∞, for values of the independent variables X, or more immediately, the transformed quantities fi(X) in the range −∞ to +∞. This may be contrasted to logistic models, similar to the logistic function, for which the output quantity lies in the range 0 to 1. Thus the contexts where these models are useful or realistic often depends on the range of the values being modelled.
Sigmoid function
A sigmoid function is any mathematical function whose graph has a characteristic S-shaped or sigmoid curve. A common example of a sigmoid function is the logistic function. Other sigmoid functions are given in the Examples section. In some fields, most notably in the context of artificial neural networks, the term "sigmoid function" is used as a synonym for "logistic function". Special cases of sigmoid functions include the Gompertz curve (used in modeling systems that saturate at large values of x) and the ogee curve (used in the spillway of some dams). Sigmoid functions have domain of all real numbers, with return (response) value commonly monotonically increasing but could be decreasing. Sigmoid functions most often show a return value (y axis) in the range 0 to 1. Another commonly used range is from −1 to 1. There is also the Heaviside step function, which instantaneously transitions between 0 and 1. A wide variety of sigmoid functions including the logistic and hyperbolic tangent functions have been used as the activation function of artificial neurons. Sigmoid curves are also common in statistics as cumulative distribution functions (which go from 0 to 1), such as the integrals of the logistic density, the normal density, and Student's t probability density functions. The logistic sigmoid function is invertible, and its inverse is the logit function. == Theory == In mathematics, a unitary sigmoid function is a bounded sigmoid-type function normalized to the unit range, typically with lower and upper asymptotes at 0 and 1. The theory proposed by Grebenc distinguishes three kinds of unitary sigmoid functions according to their asymptotic behavior and the presence or absence of oscillation near the asymptotes. A general form of a unitary sigmoid function is y = A S ( f ( x ) ) + B , {\displaystyle y=A\,S(f(x))+B,} where S {\displaystyle S} is an increasing sigmoid function, f ( x ) {\displaystyle f(x)} is a transformation of the independent variable, and A {\displaystyle A} and B {\displaystyle B} are constants controlling scaling and translation. === Classification === ==== 1st kind ==== A unitary sigmoid function of the first kind is a bounded increasing function that approaches its lower and upper asymptotes monotonically, without oscillation. This class includes many of the standard sigmoid functions used in statistics, biomathematics, and engineering, such as the logistic function and related generalizations. ==== 2nd kind ==== A unitary sigmoid function of the second kind is a bounded increasing function that oscillates near the upper asymptote while preserving an overall sigmoid transition. ==== 3rd kind ==== A unitary sigmoid function of the third kind is a bounded increasing function that oscillates near both the lower and upper asymptotes. These functions retain the global shape of a sigmoid curve but exhibit oscillatory behavior in the vicinity of both limiting states. === Taxonomy === The tables below show the taxonomy of unitary sigmoid functions of all three kinds. Table 1. Taxonomy matrix with examples of sigmoid functions of the 1st kind Table 2. Taxonomy matrix with examples of sigmoid functions of the 2nd kind on the unbounded interval Table 3. Taxonomy matrix with examples of sigmoid functions of the 3rd kind === Construction methods === The same theory presents a list of 30 methods for constructing sigmoid functions.. These include algebraic transformations, integration and convolution methods, constructions from bell-shaped functions, solutions of ordinary and partial differential equations, recursive schemes, stochastic differential equations, feedback systems, and chaotic systems. M0: Construction method for sigmoid functions not evident or intuitive M1: Inverse of singularity functions M2: Sigmoid functions of embedded positive functions M3: Rising a sigmoid function to the power M4: Exponentiating a