AI Chatbot Interface

AI Chatbot Interface — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

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  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.

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  Ordinal regression

  In statistics, ordinal regression, also called ordinal classification, is a type of regression analysis used for predicting an ordinal variable, i.e. a variable whose value exists on an arbitrary scale where only the relative ordering between different values is significant. It can be considered an intermediate problem between regression and classification. Examples of ordinal regression are ordered logit and ordered probit. Ordinal regression turns up often in the social sciences, for example in the modeling of human levels of preference (on a scale from, say, 1–5 for "very poor" through "excellent"), as well as in information retrieval. In machine learning, ordinal regression may also be called ranking learning. == Linear models for ordinal regression == Ordinal regression can be performed using a generalized linear model (GLM) that fits both a coefficient vector and a set of thresholds to a dataset. Suppose one has a set of observations, represented by length-p vectors x1 through xn, with associated responses y1 through yn, where each yi is an ordinal variable on a scale 1, ..., K. For simplicity, and without loss of generality, we assume y is a non-decreasing vector, that is, yi ≤ {\displaystyle \leq } yi+1. To this data, one fits a length-p coefficient vector w and a set of thresholds θ1, ..., θK−1 with the property that θ1 < θ2 < ... < θK−1. This set of thresholds divides the real number line into K disjoint segments, corresponding to the K response levels. The model can now be formulated as Pr ( y ≤ i ∣ x ) = σ ( θ i − w ⋅ x ) {\displaystyle \Pr(y\leq i\mid \mathbf {x} )=\sigma (\theta _{i}-\mathbf {w} \cdot \mathbf {x} )} or, the cumulative probability of the response y being at most i is given by a function σ (the inverse link function) applied to a linear function of x. Several choices exist for σ; the logistic function σ ( θ i − w ⋅ x ) = 1 1 + e − ( θ i − w ⋅ x ) {\displaystyle \sigma (\theta _{i}-\mathbf {w} \cdot \mathbf {x} )={\frac {1}{1+e^{-(\theta _{i}-\mathbf {w} \cdot \mathbf {x} )}}}} gives the ordered logit model, while using the CDF of the standard normal distribution gives the ordered probit model. A third option is to use an exponential function σ ( θ i − w ⋅ x ) = 1 − exp ⁡ ( − exp ⁡ ( θ i − w ⋅ x ) ) {\displaystyle \sigma (\theta _{i}-\mathbf {w} \cdot \mathbf {x} )=1-\exp(-\exp(\theta _{i}-\mathbf {w} \cdot \mathbf {x} ))} which gives the proportional hazards model. === Latent variable model === The probit version of the above model can be justified by assuming the existence of a real-valued latent variable (unobserved quantity) y, determined by y ∗ = w ⋅ x + ε {\displaystyle y^{}=\mathbf {w} \cdot \mathbf {x} +\varepsilon } where ε is normally distributed with zero mean and unit variance, conditioned on x. The response variable y results from an "incomplete measurement" of y, where one only determines the interval into which y falls: y = { 1 if y ∗ ≤ θ 1 , 2 if θ 1 < y ∗ ≤ θ 2 , 3 if θ 2 < y ∗ ≤ θ 3 ⋮ K if θ K − 1 < y ∗ . {\displaystyle y={\begin{cases}1&{\text{if}}~~y^{}\leq \theta _{1},\\2&{\text{if}}~~\theta _{1} Read more →

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  Plate notation

  In Bayesian inference, plate notation is a method of representing variables that repeat in a graphical model. Instead of drawing each repeated variable individually, a plate or rectangle is used to group variables into a subgraph that repeat together, and a number is drawn on the plate to represent the number of repetitions of the subgraph in the plate. The assumptions are that the subgraph is duplicated that many times, the variables in the subgraph are indexed by the repetition number, and any links that cross a plate boundary are replicated once for each subgraph repetition. == Example == In this example, we consider Latent Dirichlet allocation, a Bayesian network that models how documents in a corpus are topically related. There are two variables not in any plate; α is the parameter of the uniform Dirichlet prior on the per-document topic distributions, and β is the parameter of the uniform Dirichlet prior on the per-topic word distribution. The outermost plate represents all the variables related to a specific document, including θ i {\displaystyle \theta _{i}} , the topic distribution for document i. The M in the corner of the plate indicates that the variables inside are repeated M times, once for each document. The inner plate represents the variables associated with each of the N i {\displaystyle N_{i}} words in document i: z i j {\displaystyle z_{ij}} is the topic distribution for the jth word in document i, and w i j {\displaystyle w_{ij}} is the actual word used. The N in the corner represents the repetition of the variables in the inner plate N j {\displaystyle N_{j}} times, once for each word in document i. The circle representing the individual words is shaded, indicating that each w i j {\displaystyle w_{ij}} is observable, and the other circles are empty, indicating that the other variables are latent variables. The directed edges between variables indicate dependencies between the variables: for example, each w i j {\displaystyle w_{ij}} depends on z i j {\displaystyle z_{ij}} and β. == Extensions == A number of extensions have been created by various authors to express more information than simply the conditional relationships. However, few of these have become standard. Perhaps the most commonly used extension is to use rectangles in place of circles to indicate non-random variables—either parameters to be computed, hyperparameters given a fixed value (or computed through empirical Bayes), or variables whose values are computed deterministically from a random variable. The diagram on the right shows a few more non-standard conventions used in some articles in Wikipedia (e.g. variational Bayes): Variables that are actually random vectors are indicated by putting the vector size in brackets in the middle of the node. Variables that are actually random matrices are similarly indicated by putting the matrix size in brackets in the middle of the node, with commas separating row size from column size. Categorical variables are indicated by placing their size (without a bracket) in the middle of the node. Categorical variables that act as "switches", and which pick one or more other random variables to condition on from a large set of such variables (e.g. mixture components), are indicated with a special type of arrow containing a squiggly line and ending in a T junction. Boldface is consistently used for vector or matrix nodes (but not categorical nodes). == Software implementation == Plate notation has been implemented in various TeX/LaTeX drawing packages, but also as part of graphical user interfaces to Bayesian statistics programs such as BUGS and BayesiaLab and PyMC.

