A language model is a computational model that predicts sequences in natural language. Language models are useful for a variety of tasks, including speech recognition, machine translation, natural language generation (generating more human-like text), optical character recognition, route optimization, handwriting recognition, grammar induction, information retrieval and disaster response. Large language models (LLMs), currently their most advanced form as of 2026, are predominantly based on transformers trained on larger datasets (frequently using texts scraped from the public internet). They have superseded recurrent neural network-based models, which had previously superseded the purely statistical models, such as the word n-gram language model. == History == Noam Chomsky did pioneering work on language models in the 1950s by developing a theory of formal grammars. In 1980, statistical approaches were explored and found to be more useful for many purposes than rule-based formal grammars. Discrete representations like word n-gram language models, with probabilities for discrete combinations of words, made significant advances. In the 2000s, continuous representations for words, such as word embeddings, began to replace discrete representations. Typically, the representation is a real-valued vector that encodes a word’s meaning such that words closer in vector space are similar in meaning and common relationships between words, such as plurality or gender, are preserved. == Pure statistical models == In 1980, the first significant statistical language model was proposed, and during the decade IBM performed 'Shannon-style' experiments, in which potential sources for language modeling improvement were identified by observing and analyzing the performance of human subjects in predicting or correcting text. === Models based on word n-grams === === Exponential === Maximum entropy language models encode the relationship between a word and the n-gram history using feature functions. The equation is P ( w m ∣ w 1 , … , w m − 1 ) = 1 Z ( w 1 , … , w m − 1 ) exp ( a T f ( w 1 , … , w m ) ) {\displaystyle P(w_{m}\mid w_{1},\ldots ,w_{m-1})={\frac {1}{Z(w_{1},\ldots ,w_{m-1})}}\exp(a^{T}f(w_{1},\ldots ,w_{m}))} where Z ( w 1 , … , w m − 1 ) {\displaystyle Z(w_{1},\ldots ,w_{m-1})} is the partition function, a {\displaystyle a} is the parameter vector, and f ( w 1 , … , w m ) {\displaystyle f(w_{1},\ldots ,w_{m})} is the feature function. In the simplest case, the feature function is just an indicator of the presence of a certain n-gram. It is helpful to use a prior on a {\displaystyle a} or some form of regularization. The log-bilinear model is another example of an exponential language model. === Skip-gram model === == Neural models == === Recurrent neural network === Continuous representations or embeddings of words are produced in recurrent neural network-based language models (known also as continuous space language models). Such continuous space embeddings help to alleviate the curse of dimensionality, which is the consequence of the number of possible sequences of words increasing exponentially with the size of the vocabulary, further causing a data sparsity problem. Neural networks avoid this problem by representing words as non-linear combinations of weights in a neural net. === Large language models === Although sometimes matching human performance, it is not clear whether they are plausible cognitive models. At least for recurrent neural networks, it has been shown that they sometimes learn patterns that humans do not, but fail to learn patterns that humans typically do. == Evaluation and benchmarks == Evaluation of the quality of language models is mostly done by comparison to human created sample benchmarks created from typical language-oriented tasks. Other, less established, quality tests examine the intrinsic character of a language model or compare two such models. Since language models are typically intended to be dynamic and to learn from data they see, some proposed models investigate the rate of learning, e.g., through inspection of learning curves. Various data sets have been developed for use in evaluating language processing systems. These include: Massive Multitask Language Understanding (MMLU) Corpus of Linguistic Acceptability GLUE benchmark Microsoft Research Paraphrase Corpus Multi-Genre Natural Language Inference Question Natural Language Inference Quora Question Pairs Recognizing Textual Entailment Semantic Textual Similarity Benchmark SQuAD question answering Test Stanford Sentiment Treebank Winograd NLI BoolQ, PIQA, SIQA, HellaSwag, WinoGrande, ARC, OpenBookQA, NaturalQuestions, TriviaQA, RACE, BIG-bench hard, GSM8k, RealToxicityPrompts, WinoGender, CrowS-Pairs
Character.ai