sigmoid function M5: Symmetric sigmoid functions derived from asymmetric ones M6: Sigmoid functions of the reciprocal independent variable M7: Embedding a sigmoid function into other function M8: Sum of sigmoid functions M9: Multiplication of sigmoid functions M10: Integral of the product of an increasing and a decreasing function M11: Derivation from lambda (bell-shaped) functions M12: Integration of lambda (bell-shaped) function M13: Integration of the sum of lambda (bell-shaped) functions M14: Integration of the product of two lambda (bell-shaped) functions M15: Integration of the difference of two shifted sigmoid functions M16: Integration of the product of two shifted sigmoid functions M17: Convolution of sigmoid functions M18: Integration of the product of lambda and sigmoid function M19: Solutions of ordinary differential equations M20: Solutions of partial differential equation (PDE) M21: Solutions of functional differential equation (FDE) M22: Sum of a sigmoid function and some derivatives M23: Combination of sigmoid functions, its derivative and integral M24: Filtering sigmoid functions M25: Special cases of Gauss hypergeometric functions M26: Feedback closed-loop systems M27: Recursive functions M28: Recursive time-delayed feed-forward loops M29: Solutions of stochastic differential equation M30: Chaotic sigmoid functions Consult reference for more details. == Definition == A sigmoid function is a bounded, differentiable, real function that is defined for all real input values and has a positive derivative at each point. == Properties == In general, a sigmoid function is monotonic, and has a first derivative which is bell shaped. Conversely, the integral of any continuous, non-negative, bell-shaped function (with one local maximum and no local minimum, unless degenerate) will be sigmoidal. Thus the cumulative distribution functions for many common probability distributions are sigmoidal. One such example is the error function, which is related to the cumulative distribution function of a normal distribution; another is the arctan function, which is related to the cumulative distribution function of a Cauchy distribution. A sigmoid function is constrained by a pair of horizontal asymptotes as x → ± ∞ {\displaystyle x\rightarrow \pm \infty } . A sigmoid function is convex for values less than a particular point, and it is concave for values greater than that point: in many of the examples here, that point is 0. == Examples == Logistic function f ( x ) = 1 1 + e − x {\displaystyle f(x)={\frac {1}{1+e^{-x}}}} Hyperbolic tangent (shifted and scaled version of the logistic function, above) f ( x ) = tanh x = e x − e − x e x + e − x {\displaystyle f(x)=\tanh x={\frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}} Arctangent function f ( x ) = arctan x {\displaystyle f(x)=\arctan x} Gudermannian function f ( x ) = gd ( x ) = ∫ 0 x d t cosh t = 2 arctan ( tanh ( x 2 ) ) {\displaystyle f(x)=\operatorname {gd} (x)=\int _{0}^{x}{\frac {dt}{\cosh t}}=2\arctan \left(\tanh \left({\frac {x}{2}}\right)\right)} Error function f ( x ) = erf ( x ) = 2 π ∫ 0 x e − t 2 d t {\displaystyle f(x)=\operatorname {erf} (x)={\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{-t^{2}}\,dt} Generalised logistic function f ( x ) = ( 1 + e − x ) − α , α > 0 {\displaystyle f(x)=\left(1+e^{-x}\right)^{-\alpha },\quad \alpha >0} Smoothstep function f ( x ) = { ( ∫ 0 1 ( 1 − u 2 ) N d u ) − 1 ∫ 0 x ( 1 − u 2 ) N d u , | x | ≤ 1 sgn ( x ) | x | ≥ 1 N ∈ Z ≥ 1 {\displaystyle f(x)={\begin{cases}{\displaystyle \left(\int _{0}^{1}\left(1-u^{2}\right)^{N}du\right)^{-1}\int _{0}^{x}\left(1-u^{2}\right)^{N}\ du},&|x|\leq 1\\\\\operatorname {sgn}(x)&|x|\geq 1\\\end{cases}}\quad N\in \mathbb {Z} \geq 1} Some algebraic functions, for example f ( x ) = x 1 + x 2 {\displaystyle f(x)={\frac {x}{\sqrt {1+x^{2}}}}} and in a more general form f ( x ) = x ( 1 + | x | k ) 1 / k {\displaystyle f(x)={\frac {x}{\left(1+|x|^{k}\right)^{1/k}}}} Up to shifts and scaling, many sigmoids are special cases of f ( x ) = φ ( φ ( x , β ) , α ) , {\displaystyle f(x)=\varphi (\varphi (x,\beta ),\alpha ),} where φ ( x , λ ) = { ( 1 − λ x ) 1 / λ λ ≠ 0 e − x λ = 0 {\displaystyle \varphi (x,\lambda )={\begin{cases}(1-\lambda x)^{1/\lambda }&\lambda \neq 0\\e^{-x}&\lambda =0\\\end{cases}}} is the inverse of the negative Box–Cox transformation, and α < 1 {\displaystyle \alpha <1} and β < 1 {\displaystyle \beta <1} are shape parameters. Smooth transition function normalized to (−1,1): f ( x ) = { 2 1 + e − 2 m x 1 − x 2 − 1 , | x | < 1 sgn ( x ) | x | ≥ 1 = { tanh ( m x 1 − x 2 ) , | x | < 1 sgn ( x ) | x | ≥ 1 {\displaystyle {\begin{aligned}f(x)&={\begin{cases}{\displaystyle {\frac {2}{1+e^{-2m{\frac {x}{1-x^{2}}}}}}-1},&|x|<1\\\\\operatorname {sgn}(x)&|x|\geq 1\\\end{cases}}\\&={\begin{cases}{\displaystyle \tanh \left(m{\frac {x}{1-x^{2}}}\right)},&|x|<1\\\\\operatorname {sgn}(x)&|x|\geq 1\\\end{cases}}\end{aligned}}} using the hyperbolic tangent mentioned above. Here, m {\displaystyle m} is a free parameter encoding the slope at x = 0 {\displaystyle x=0} , which must be great
Transkribus
Transkribus is a platform for the text recognition, image analysis and structure recognition of historical documents. The platform was created in the context of the two EU projects "tranScriptorium" (2013–2015) and "READ" (Recognition and Enrichment of Archival Documents – 2016–2019). It was developed by the University of Innsbruck. Since July 1, 2019 the platform has been directed and further developed by the READ-COOP, a non-profit cooperative. The platform integrates tools developed by research groups throughout Europe, including the Pattern Recognition and Human Language Technology (PRHLT) group of the Technical University of Valencia and the Computational Intelligence Technology Lab (CITlab) group of University of Rostock. Comparable programs that offer similar functions are eScriptorium and OCR4All.
SemEval
SemEval (Semantic Evaluation) is an ongoing series of evaluations of computational semantic analysis systems; it evolved from the Senseval word sense evaluation series. The evaluations are intended to explore the nature of meaning in language. While meaning is intuitive to humans, transferring those intuitions to computational analysis has proved elusive. This series of evaluations provides a mechanism to characterize in more precise terms exactly what is necessary to compute in meaning. As such, the evaluations provide an emergent mechanism to identify the problems and solutions for computations with meaning. These exercises have evolved to articulate more of the dimensions that are involved in our use of language. They began with apparently simple attempts to identify word senses computationally. They have evolved to investigate the interrelationships among the elements in a sentence (e.g., semantic role labeling), relations between sentences (e.g., coreference), and the nature of what we are saying (semantic relations and sentiment analysis). The purpose of the SemEval and Senseval exercises is to evaluate semantic analysis systems. "Semantic Analysis" refers to a formal analysis of meaning, and "computational" refer to approaches that in principle support effective implementation. The first three evaluations, Senseval-1 through Senseval-3, were focused on word sense disambiguation (WSD), each time growing in the number of languages offered in the tasks and in the number of participating teams. Beginning with the fourth workshop, SemEval-2007 (SemEval-1), the nature of the tasks evolved to include semantic analysis tasks outside of word sense disambiguation. Triggered by the conception of the SEM conference, the SemEval community had decided to hold the evaluation workshops yearly in association with the SEM conference. It was also the decision that not every evaluation task will be run every year, e.g. none of the WSD tasks were included in the SemEval-2012 workshop. == History == === Early evaluation of algorithms for word sense disambiguation === From the earliest days, assessing the quality of word sense disambiguation algorithms had been primarily a matter of intrinsic evaluation, and “almost no attempts had been made to evaluate embedded WSD components”. Only very recently (2006) had extrinsic evaluations begun to provide some evidence for the value of WSD in end-user applications. Until 1990 or so, discussions of the sense disambiguation task focused mainly on illustrative examples rather than comprehensive evaluation. The early 1990s saw the beginnings of more systematic