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  Relationship square

  In statistics, the relationship square is a graphical representation for use in the factorial analysis of a table individuals x variables. This representation completes classical representations provided by principal component analysis (PCA) or multiple correspondence analysis (MCA), namely those of individuals, of quantitative variables (correlation circle) and of the categories of qualitative variables (at the centroid of the individuals who possess them). It is especially important in factor analysis of mixed data (FAMD) and in multiple factor analysis (MFA). == Definition of relationship square in the MCA frame == The first interest of the relationship square is to represent the variables themselves, not their categories, which is all the more valuable as there are many variables. For this, we calculate for each qualitative variable j {\displaystyle j} and each factor F s {\displaystyle F_{s}} ( F s {\displaystyle F_{s}} , rank s {\displaystyle s} factor, is the vector of coordinates of the individuals along the axis of rank s {\displaystyle s} ; in PCA, F s {\displaystyle F_{s}} is called principal component of rank s {\displaystyle s} ), the square of the correlation ratio between the F s {\displaystyle F_{s}} and the variable j {\displaystyle j} , usually denoted : η 2 ( j , F s ) {\displaystyle \eta ^{2}(j,F_{s})} Thus, to each factorial plane, we can associate a representation of qualitative variables themselves. Their coordinates being between 0 and 1, the variables appear in the square having as vertices the points (0,0), ( 0,1), (1,0) and (1,1). == Example in MCA == Six individuals ( i 1 , … , i 6 ) {\displaystyle i_{1},\ldots ,i_{6})} are described by three variables ( q 1 , q 2 , q 3 ) {\displaystyle (q_{1},q_{2},q_{3})} having respectively 3, 2 and 3 categories. Example : the individual i 1 {\displaystyle i_{1}} possesses the category a {\displaystyle a} of q 1 {\displaystyle q_{1}} , d {\displaystyle d} of q 2 {\displaystyle q_{2}} and f {\displaystyle f} of q 3 {\displaystyle q_{3}} . Applied to these data, the MCA function included in the R Package FactoMineR provides to the classical graph in Figure 1. The relationship square (Figure 2) makes easier the reading of the classic factorial plane. It indicates that: The first factor is related to the three variables but especially q 3 {\displaystyle q_{3}} (which have a very high coordinate along the first axis) and then q 2 {\displaystyle q_{2}} . The second factor is related only to q 1 {\displaystyle q_{1}} and q 3 {\displaystyle q_{3}} (and not to q 2 {\displaystyle q_{2}} which has a coordinate along axis 2 equal to 0) and that in a strong and equal manner. All this is visible on the classic graphic but not so clearly. The role of the relationship square is first to assist in reading a conventional graphic. This is precious when the variables are numerous and possess numerous coordinates. == Extensions == This representation may be supplemented with those of quantitative variables, the coordinates of the latter being the square of correlation coefficients (and not of correlation ratios). Thus, the second advantage of the relationship square lies in the ability to represent simultaneously quantitative and qualitative variables. The relationship square can be constructed from any factorial analysis of a table individuals x variables. In particular, it is (or should be) used systematically: in multiple correspondences analysis (MCA); in principal components analysis (PCA) when there are many supplementary variables; in factor analysis of mixed data (FAMD). An extension of this graphic to groups of variables (how to represent a group of variables by a single point ?) is used in Multiple Factor Analysis (MFA) == History == The idea of representing the qualitative variables themselves by a point (and not the categories) is due to Brigitte Escofier. The graphic as it is used now has been introduced by Brigitte Escofier and Jérôme Pagès in the framework of multiple factor analysis == Conclusion == In MCA, the relationship square provides a synthetic view of the connections between mixed variables, all the more valuable as there are many variables having many categories. This representation iscan be useful in any factorial analysis when there are numerous mixed variables, active and/or supplementary.

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  Dominant resource fairness