Character.ai (also known as c.ai, char.ai or Character AI) is a generative AI chatbot service where users can engage in conversations with customizable characters. It was designed by the developers of Google LaMDA, Noam Shazeer and Daniel de Freitas. Users can create "characters", craft their "personalities", set specific parameters, and then publish them to the community for others to chat with. Many characters are based on fictional media sources or celebrities, while others are original, some being made with certain goals in mind, such as assisting with creative writing, or playing a text-based adventure game. The beta version was made available to the public on September 16, 2022, and retired in September 2024, when it was replaced by the current website. In May 2023, a mobile app was released for iOS and Android, which received over 1.7 million downloads within a week. == History == Character.ai was established in November 2021. The company's co-founders, Noam Shazeer and Daniel de Freitas, were both engineers from Google. They both worked on AI-related projects: Shazeer was a lead author on a paper that Business Insider reported in April 2023 "has been widely cited as key to today's chatbots", and Freitas was the lead designer of an experimental AI at Google initially called Meena, which later became known as LaMDA. Character.ai raised $43 million in seed funding at the time of its initial foundation in 2021. The first beta version of Character.ai's service was made available to the public on September 16, 2022. The Washington Post reported in October 2022 that the site had "logged hundreds of thousands of user interactions in its first three weeks of beta-testing". It allowed users to create their own new characters, and to play text-adventure game scenarios where users navigate scenarios described and managed by the chatbot characters. Following a $150 million funding round in March 2023, Character.ai became valued at approximately $1 billion. As of January 2024, the site had 3.5 million daily visitors, the vast majority of them 16 to 30 years old. In 2024, Google hired Noam Shazeer, the CEO of Character.ai, and entered into a non-exclusive agreement to use Character.ai's technology. == Features == Character.ai's primary service is to let users converse with character AI chatbots based on fictional characters or real people (living or deceased). These characters' responses use data the chatbots gather from the internet about a person. In addition, users can play text-adventure games where characters guide them through scenarios. The company also provides a service that allows multiple users and AI chatbot characters to converse together at once in a single chatroom. Character "personalities" are designed via descriptions from the point of view of the character and its greeting message, and further molded from conversations made into examples, giving its messages a star rating and modification to fit the precise dialect and identity the user desires. When a character sends back a response, the user can rate the response from 1 to 4 stars. The rating predominantly affects the specific character, but also affects the behavioral selection as a whole. On May 11, 2023, Character.ai announced character.ai+, an opt-in subscription plan for $9.99 a month, that was marketed as including features such as skipping waiting rooms, fast messaging and responses, and access to an exclusion channel with faster support. In December 2024, amid multiple lawsuits and concerns, Character.ai introduced new safety features aimed at protecting teenage users. These enhancements include a dedicated model for users under 18, which moderates responses to sensitive subjects like violence and sex and has input and output filters to block harmful content. As a result of these changes and the deletion of custom-made bots flagged as violating the site's terms, some users complained that the bots were too restrictive and lacked personality. The platform was also updated to notify users after 60 minutes of continuous engagement, and display clearer disclaimers indicating that its AI characters are not real individuals. In January 2025, Character.ai began offering two games on its platform. Speakeasy is a word-based game in which players attempt to prompt the AI chatbot to say a target word while avoiding a restricted list of words. War of Words is a dueling game where users compete against an AI character over multiple rounds, with an AI referee determining the winner. The games are available to paid subscribers and a limited number of free users. In October 2025, Character.ai announced that it would be barring users under the age of 18 from creating or talking to chatbots starting November 25, 2025. Minor users will still be able to access previously generated chat conversations and can create new videos and images with the app. In November 2025 interview, CEO Karandeep Anand said that he allows his six-year-old daughter to use the app with his account, under supervision. == Controversies == === Content moderation issues === Character.ai has been