and rigorous intrinsic evaluations, including more formal experimentation on small sets of ambiguous words. === Senseval to SemEval === In April 1997, Martha Palmer and Marc Light organized a workshop entitled Tagging with Lexical Semantics: Why, What, and How? in conjunction with the Conference on Applied Natural Language Processing. At the time, there was a clear recognition that manually annotated corpora had revolutionized other areas of NLP, such as part-of-speech tagging and parsing, and that corpus-driven approaches had the potential to revolutionize automatic semantic analysis as well. Kilgarriff recalled that there was "a high degree of consensus that the field needed evaluation", and several practical proposals by Resnik and Yarowsky kicked off a discussion that led to the creation of the Senseval evaluation exercises. === SemEval's 3, 2 or 1 year(s) cycle === After SemEval-2010, many participants feel that the 3-year cycle is a long wait. Many other shared tasks such as Conference on Natural Language Learning (CoNLL) and Recognizing Textual Entailments (RTE) run annually. For this reason, the SemEval coordinators gave the opportunity for task organizers to choose between a 2-year or a 3-year cycle. The SemEval community favored the 3-year cycle. Although the votes within the SemEval community favored a 3-year cycle, organizers and coordinators had settled to split the SemEval task into 2 evaluation workshops. This was triggered by the introduction of the new SEM conference. The SemEval organizers thought it would be appropriate to associate our event with the SEM conference and collocate the SemEval workshop with the SEM conference. The organizers got very positive responses (from the task coordinators/organizers and participants) about the association with the yearly SEM, and 8 tasks were willing to switch to 2012. Thus was born SemEval-2012 and SemEval-2013. The current plan is to switch to a yearly SemEval schedule to associate it with the SEM conference but not every task needs to run every year. ==== List of Senseval and SemEval Workshops ==== Senseval-1 took place in the summer of 1998 for English, French, and Italian, culminating in a workshop held at Herstmonceux Castle, Sussex, England on September 2–4. Senseval-2 took place in the summer of 2001, and was followed by a workshop held in July 2001 in Toulouse, in conjunction with ACL 2001. Senseval-2 included tasks for Basque, Chinese, Czech, Danish, Dutch, English, Estonian, Italian, Japanese, Korean, Spanish and Swedish. Senseval-3 took place in March–April 2004, followed by a workshop held in July 2004 in Barcelona, in conjunction with ACL 2004. Senseval-3 included 14 different tasks for core word sense disambiguation, as well as identification of semantic roles, multilingual annotations, logic forms, subcategorization acquisition. SemEval-2007 (Senseval-4) took place in 2007, followed by a workshop held in conjunction with ACL in Prague. SemEval-2007 included 18 different tasks targeting the evaluation of systems for the semantic analysis of text. A special issue of Language Resources and Evaluation is devoted to the result. SemEval-2010 took place in 2010, followed by a workshop held in conjunction with ACL in Uppsala. SemEval-2010 included 18 different tasks targeting the evaluation of semantic analysis systems. SemEval-2012 took place in 2012; it was associated with the new SEM, First Joint Conference on Lexical and Computational Semantics, and co-located with NAACL, Montreal, Canada. SemEval-2012 included 8 different tasks targeting at evaluating computational semantic systems. However, there was no WSD task involved in SemEval-2012, the WSD related tasks were scheduled in the upcoming SemEval-2013. SemEval-2013 was associated with NAACL 2013, North American Association of Computational Linguistics, Georgia, USA and took place in 2013. It included 13 different tasks targeting at evaluating computational semantic systems. SemEval-2014 took place in 2014. It was co-located with COLING 2014, 25th International Conference on Computational Linguistics and SEM 2014, Second Joint Conference on Lexical and Computational Semantics, Dublin, Ireland. There were 10 different tasks in SemEval-2014 evaluating various computational semantic systems. SemEval-2015 