  Dominant resource fairness (DRF) is a rule for fair division. It is particularly useful for dividing computing resources in among users in cloud computing environments, where each user may require a different combination of resources. DRF was presented by Ali Ghodsi, Matei Zaharia, Benjamin Hindman, Andy Konwinski, Scott Shenker and Ion Stoica in 2011. == Motivation == In an environment with a single resource, a widely used criterion is max-min fairness, which aims to maximize the minimum amount of resource given to a user. But in cloud computing, it is required to share different types of resource, such as: memory, CPU, bandwidth and disk-space. Previous fair schedulers, such as in Apache Hadoop, reduced the multi-resource setting to a single-resource setting by defining nodes with a fixed amount of each resource (e.g. 4 CPU, 32 MB memory, etc.), and dividing slots which are fractions of nodes. But this method is inefficient, since not all users need the same ratio of resources. For example, some users need more CPU whereas other users need more memory. As a result, most tasks either under-utilize or over-utilize their resources. DRF solves the problem by maximizing the minimum amount of the dominant resource given to a user (then the second-minimum etc., in a leximin order). The dominant resource may be different for different users. For example, if user A runs CPU-heavy tasks and user B runs memory-heavy tasks, DRF will try to equalize the CPU share given to user A and the memory share given to user B. == Definition == There are m resources. The total capacities of the resources are r1,...,rm. There are n users. Each users runs individual tasks. Each task has a demand-vector (d1,..,dm), representing the amount it needs of each resource. It is implicitly assumed that the utility of a user equals the number of tasks he can perform. For example, if user A runs tasks with demand-vector [1 CPU, 4 GB RAM], and receives 3 CPU and 8 GB RAM, then his utility is 2, since he can perform only 2 tasks. More generally, the utility of a user receiving x1,...,xm resources is minj(xj/dj), that is, the users have Leontief utilities. The demand-vectors are normalized to fractions of the capacities. For example, if the system has 9 CPUs and 18 GB RAM, then the above demand-vector is normalized to [1/9 CPU, 2/9 GB]. For each user, the resource with the highest demand-fraction is called the dominant resource. In the above example, the dominant resource is memory, as 2/9 is the largest fraction. If user B runs a task with demand-vector [3 CPU, 1 GB], which is normalized to [1/3 CPU, 1/18 GB], then his dominant resource is CPU. DRF aims to find the maximum x such that all agents can receive at least x of their dominant resource. In the above example, this maximum x is 2/3: User A gets 3 tasks, which require 3/9 CPU and 2/3 GB. User B gets 2 tasks, which require 2/3 CPU and 1/9 GB. The maximum x can be found by solving a linear program; see Lexicographic max-min optimization. Alternatively, the DRF can be computed sequentially. The algorithm tracks the amount of dominant resource used by each user. At each round, it finds a user with the smallest allocated dominant resource so far, and allocates the next task of this user. Note that this procedure allows the same user to run tasks with different demand vectors. == Properties == DRF has several advantages over other policies for resource allocation. Proportionality: each user receives at least as much resources as they could get in a system in which all resources are partitioned equally among users (the authors call this condition "sharing incentive"). Strategyproofness: a user cannot get a larger allocation by lying about his needs. Strategyproofness is important, as evidence from cloud operators show that users try to manipulate the servers in order to get better allocations. Envy-freeness: no user would prefer the allocation of another user. Pareto efficiency: no other allocation is better for some users and not worse for anyone. Population monotonicity: when a user leaves the system, the allocations of remaining users do not decrease. When there is a single resource that is a bottleneck resource (highly demanded by all users), DRF reduces to max-min fairness. However, DRF violates resource monotonicity: when resources are added to the system, some allocations might decrease. == Extensions == Weighted DRF is an extension of DRF to settings in which different users have different weights (representing their different entitlements). Parkes, Procaccia and Shah formally extend weighted DRF to a setting in which some users do not need all resources (that is, they may have demand 0 to some resource). They prove that the extended version still satisfies proportionality, Pareto-efficiency, envy-freeness, strategyproofness, and even Group strategyproofness. On the other hand, they show that DRF may yield poor utilitarian social welfare, that is, the sum of utilities may be only 1/m of the optimum. However, they prove that any mechanism satisfying one of proportionality, envy-freeness or strategyproofness may suffers from the same low utilitarian welfare. They also extend DRF to the setting in which the users' demands are indivisible (as in fair item allocation). For the indivisible setting, they relax envy-freeness to EF1. They show that strategyproofness is incompatible with PO+EF1 or with PO+proportionality. However, a mechanism called SequentialMinMax satisfies efficiency, proportionality and EF1. Wang, Li and Liang present DRFH - an extension of DRF to a system with several heterogeneous servers. == Implementation == DRF was first implemented in Apache Mesos - a cluster resource manager, and it led to better throughput and fairness than previously used fair-sharing schemes.