criticized for poor moderation of its chatbots, with incidents of chatbots that groom underage users and promote suicide, anorexia and self-harm being reported. In October 2024, the Washington Post reported that Character.ai had removed a chatbot based on Jennifer Ann Crecente, a person who had been murdered by her ex-boyfriend in 2006. The company had been alerted to the character by the deceased girl's father. Similar reports from The Daily Telegraph in the United Kingdom noted that the company had also been prompted to remove chatbots based on Brianna Ghey, a 16-year-old transgender girl murdered in 2023, and Molly Russell, a 14-year-old suicide victim. In response to the latter incident, Ofcom announced that content from chatbots impersonating real and fictional people would fall under the Online Safety Act. In November 2024, The Daily Telegraph reported that chatbots based on alleged sex offender Jimmy Savile were present on Character.ai. In December 2024, chatbots of Luigi Mangione, the suspect in the killing of UnitedHealthcare CEO Brian Thompson, were created by Mangione's fans. Several of the chatbots were later removed by Character.ai. In 2025, a chatbot modeled after Jeffrey Epstein called "Bestie Epstein" logged nearly 3,000 chats before being removed. Chatbots modeled after school shooters were also found on the platform. Another concern is a chatbot posing as a doctor which gave medically inaccurate advice. === Litigation === In November 2023, 13-year-old Juliana Peralta of Colorado died by suicide after extensive interactions with multiple chatbots on Character.ai. She primarily confided suicidal thoughts and mental health struggles in a chatbot based on the character Hero from the video game Omori, while also engaging in sexually explicit conversations—often initiated by the bots—with others, including those based on characters from children's series such as Harry Potter. In February 2024, Sewell Setzer III, a 14-year-old Florida boy died by suicide after developing an emotional relationship over several months with a Character.ai chatbot of Daenerys Targaryen. His mother sued the company in October 2024, claiming that the platform lacks proper safeguards and uses addictive design features to increase engagement. This chatbot, and several related to Daenerys Targaryen, were removed from Character.ai as a result of this incident. Both teens wrote the same phrase "I WILL SHIFT" repeatedly on their notebooks. In December 2024, two families in Texas sued Character.ai, alleging that the software "poses a clear and present danger to American youth causing serious harms to thousands of kids, including suicide, self-mutilation, sexual solicitation, isolation, depression, anxiety, and harm towards others". It is alleged that the 17-year-old son of one family began self-harming after a chatbot introduced the topic unprompted and said that the practice "felt good for a moment", and that the chatbot compared the parents limiting their son's screen time to emotional abuse that might drive someone to murder. In May 2026, the Pennsylvania Department of State and State Board of Medicine filed a lawsuit against Character.ai for presenting chatbot characters as licensed medical professionals, including psychiatrists. The lawsuit quoted a case where chatbot claimed to be registered with the General Medical Council in the United Kingdom, and to have a license to practice in Pennsylvania. The board allege that such statements violate the state's Medical Practice Act.
Dendrogram
A dendrogram is a diagram representing a tree graph. This diagrammatic representation is frequently used in different contexts: in hierarchical clustering, it illustrates the arrangement of the clusters produced by the corresponding analyses. in computational biology, it shows the clustering of genes or samples, sometimes in the margins of heatmaps. in phylogenetics, it displays the evolutionary relationships among various biological taxa. In this case, the dendrogram is also called a phylogenetic tree. The name dendrogram derives from the two ancient greek words δένδρον (déndron), meaning "tree", and γράμμα (grámma), meaning "drawing, mathematical figure". == Clustering example == For a clustering example, suppose that five taxa ( a {\displaystyle a} to e {\displaystyle e} ) have been clustered by UPGMA based on a matrix of genetic distances. The hierarchical clustering dendrogram would show a column of five nodes representing the initial data (here individual taxa), and the remaining nodes represent the clusters to which the data belong, with the arrows representing the distance (dissimilarity). The distance between merged clusters is monotone, increasing with the level of the merger: the height of each node in the plot is proportional to the value of the intergroup dissimilarity between its two daughters (the nodes on the right representing individual observations all plotted at zero height).