took place in 2015. It was co-located with NAACL-HLT 2015, 2015 Conference of the North American Chapter of the Association for Computational Linguistics – Human Language Technologies and SEM 2015, Third Joint Conference on Lexical and Computational Semantics, Denver, USA. There were 17 different tasks in SemEval-2015 evaluating various computational semantic systems. == SemEval Workshop framework == The framework of the SemEval/Senseval evaluation workshops emulates the Message Understanding Conferences (MUCs) and other evaluation workshops ran by ARPA (Advanced Research Projects Agency, renamed the Defense Advanced Research Projects Agency (DARPA)). Stages of SemEval/Senseval evaluation workshops Firstly, all likely participants were invited to express their interest and participate in the exercise design. A timetable towards a final workshop was worked out. A plan for selecting evaluation materials was agreed. 'Gold standards' for the individual tasks were acquired, often human annotators were considered as a gold standard to measure precision and recall scores of computer systems. These 'gold standards' are what the computational systems strive towards. In WSD tasks, human annotators were set on the task of generating a set of correct WSD answers (i.e. the correct sense for a given word in a given context) The gold standard materials, without answers, were released to participants, who then had a short time to run their programs over them and return their sets of answers to the organizers. The organizers then scored the answers and the scores were announced and discussed at a workshop. == Semantic evaluation tasks == Senseval-1 & Senseval-2 focused on evaluation WSD systems on major languages that were available corpus and computerized dictionary. Senseval-3 looked beyond the lexemes and started to evaluate systems that looked into wider areas of semantics, such as Semantic Roles (technically known as Theta roles in formal semantics), Logic Form Transformation (commonly semantics of phrases, clauses or sentences were represented
Wake-sleep algorithm
The wake-sleep algorithm is an unsupervised learning algorithm for deep generative models, especially Helmholtz Machines. The algorithm is similar to the expectation-maximization algorithm, and optimizes the model likelihood for observed data. The name of the algorithm derives from its use of two learning phases, the “wake” phase and the “sleep” phase, which are performed alternately. It can be conceived as a model for learning in the brain, but is also being applied for machine learning. == Description == The goal of the wake-sleep algorithm is to find a hierarchical representation of observed data. In a graphical representation of the algorithm, data is applied to the algorithm at the bottom, while higher layers form gradually more abstract representations. Between each pair of layers are two sets of weights: Recognition weights, which define how representations are inferred from data, and generative weights, which define how these representations relate to data. == Training == Training consists of two phases – the “wake” phase and the “sleep” phase. It has been proven that this learning algorithm is convergent. === The "wake" phase === Neurons are fired by recognition connections (from what would be input to what would be output). Generative connections (leading from outputs to inputs) are then modified to increase probability that they would recreate the correct activity in the layer below – closer to actual data from sensory input. === The "sleep" phase === The process is reversed in the “sleep” phase – neurons are fired by generative connections while recognition connections are being modified to increase probability that they would recreate the correct activity in the layer above – further to actual data from sensory input. == Extensions == Since the recognition network is limited in its flexibility, it might not be able to approximate the posterior distribution of latent variables well. To better approximate the posterior distribution, it is possible to employ importance sampling, with the recognition network as the proposal distribution. This improved approximation of the posterior distribution also improves the overall performance of the model.