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  Tucker decomposition

  In mathematics, Tucker decomposition decomposes a tensor into a set of matrices and one small core tensor. It is named after Ledyard R. Tucker although it goes back to Hitchcock in 1927. Initially described as a three-mode extension of factor analysis and principal component analysis it may actually be generalized to higher mode analysis, which is also called higher-order singular value decomposition (HOSVD) or the M-mode SVD. The algorithm to which the literature typically refers when discussing the Tucker decomposition or the HOSVD is the M-mode SVD algorithm introduced by Vasilescu and Terzopoulos, but misattributed to Tucker or De Lathauwer etal. It may be regarded as a more flexible PARAFAC (parallel factor analysis) model. In PARAFAC the core tensor is restricted to be "diagonal". In practice, Tucker decomposition is used as a modelling tool. For instance, it is used to model three-way (or higher way) data by means of relatively small numbers of components for each of the three or more modes, and the components are linked to each other by a three- (or higher-) way core array. The model parameters are estimated in such a way that, given fixed numbers of components, the modelled data optimally resemble the actual data in the least squares sense. The model gives a summary of the information in the data, in the same way as principal components analysis does for two-way data. For a 3rd-order tensor T ∈ F n 1 × n 2 × n 3 {\displaystyle T\in F^{n_{1}\times n_{2}\times n_{3}}} , where F {\displaystyle F} is either R {\displaystyle \mathbb {R} } or C {\displaystyle \mathbb {C} } , Tucker Decomposition can be denoted as follows, T = T × 1 U ( 1 ) × 2 U ( 2 ) × 3 U ( 3 ) {\displaystyle T={\mathcal {T}}\times _{1}U^{(1)}\times _{2}U^{(2)}\times _{3}U^{(3)}} where T ∈ F d 1 × d 2 × d 3 {\displaystyle {\mathcal {T}}\in F^{d_{1}\times d_{2}\times d_{3}}} is the core tensor, a 3rd-order tensor that contains the 1-mode, 2-mode and 3-mode singular values of T {\displaystyle T} , which are defined as the Frobenius norm of the 1-mode, 2-mode and 3-mode slices of tensor T {\displaystyle {\mathcal {T}}} respectively. U ( 1 ) , U ( 2 ) , U ( 3 ) {\displaystyle U^{(1)},U^{(2)},U^{(3)}} are unitary matrices in F d 1 × n 1 , F d 2 × n 2 , F d 3 × n 3 {\displaystyle F^{d_{1}\times n_{1}},F^{d_{2}\times n_{2}},F^{d_{3}\times n_{3}}} respectively. The k-mode product (k = 1, 2, 3) of T {\displaystyle {\mathcal {T}}} by U ( k ) {\displaystyle U^{(k)}} is denoted as T × U ( k ) {\displaystyle {\mathcal {T}}\times U^{(k)}} with entries as ( T × 1 U ( 1 ) ) ( i 1 , j 2 , j 3 ) = ∑ j 1 = 1 d 1 T ( j 1 , j 2 , j 3 ) U ( 1 ) ( j 1 , i 1 ) ( T × 2 U ( 2 ) ) ( j 1 , i 2 , j 3 ) = ∑ j 2 = 1 d 2 T ( j 1 , j 2 , j 3 ) U ( 2 ) ( j 2 , i 2 ) ( T × 3 U ( 3 ) ) ( j 1 , j 2 , i 3 ) = ∑ j 3 = 1 d 3 T ( j 1 , j 2 , j 3 ) U ( 3 ) ( j 3 , i 3 ) {\displaystyle {\begin{aligned}({\mathcal {T}}\times _{1}U^{(1)})(i_{1},j_{2},j_{3})&=\sum _{j_{1}=1}^{d_{1}}{\mathcal {T}}(j_{1},j_{2},j_{3})U^{(1)}(j_{1},i_{1})\\({\mathcal {T}}\times _{2}U^{(2)})(j_{1},i_{2},j_{3})&=\sum _{j_{2}=1}^{d_{2}}{\mathcal {T}}(j_{1},j_{2},j_{3})U^{(2)}(j_{2},i_{2})\\({\mathcal {T}}\times _{3}U^{(3)})(j_{1},j_{2},i_{3})&=\sum _{j_{3}=1}^{d_{3}}{\mathcal {T}}(j_{1},j_{2},j_{3})U^{(3)}(j_{3},i_{3})\end{aligned}}} Altogether, the decomposition may also be written more directly as T ( i 1 , i 2 , i 3 ) = ∑ j 1 = 1 d 1 ∑ j 2 = 1 d 2 ∑ j 3 = 1 d 3 T ( j 1 , j 2 , j 3 ) U ( 1 ) ( j 1 , i 1 ) U ( 2 ) ( j 2 , i 2 ) U ( 3 ) ( j 3 , i 3 ) {\displaystyle T(i_{1},i_{2},i_{3})=\sum _{j_{1}=1}^{d_{1}}\sum _{j_{2}=1}^{d_{2}}\sum _{j_{3}=1}^{d_{3}}{\mathcal {T}}(j_{1},j_{2},j_{3})U^{(1)}(j_{1},i_{1})U^{(2)}(j_{2},i_{2})U^{(3)}(j_{3},i_{3})} Taking d i = n i {\displaystyle d_{i}=n_{i}} for all i {\displaystyle i} is always sufficient to represent T {\displaystyle T} exactly, but often T {\displaystyle T} can be compressed or efficiently approximately by choosing d i < n i {\displaystyle d_{i} Read more →