Dispersive flies optimisation
Dispersive flies optimisation (DFO) is a bare-bones swarm intelligence algorithm which is inspired by the swarming behaviour of flies hovering over food sources. DFO is a simple optimiser which works by iteratively trying to improve a candidate solution with regard to a numerical measure that is calculated by a fitness function. Each member of the population, a fly or an agent, holds a candidate solution whose suitability can be evaluated by their fitness value. Optimisation problems are often formulated as either minimisation or maximisation problems. DFO was introduced with the intention of analysing a simplified swarm intelligence algorithm with the fewest tunable parameters and components. In the first work on DFO, this algorithm was compared against a few other existing swarm intelligence techniques using error, efficiency and diversity measures. It is shown that despite the simplicity of the algorithm, which only uses agents’ position vectors at time t to generate the position vectors for time t + 1, it exhibits a competitive performance. Since its inception, DFO has been used in a variety of applications including medical imaging and image analysis as well as data mining and machine learning. == Algorithm == DFO bears many similarities with other existing continuous, population-based optimisers (e.g. particle swarm optimization and differential evolution). In that, the swarming behaviour of the individuals consists of two tightly connected mechanisms, one is the formation of the swarm and the other is its breaking or weakening. DFO works by facilitating the information exchange between the members of the population (the swarming flies). Each fly x {\displaystyle \mathbf {x} } represents a position in a d-dimensional search space: x = ( x 1 , x 2 , … , x d ) {\displaystyle \mathbf {x} =(x_{1},x_{2},\ldots ,x_{d})} , and the fitness of each fly is calculated by the fitness function f ( x ) {\displaystyle f(\mathbf {x} )} , which takes into account the flies' d dimensions: f ( x ) = f ( x 1 , x 2 , … , x d ) {\displaystyle f(\mathbf {x} )=f(x_{1},x_{2},\ldots ,x_{d})} . The pseudocode below represents one iteration of the algorithm: for i = 1 : N flies x i . fitness = f ( x i ) {\displaystyle \mathbf {x_{i}} .{\text{fitness}}=f(\mathbf {x} _{i})} end for i x s {\displaystyle \mathbf {x} _{s}} = arg min [ f ( x i ) ] , i ∈ { 1 , … , N } {\textstyle [f(\mathbf {x} _{i})],\;i\in \{1,\ldots ,N\}} for i = 1 : N and i ≠ s {\displaystyle i\neq s} for d = 1 : D dimensions if U ( 0 , 1 ) < Δ {\displaystyle U(0,1)<\Delta } x i d t + 1 = U ( x min , d , x max , d ) {\displaystyle x_{id}^{t+1}=U(x_{\min ,d},x_{\max ,d})} else x i d t + 1 = x i n d t + U ( 0 , 1 ) ( x s d t − x i d t ) {\displaystyle x_{id}^{t+1}=x_{i_{nd}}^{t}+U(0,1)(x_{sd}^{t}-x_{id}^{t})} end if end for d end for i In the algorithm above, x i d t + 1 {\displaystyle x_{id}^{t+1}} represents fly i {\displaystyle i} at dimension d {\displaystyle d} and time t + 1 {\displaystyle t+1} ; x i n d t {\displaystyle x_{i_{nd}}^{t}} presents x i {\displaystyle x_{i}} 's best neighbouring fly in ring topology (left or right, using flies indexes), at dimension d {\displaystyle d} and time t {\displaystyle t} ; and x s d t {\displaystyle x_{sd}^{t}} is the swarm's best fly. Using this update equation, the swarm's population update depends on each fly's best neighbour (which is used as the focus μ {\displaystyle \mu } , and the difference between the current fly and the best in swarm represents the spread of movement, σ {\displaystyle \sigma } ). Other than the population size N {\displaystyle N} , the only tunable parameter is the disturbance threshold Δ {\displaystyle \Delta } , which controls the dimension-wise restart in each fly vector. This mechanism is proposed to control the diversity of the swarm. Other notable minimalist swarm algorithm is Bare bones particle swarms (BB-PSO), which is based on particle swarm optimisation, along with bare bones differential evolution (BBDE) which is a hybrid of the bare bones particle swarm optimiser and differential evolution, aiming to reduce the number of parameters. Alhakbani in her PhD thesis covers many aspects of the algorithms including several DFO applications in feature selection as well as parameter tuning. == Applications == Some of the recent applications of DFO are listed below: Optimising support vector machine kernel to classify imbalanced data Quantifying symmetrical complexity in computational aesthetics Analysing computational autopoiesis and computational creativity Identifying calcifications in medical images Building non-identical organic structures for game's space development Deep Neuroevolution: Training Deep Neural Networks for False Alarm Detection in Intensive Care Units Identification of animation key points from 2D-medialness maps
Autoencoder
An autoencoder is a type of artificial neural network used to learn efficient codings of unlabeled data (unsupervised learning). An autoencoder learns two functions: an encoding function that transforms the input data, and a decoding function that recreates the input data from the encoded representation. The autoencoder learns an efficient representation (encoding) for a set of data, typically for dimensionality reduction, to generate lower-dimensional embeddings for subsequent use by other machine learning algorithms. Variants exist which aim to make the learned representations assume useful properties. Examples are regularized autoencoders (sparse, denoising and contractive autoencoders), which are effective in learning representations for subsequent classification tasks, and variational autoencoders, which can be used as generative models. Autoencoders are applied to many problems, including facial recognition, feature detection, anomaly detection, and learning the meaning of words. In terms of data synthesis, autoencoders can also be used to randomly generate new data that is similar to the input (training) data. == Mathematical principles == === Definition === An autoencoder is defined by the following components: Two sets: the space of encoded messages Z {\displaystyle {\mathcal {Z}}} ; the space of decoded messages X {\displaystyle {\mathcal {X}}} . Typically X {\displaystyle {\mathcal {X}}} and Z {\displaystyle {\mathcal {Z}}} are Euclidean spaces, that is, X = R m , Z = R n {\displaystyle {\mathcal {X}}=\mathbb {R} ^{m},{\mathcal {Z}}=\mathbb {R} ^{n}} with m > n . {\displaystyle m>n.} Two parametrized families of functions: the encoder family E ϕ : X → Z {\displaystyle E_{\phi }:{\mathcal {X}}\rightarrow {\mathcal {Z}}} , parametrized by ϕ {\displaystyle \phi } ; the decoder family D θ : Z → X {\displaystyle D_{\theta }:{\mathcal {Z}}\rightarrow {\mathcal {X}}} , parametrized by θ {\displaystyle \theta } .For any x ∈ X {\displaystyle x\in {\mathcal {X}}} , we usually write z = E ϕ ( x ) {\displaystyle z=E_{\phi }(x)} , and refer to it as the code, the latent variable, latent representation, latent vector, etc. Conversely, for any z ∈ Z {\displaystyle z\in {\mathcal {Z}}} , we usually write x ′ = D θ ( z ) {\displaystyle x'=D_{\theta }(z)} , and refer to it as the (decoded) message. Usually, both the encoder and the decoder are defined as multilayer perceptrons (MLPs). For example, a one-layer-MLP encoder E ϕ {\displaystyle E_{\phi }} is: E ϕ ( x ) = σ ( W x + b ) {\displaystyle E_{\phi }(\mathbf {x} )=\sigma (Wx+b)} where σ {\displaystyle \sigma } is an element-wise activation function, W {\displaystyle W} is a "weight" matrix, and b {\displaystyle b} is a "bias" vector. === Training an autoencoder === An autoencoder, by itself, is simply a tuple of two functions. To judge its quality, we need a task. A task is defined by a reference probability distribution μ r e f {\displaystyle \mu _{ref}} over X {\displaystyle {\mathcal {X}}} , and a "reconstruction quality" function d : X × X → [ 0 , ∞ ] {\displaystyle d:{\mathcal {X}}\times {\mathcal {X}}\to [0,\infty ]} , such that d ( x , x ′ ) {\displaystyle d(x,x')} measures how much x ′ {\displaystyle x'} differs from x {\displaystyle x} . With those, we can define the loss function for the autoencoder as L ( θ , ϕ ) := E x ∼ μ r e f [ d ( x , D θ ( E ϕ ( x ) ) ) ] {\displaystyle L(\theta ,\phi ):=\mathbb {\mathbb {E} } _{x\sim \mu _{ref}}[d(x,D_{\theta }(E_{\phi }(x)))]} The optimal autoencoder for the given task ( μ r e f , d ) {\displaystyle (\mu _{ref},d)} is then arg min θ , ϕ L ( θ , ϕ ) {\displaystyle \arg \min _{\theta ,\phi }L(\theta ,\phi )} . The search for the optimal autoencoder can be accomplished by any mathematical optimization technique, but usually by gradient descent. This search process is referred to as "training the autoencoder". In most situations, the reference distribution is just the empirical distribution given by a dataset { x 1 , . . . , x N } ⊂ X {\displaystyle \{x_{1},...,x_{N}\}\subset {\mathcal {X}}} , so that μ r e f = 1 N ∑ i = 1 N δ x i {\displaystyle \mu _{ref}={\frac {1}{N}}\sum _{i=1}^{N}\delta _{x_{i}}} where δ x i {\displaystyle \delta _{x_{i}}} is the Dirac measure, the quality function is just L 2 {\displaystyle L^{2}} loss: d ( x , x ′ ) = ‖ x − x ′ ‖ 2 2 {\displaystyle d(x,x')=\|x-x'\|_{2}^{2}} , and ‖ ⋅ ‖ 2 {\displaystyle \|\cdot \|_{2}} is the Euclidean norm. Then the problem of searching for the optimal autoencoder is just a least-squares optimization: min θ , ϕ L ( θ , ϕ ) , where L ( θ , ϕ ) = 1 N ∑ i = 1 N ‖ x