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  Correlation clustering

  Clustering is the problem of partitioning data points into groups based on similarity or dissimilarity. Correlation clustering is a clustering framework in which a set of objects is partitioned into clusters based on pairwise similarity and dissimilarity information, without requiring the number of clusters to be specified in advance. == Description of the problem == In machine learning, correlation clustering (also known as cluster editing) considers settings in which pairwise similarity or dissimilarity relationships between objects are known. A standard formulation models the input as an unweighted complete graph G = ( V , E ) {\displaystyle G=(V,E)} , where each edge is labeled either + {\displaystyle +} or − {\displaystyle -} (that is, the graph is a signed graph), indicating whether the corresponding endpoints are similar or dissimilar. The goal is to find a clustering (that is, a partition of V {\displaystyle V} ) that either maximizes the number of agreements—the sum of positive edges whose endpoints lie in the same cluster and negative edges whose endpoints lie in different clusters—or minimizes the number of disagreements—the sum of positive edges whose endpoints are separated and negative edges whose endpoints lie in the same cluster. Unlike other clustering methods such as k-means, correlation clustering does not require choosing the number of clusters k {\displaystyle k} in advance. It is not always possible to find a clustering with zero disagreements. For example, consider a triangle graph containing two positive edges and one negative edge. In this case, every clustering incurs at least one disagreement. Such configurations are referred to in the literature as bad triangles. From a computational perspective, optimizing the correlation clustering objective is challenging. The (decision version of the) problem is NP-complete. A large body of subsequent work has developed approximation algorithms for correlation clustering under various assumptions, including complete or general graphs and unweighted or weighted graphs, for both minimization and maximization objectives. This problem is considered one of the fundamental combinatorial optimization problems, and many algorithmic techniques have been developed to address it. The problem has also been studied extensively across multiple disciplines. A comprehensive literature review of early correlation clustering research is provided by Wahid and Hassini. == Formal Definitions == Let G = ( V , E ) {\displaystyle G=(V,E)} be a graph with nodes V {\displaystyle V} and edges E {\displaystyle E} . A clustering of G {\displaystyle G} is a partition of its node set Π = { π 1 , … , π k } {\displaystyle \Pi =\{\pi _{1},\dots ,\pi _{k}\}} with V = π 1 ∪ ⋯ ∪ π k {\displaystyle V=\pi _{1}\cup \dots \cup \pi _{k}} and π i ∩ π j = ∅ {\displaystyle \pi _{i}\cap \pi _{j}=\emptyset } for i ≠ j {\displaystyle i\neq j} . For a given clustering Π {\displaystyle \Pi } , let δ ( Π ) = { { u , v } ∈ E ∣ { u , v } ⊈ π ∀ π ∈ Π } {\displaystyle \delta (\Pi )=\{\{u,v\}\in E\mid \{u,v\}\not \subseteq \pi \;\forall \pi \in \Pi \}} denote the subset of edges of G {\displaystyle G} whose endpoints are in different subsets of the clustering Π {\displaystyle \Pi } . Now, let w : E → R ≥ 0 {\displaystyle w\colon E\to \mathbb {R} _{\geq 0}} be a function that assigns a non-negative weight to each edge of the graph and let E = E + ∪ E − {\displaystyle E=E^{+}\cup E^{-}} be a partition of the edges into attractive ( E + {\displaystyle E^{+}} ) and repulsive ( E − {\displaystyle E^{-}} ) edges; that is, the edges are signed. The minimum disagreement correlation clustering problem is the following optimization problem: minimize Π ∑ e ∈ E + ∩ δ ( Π ) w e + ∑ e ∈ E − ∖ δ ( Π ) w e . {\displaystyle {\begin{aligned}&{\underset {\Pi }{\operatorname {minimize} }}&&\sum _{e\in E^{+}\cap \delta (\Pi )}w_{e}+\sum _{e\in E^{-}\setminus \delta (\Pi )}w_{e}\;.\end{aligned}}} Here, the set E + ∩ δ ( Π ) {\displaystyle E^{+}\cap \delta (\Pi )} contains the attractive edges whose endpoints are in different components with respect to the clustering Π {\displaystyle \Pi } and the set E − ∖ δ ( Π ) {\displaystyle E^{-}\setminus \delta (\Pi )} contains the repulsive edges whose endpoints are in the same component with respect to the clustering Π {\displaystyle \Pi } . Together these two sets contain all edges that disagree with the clustering Π {\displaystyle \Pi } . Similarly to the minimum disagreement correlation clustering problem, the maximum agreement correlation clustering problem is defined as maximize Π ∑ e ∈ E + ∖ δ ( Π ) w e + ∑ e ∈ E − ∩ δ ( Π ) w e . {\displaystyle {\begin{aligned}&{\underset {\Pi }{\operatorname {maximize} }}&&\sum _{e\in E^{+}\setminus \delta (\Pi )}w_{e}+\sum _{e\in E^{-}\cap \delta (\Pi )}w_{e}\;.\end{aligned}}} Here, the set E + ∖ δ ( Π ) {\displaystyle E^{+}\setminus \delta (\Pi )} contains the attractive edges whose endpoints are in the same component with respect to the clustering Π {\displaystyle \Pi } and the set E − ∩ δ ( Π ) {\displaystyle E^{-}\cap \delta (\Pi )} contains the repulsive edges whose endpoints are in different components with respect to the clustering Π {\displaystyle \Pi } . Together these two sets contain all edges that agree with the clustering Π {\displaystyle \Pi } . Instead of formulating the correlation clustering problem in terms of non-negative edge weights and a partition of the edges into attractive and repulsive edges the problem is also formulated in terms of positive and negative edge costs without partitioning the set of edges explicitly. For given weights w : E → R ≥ 0 {\displaystyle w\colon E\to \mathbb {R} _{\geq 0}} and a given partition E = E + ∪ E − {\displaystyle E=E^{+}\cup E^{-}} of the edges into attractive and repulsive edges, the edge costs can be defined by c e = { w e if e ∈ E + − w e if e ∈ E − {\displaystyle {\begin{aligned}c_{e}={\begin{cases}\;\;w_{e}&{\text{if }}e\in E^{+}\\-w_{e}&{\text{if }}e\in E^{-}\end{cases}}\end{aligned}}} for all e ∈ E {\displaystyle e\in E} . An edge whose endpoints are in different clusters is said to be cut. The set δ ( Π ) {\displaystyle \delta (\Pi )} of all edges that are cut is often called a multicut of G {\displaystyle G} . The minimum cost multicut problem is the problem of finding a clustering Π {\displaystyle \Pi } of G {\displaystyle G} such that the sum of the costs of the edges whose endpoints are in different clusters is minimal: minimize Π ∑ e ∈ δ ( Π ) c e . {\displaystyle {\begin{aligned}&{\underset {\Pi }{\operatorname {minimize} }}&&\sum _{e\in \delta (\Pi )}c_{e}\;.\end{aligned}}} Similar to the minimum cost multicut problem, coalition structure generation in weighted graph games is the problem of finding a clustering such that the sum of the costs of the edges that are not cut is maximal: maximize Π ∑ e ∈ E ∖ δ ( Π ) c e . {\displaystyle {\begin{aligned}&{\underset {\Pi }{\operatorname {maximize} }}&&\sum _{e\in E\setminus \delta (\Pi )}c_{e}\;.\end{aligned}}} This formulation is also known as the clique partitioning problem. It can be shown that all four problems that are formulated above are equivalent. This means that a clustering that is optimal with respect to any of the four objectives is optimal for all of the four objectives. == Algorithms == If the graph admits a clustering with zero disagreements, then deleting all negative edges and computing the connected components of the remaining graph yields an optimal clustering. A necessary and sufficient condition for the existence of such a clustering was given by Davis: no cycle in the graph may contain exactly one negative edge. Bansal et al. discuss the NP-completeness proof and also present both a constant factor approximation algorithm and polynomial-time approximation scheme to find the clusters in this setting. Ailon et al. propose a randomized 3-approximation algorithm for the same problem. CC-Pivot(G=(V,E+,E−)) Pick random pivot i ∈ V Set C = { i } {\displaystyle C=\{i\}} , V'=Ø For all j ∈ V, j ≠ i; If (i,j) ∈ E+ then Add j to C Else (If (i,j) ∈ E−) Add j to V' Let G' be the subgraph induced by V' Return clustering C,CC-Pivot(G') The authors show that the above algorithm is a 3-approximation algorithm for correlation clustering. The best polynomial-time approximation algorithm known at the moment for this problem achieves a ~2.06 approximation by rounding a linear program, as shown by Chawla, Makarychev, Schramm, and Yaroslavtsev. Karpinski and Schudy proved existence of a polynomial time approximation scheme (PTAS) for that problem on complete graphs and fixed number of clusters. == Optimal number of clusters == In 2011, it was shown by Bagon and Galun that the optimization of the correlation clustering functional is closely related to well known discrete optimization methods. In their work they proposed a probabilistic analysis of the underlying implicit model that allows the correlation clustering functional to estimate the