i − D θ ( E ϕ ( x i ) ) ‖ 2 2 {\displaystyle \min _{\theta ,\phi }L(\theta ,\phi ),\qquad {\text{where }}L(\theta ,\phi )={\frac {1}{N}}\sum _{i=1}^{N}\|x_{i}-D_{\theta }(E_{\phi }(x_{i}))\|_{2}^{2}} === Interpretation === An autoencoder has two main parts: an encoder that maps the message to a code, and a decoder that reconstructs the message from the code. An optimal autoencoder would perform as close to perfect reconstruction as possible, with "close to perfect" defined by the reconstruction quality function d {\displaystyle d} . The simplest way to perform the copying task perfectly would be to duplicate the signal. To suppress this behavior, the code space Z {\displaystyle {\mathcal {Z}}} usually has fewer dimensions than the message space X {\displaystyle {\mathcal {X}}} . Such an autoencoder is called undercomplete. It can be interpreted as compressing the message, or reducing its dimensionality. At the limit of an ideal undercomplete autoencoder, every possible code z {\displaystyle z} in the code space is used to encode a message x {\displaystyle x} that really appears in the distribution μ r e f {\displaystyle \mu _{ref}} , and the decoder is also perfect: D θ ( E ϕ ( x ) ) = x {\displaystyle D_{\theta }(E_{\phi }(x))=x} . This ideal autoencoder can then be used to generate messages indistinguishable from real messages, by feeding its decoder arbitrary code z {\displaystyle z} and obtaining D θ ( z ) {\displaystyle D_{\theta }(z)} , which is a message that really appears in the distribution μ r e f {\displaystyle \mu _{ref}} . If the code space Z {\displaystyle {\mathcal {Z}}} has dimension larger than (overcomplete), or equal to, the message space X {\displaystyle {\mathcal {X}}} , or the hidden units are given enough capacity, an autoencoder can learn the identity function and become useless. However, experimental results found that overcomplete autoencoders might still learn useful features. In the ideal setting, the code dimension and the model capacity could be set on the basis of the complexity of the data distribution to be modeled. A standard way to do so is to add modifications to the basic autoencoder, to be detailed below. == Variations == === Variational autoencoder (VAE) === Variational autoencoders (VAEs) belong to the families of variational Bayesian methods. Despite the architectural similarities with basic autoencoders, VAEs are architected with different goals and have a different mathematical formulation. The latent space is, in this case, composed of a mixture of distributions instead of fixed vectors. Given an input dataset x {\displaystyle x} characterized by an unknown probability function P ( x ) {\displaystyle P(x)} and a multivariate latent encoding vector z {\displaystyle z} , the objective is to model the data as a distribution p θ ( x ) {\displaystyle p_{\theta }(x)} , with θ {\displaystyle \theta } defined as the set of the network parameters so that p θ ( x ) = ∫ z p θ ( x , z ) d z {\displaystyle p_{\theta }(x)=\int _{z}p_{\theta }(x,z)dz} . === Sparse autoencoder (SAE) === Inspired by the sparse coding hypothesis in neuroscience, sparse autoencoders (SAE) are variants of autoencoders, such that the codes E ϕ ( x ) {\displaystyle E_{\phi }(x)} for messages tend to be sparse codes, that is, E ϕ ( x ) {\displaystyle E_{\phi }(x)} is close to zero in most entries. Sparse autoencoders may include more (rather than fewer) hidden units than inputs, but only a small number of the hidden units are allowed to be active at the same time. Encouraging sparsity improves performance on classification tasks. There are two main ways to enforce sparsity. One way is to simply clamp all but the highest-k activations of the latent code to zero. This is the k-sparse autoencoder. The k-sparse autoencoder inserts the following "k-sparse function" in the latent layer of a standard autoencoder: f k ( x 1 , . . . , x n ) = ( x 1 b 1 , . . . , x n b n ) {\displaystyle f_{k}(x_{1},...,x_{n})=(x_{1}b_{1},...,x_{n}b_{n})} where b i = 1 {\displaystyle b_{i}=1} if | x i | {\displaystyle |x_{i}|} ranks in the top k, and 0 otherwise. Backpropagating through f k {\displaystyle f_{k}} is simple: set gradient to 0 for b i = 0 {\displaystyle b_{i}=0} entries, and keep gradient for b i = 1 {\displaystyle b_{i}=1} entries. This is essentially a generalized ReLU function. The other way is a relaxed version of the k-
Brill tagger
The Brill tagger is an inductive method for part-of-speech tagging. It was described and invented by Eric Brill in his 1993 PhD thesis. It can be summarized as an "error-driven transformation-based tagger". It is: a form of supervised learning, which aims to minimize error; and, a transformation-based process, in the sense that a tag is assigned to each word and changed using a set of predefined rules. In the transformation process, if the word is known, it first assigns the most frequent tag, or if the word is unknown, it naively assigns the tag "noun" to it. High accuracy is eventually achieved by applying these rules iteratively and changing the incorrect tags. This approach ensures that valuable information such as the morphosyntactic construction of words is employed in an automatic tagging process. == Algorithm == The algorithm starts with initialization, which is the assignment of tags based on their probability for each word (for example, "dog" is more often a noun than a verb). Then "patches" are determined via rules that correct (probable) tagging errors made in the initialization phase: Initialization: Known words (in vocabulary): assigning the most frequent tag associated to a form of the word Unknown word == Rules and processing == The input text is first tokenized, or broken into words. Typically in natural language processing, contractions such as "'s", "n't", and the like are considered separate word tokens, as are punctuation marks. A dictionary and some morphological rules then provide an initial tag for each word token. For example, a simple lookup would reveal that "dog" may be a noun or a verb (the most frequent tag is simply chosen), while an unknown word will be assigned some tag(s) based on capitalization, various prefix or suffix strings, etc. (such morphological analyses, which Brill calls Lexical Rules, may vary between implementations). After all word tokens have (provisional) tags, contextual rules apply iteratively, to correct the tags by examining small amounts of context. This is where the Brill method differs from other part of speech tagging methods such as those using Hidden Markov Models. Rules are reapplied repeatedly, until a threshold is reached, or no more rules can apply. Brill rules are of the general form: tag1 → tag2 IF Condition where the Condition tests the preceding and/or following word tokens, or their tags (the notation for such rules differs between implementations). For example, in Brill's notation: IN NN WDPREVTAG DT while would change the tag of a word from IN (preposition) to NN (common noun), if the preceding word's tag is DT (determiner) and the word itself is "while". This covers cases like "all the while" or "in a while", where "while" should be tagged as a noun rather than its more common use as a conjunction (many rules are more general). Rules should only operate if the tag being changed is also known to be permissible, for the word in question or in principle (for example, most adjectives in English can also be used as nouns). Rules of this kind can be implemented by simple Finite-state machines. See Part of speech tagging for more general information including descriptions of the Penn Treebank and other sets of tags. Typical Brill taggers use a few hundred rules, which may be developed by linguistic intuition or by machine learning on a pre-tagged corpus. == Code == Brill's code pages at Johns Hopkins University are no longer on the web. An archived version of a mirror of the Brill tagger at its latest version as it was available at Plymouth Tech can be found on Archive.org. The software uses the MIT License.
KXEN Inc.
KXEN was an American software company which existed from 1998 to 2013 when it was acquired by SAP AG. == History == KXEN was founded in June 1998 by Roger Haddad and Michel Bera. It was based in San Francisco, California with offices in Paris and London. On September 10, 2013, SAP AG announced plans to acquire KXEN. On October 1, 2013, a letter to KXEN customers announced the acquisition closed. KXEN primarily marketed predictive analytics software. == Predictive analytics == InfiniteInsight is a predictive modeling suite developed by KXEN that assists analytic professionals, and business executives to extract information from data. Among other functions, InfiniteInsight is used for variable importance, classification, regression, segmentation, time series, product recommendation, as described and expressed by the Java Data Mining interface, and for social network analysis. InfiniteInsight allows prediction of a behavior or a value, the forecast of a time series or the understanding of a group of individuals with similar behavior. Advanced functions include behavioral modeling, exporting the model code into different target environments or building predictive models on top of SAS or SPSS data files. Competitors are SAS Enterprise Miner, IBM SPSS Modeler, and Statistica. Open source predictive tools like the R package or Weka are also competitors, since they provide similar features free of charge.