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  Optimal discriminant analysis and classification tree analysis

  Optimal Discriminant Analysis (ODA) and the related classification tree analysis (CTA) are exact statistical methods that maximize predictive accuracy. For any specific sample and exploratory or confirmatory hypothesis, optimal discriminant analysis (ODA) identifies the statistical model that yields maximum predictive accuracy, assesses the exact Type I error rate, and evaluates potential cross-generalizability. Optimal discriminant analysis may be applied to > 0 dimensions, with the one-dimensional case being referred to as UniODA and the multidimensional case being referred to as MultiODA. Optimal discriminant analysis is an alternative to ANOVA (analysis of variance) and regression analysis.

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  Feature hashing

  In machine learning, feature hashing, also known as the hashing trick (by analogy to the kernel trick), is a fast and space-efficient way of vectorizing features, i.e. turning arbitrary features into indices in a vector or matrix. It works by applying a hash function to the features and using their hash values as indices directly (after a modulo operation), rather than looking the indices up in an associative array. In addition to its use for encoding non-numeric values, feature hashing can also be used for dimensionality reduction. This trick is often attributed to Weinberger et al. (2009), but there exists a much earlier description of this method published by John Moody in 1989. == Motivation == === Motivating example === In a typical document classification task, the input to the machine learning algorithm (both during learning and classification) is free text. From this, a bag of words (BOW) representation is constructed: the individual tokens are extracted and counted, and each distinct token in the training set defines a feature (independent variable) of each of the documents in both the training and test sets. Machine learning algorithms, however, are typically defined in terms of numerical vectors. Therefore, the bags of words for a set of documents is regarded as a term-document matrix where each row is a single document, and each column is a single feature/word; the entry i, j in such a matrix captures the frequency (or weight) of the j'th term of the vocabulary in document i. (An alternative convention swaps the rows and columns of the matrix, but this difference is immaterial.) Typically, these vectors are extremely sparse—according to Zipf's law. The common approach is to construct, at learning time or prior to that, a dictionary representation of the vocabulary of the training set, and use that to map words to indices. Hash tables and tries are common candidates for dictionary implementation. E.g., the three documents John likes to watch movies. Mary likes movies too. John also likes football. can be converted, using the dictionary to the term-document matrix ( John likes to watch movies Mary too also football 1 1 1 1 1 0 0 0 0 0 1 0 0 1 1 1 0 0 1 1 0 0 0 0 0 1 1 ) {\displaystyle {\begin{pmatrix}{\textrm {John}}&{\textrm {likes}}&{\textrm {to}}&{\textrm {watch}}&{\textrm {movies}}&{\textrm {Mary}}&{\textrm {too}}&{\textrm {also}}&{\textrm {football}}\\1&1&1&1&1&0&0&0&0\\0&1&0&0&1&1&1&0&0\\1&1&0&0&0&0&0&1&1\end{pmatrix}}} (Punctuation was removed, as is usual in document classification and clustering.) The problem with this process is that such dictionaries take up a large amount of storage space and grow in size as the training set grows. On the contrary, if the vocabulary is kept fixed and not increased with a growing training set, an adversary may try to invent new words or misspellings that are not in the stored vocabulary so as to circumvent a machine learned filter. To address this challenge, Yahoo! Research attempted to use feature hashing for their spam filters. Note that the hashing trick isn't limited to text classification and similar tasks at the document level, but can be applied to any problem that involves large (perhaps unbounded) numbers of features. === Mathematical motivation === Mathematically, a token is an element t {\displaystyle t} in a finite (or countably infinite) set T {\displaystyle T} . Suppose we only need to process a finite corpus, then we can put all tokens appearing in the corpus into T {\displaystyle T} , meaning that T {\displaystyle T} is finite. However, suppose we want to process all possible words made of the English letters, then T {\displaystyle T} is countably infinite. Most neural networks can only operate on real vector inputs, so we must construct a "dictionary" function ϕ : T → R n {\displaystyle \phi :T\to \mathbb {R} ^{n}} . When T {\displaystyle T} is finite, of size | T | = m ≤ n {\displaystyle |T|=m\leq n} , then we can use one-hot encoding to map it into R n {\displaystyle \mathbb {R} ^{n}} . First, arbitrarily enumerate T = { t 1 , t 2 , . . , t m } {\displaystyle T=\{t_{1},t_{2},..,t_{m}\}} , then define ϕ ( t i ) = e i {\displaystyle \phi (t_{i})=e_{i}} . In other words, we assign a unique index i {\displaystyle i} to each token, then map the token with index i {\displaystyle i} to the unit basis vector e i {\displaystyle e_{i}} . One-hot encoding is easy to interpret, but it requires one to maintain the arbitrary enumeration of T {\displaystyle T} . Given a token t ∈ T {\displaystyle t\in T} , to compute ϕ ( t ) {\displaystyle \phi (t)} , we must find out the index i {\displaystyle i} of the token t {\displaystyle t} . Thus, to implement ϕ {\displaystyle \phi } efficiently, we need a fast-to-compute bijection h : T → { 1 , . . . , m } {\displaystyle h:T\to \{1,...,m\}} , then we have ϕ ( t ) = e h ( t ) {\displaystyle \phi (t)=e_{h(t)}} . In fact, we can relax the requirement slightly: It suffices to have a fast-to-compute injection h : T → { 1 , . . . , n } {\displaystyle h:T\to \{1,...,n\}} , then use ϕ ( t ) = e h ( t ) {\displaystyle \phi (t)=e_{h(t)}} . In practice, there is no simple way to construct an efficient injection h : T → { 1 , . . . , n } {\displaystyle h:T\to \{1,...,n\}} . However, we do not need a strict injection, but only an approximate injection. That is, when t ≠ t ′ {\displaystyle t\neq t'} , we should probably have h ( t ) ≠ h ( t ′ ) {\displaystyle h(t)\neq h(t')} , so that probably ϕ ( t ) ≠ ϕ ( t ′ ) {\displaystyle \phi (t)\neq \phi (t')} . At this point, we have just specified that h {\displaystyle h} should be a hashing function. Thus we reach the idea of feature hashing. == Algorithms == === Feature hashing (Weinberger et al. 2009) === The basic feature hashing algorithm presented in (Weinberger et al. 2009) is defined as follows. First, one specifies two hash functions: the kernel hash h : T → { 1 , 2 , . . . , n } {\displaystyle h:T\to \{1,2,...,n\}} , and the sign hash ζ : T → { − 1 , + 1 } {\displaystyle \zeta :T\to \{-1,+1\}} . Next, one defines the feature hashing function: ϕ : T → R n , ϕ ( t ) = ζ ( t ) e h ( t ) {\displaystyle \phi :T\to \mathbb {R} ^{n},\quad \phi (t)=\zeta (t)e_{h(t)}} Finally, extend this feature hashing function to strings of tokens by ϕ : T ∗ → R n , ϕ ( t 1 , . . . , t k ) = ∑ j = 1 k ϕ ( t j ) {\displaystyle \phi :T^{}\to \mathbb {R} ^{n},\quad \phi (t_{1},...,t_{k})=\sum _{j=1}^{k}\phi (t_{j})} where T ∗ {\displaystyle T^{}} is the set of all finite strings consisting of tokens in T {\displaystyle T} . Equivalently, ϕ ( t 1 , . . . , t k ) = ∑ j = 1 k ζ ( t j ) e h ( t j ) = ∑ i = 1 n ( ∑ j : h ( t j ) = i ζ ( t j ) ) e i {\displaystyle \phi (t_{1},...,t_{k})=\sum _{j=1}^{k}\zeta (t_{j})e_{h(t_{j})}=\sum _{i=1}^{n}\left(\sum _{j:h(t_{j})=i}\zeta (t_{j})\right)e_{i}} ==== Geometric properties ==== We want to say something about the geometric property of ϕ {\displaystyle \phi } , but T {\displaystyle T} , by itself, is just a set of tokens, we cannot impose a geometric structure on it except the discrete topology, which is generated by the discrete metric. To make it nicer, we lift it to T → R T {\displaystyle T\to \mathbb {R} ^{T}} , and lift ϕ {\displaystyle \phi } from ϕ : T → R n {\displaystyle \phi :T\to \mathbb {R} ^{n}} to ϕ : R T → R n {\displaystyle \phi :\mathbb {R} ^{T}\to \mathbb {R} ^{n}} by linear extension: ϕ ( ( x t ) t ∈ T ) = ∑ t ∈ T x t ζ ( t ) e h ( t ) = ∑ i = 1 n ( ∑ t : h ( t ) = i x t ζ ( t ) ) e i {\displaystyle \phi ((x_{t})_{t\in T})=\sum _{t\in T}x_{t}\zeta (t)e_{h(t)}=\sum _{i=1}^{n}\left(\sum _{t:h(t)=i}x_{t}\zeta (t)\right)e_{i}} There is an infinite sum there, which must be handled at once. There are essentially only two ways to handle infinities. One may impose a metric, then take its completion, to allow well-behaved infinite sums, or one may demand that nothing is actually infinite, only potentially so. Here, we go for the potential-infinity way, by restricting R T {\displaystyle \mathbb {R} ^{T}} to contain only vectors with finite support: ∀ ( x t ) t ∈ T ∈ R T {\displaystyle \forall (x_{t})_{t\in T}\in \mathbb {R} ^{T}} , only finitely many entries of ( x t ) t ∈ T {\displaystyle (x_{t})_{t\in T}} are nonzero. Define an inner product on R T {\displaystyle \mathbb {R} ^{T}} in the obvious way: ⟨ e t , e t ′ ⟩ = { 1 , if t = t ′ , 0 , else. ⟨ x , x ′ ⟩ = ∑ t , t ′ ∈ T x t x t ′ ⟨ e t , e t ′ ⟩ {\displaystyle \langle e_{t},e_{t'}\rangle ={\begin{cases}1,{\text{ if }}t=t',\\0,{\text{ else.}}\end{cases}}\quad \langle x,x'\rangle =\sum _{t,t'\in T}x_{t}x_{t'}\langle e_{t},e_{t'}\rangle } As a side note, if T {\displaystyle T} is infinite, then the inner product space R T {\displaystyle \mathbb {R} ^{T}} is not complete. Taking its completion would get us to a Hilbert space, which allows well-behaved infinite sums. Now we have an inner product space, with enough structure to describe the geometry of the feature hashing function ϕ : R T → R n {\displaystyle \phi :\ma

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  Information Harvesting

  Information Harvesting (IH) was an early data mining product from the 1990s. It was invented by Ralphe Wiggins and produced by the Ryan Corp, later Information Harvesting Inc., of Cambridge, Massachusetts. Wiggins had a background in genetic algorithms and fuzzy logic. IH sought to infer rules from sets of data. It did this first by classifying various input variables into one of a number of bins, thereby putting some structure on the continuous variables in the input. IH then proceeds to generate rules, trading off generalization against memorization, that will infer the value of the prediction variable, possibly creating many levels of rules in the process. It included strategies for checking if overfitting took place and, if so, correcting for it. Because of its strategies for correcting for overfitting by considering more data, and refining the rules based on that data, IH might also be considered to be a form of machine learning. The advantage of IH, as compared with other data mining products of its time and even later, was that it provided a mechanism for finding multiple rules that would classify the data and determining, according to set criteria, the best rules to use.

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  Feature selection

  In machine learning, feature selection is the process of selecting a subset of relevant features (variables, predictors) for use in model construction. Feature selection techniques are used for several reasons: simplification of models to make them easier to interpret, shorter training times, to avoid the curse of dimensionality, improve the compatibility of the data with a certain learning model class, to encode inherent symmetries present in the input space. The central premise when using feature selection is that data sometimes contains features that are redundant or irrelevant, and can thus be removed without incurring much loss of information. Redundancy and irrelevance are two distinct notions, since one relevant feature may be redundant in the presence of another relevant feature with which it is strongly correlated. Feature extraction creates new features from functions of the original features, whereas feature selection finds a subset of the features. Feature selection techniques are often used in domains where there are many features and comparatively few samples (data points). == Introduction == A feature selection algorithm can be seen as the combination of a search technique for proposing new feature subsets, along with an evaluation measure which scores the different feature subsets. The simplest algorithm is to test each possible subset of features finding the one which minimizes the error rate. This is an exhaustive search of the space, and is computationally intractable for all but the smallest of feature sets. The choice of evaluation metric heavily influences the algorithm, and it is these evaluation metrics which distinguish between the three main categories of feature selection algorithms: wrappers, filters and embedded methods. Wrapper methods use a predictive model to score feature subsets. Each new subset is used to train a model, which is tested on a hold-out set. Counting the number of mistakes made on that hold-out set (the error rate of the model) gives the score for that subset. As wrapper methods train a new model for each subset, they are very computationally intensive, but usually provide the best performing feature set for that particular type of model or typical problem. Filter methods use a proxy measure instead of the error rate to score a feature subset. This measure is chosen to be fast to compute, while still capturing the usefulness of the feature set. Common measures include the mutual information, the pointwise mutual information, Pearson product-moment correlation coefficient, Relief-based algorithms, and inter/intra class distance or the scores of significance tests for each class/feature combinations. Filters are usually less computationally intensive than wrappers, but they produce a feature set which is not tuned to a specific type of predictive model. This lack of tuning means a feature set from a filter is more general than the set from a wrapper, usually giving lower prediction performance than a wrapper. However the feature set doesn't contain the assumptions of a prediction model, and so is more useful for exposing the relationships between the features. Many filters provide a feature ranking rather than an explicit best feature subset, and the cut off point in the ranking is chosen via cross-validation. Filter methods have also been used as a preprocessing step for wrapper methods, allowing a wrapper to be used on larger problems. One other popular approach is the Recursive Feature Elimination algorithm, commonly used with Support Vector Machines to repeatedly construct a model and remove features with low weights. Embedded methods are a catch-all group of techniques which perform feature selection as part of the model construction process. The exemplar of this approach is the LASSO method for constructing a linear model, which penalizes the regression coefficients with an L1 penalty, shrinking many of them to zero. Any features which have non-zero regression coefficients are 'selected' by the LASSO algorithm. Improvements to the LASSO include Bolasso which bootstraps samples; Elastic net regularization, which combines the L1 penalty of LASSO with the L2 penalty of ridge regression; and FeaLect which scores all the features based on combinatorial analysis of regression coefficients. AEFS further extends LASSO to nonlinear scenario with autoencoders. These approaches tend to be between filters and wrappers in terms of computational complexity. In traditional regression analysis, the most popular form of feature selection is stepwise regression, which is a wrapper technique. It is a greedy algorithm that adds the best feature (or deletes the worst feature) at each round. The main control issue is deciding when to stop the algorithm. In machine learning, this is typically done by cross-validation. In statistics, some criteria are optimized. This leads to the inherent problem of nesting. More robust methods have been explored, such as branch and bound and piecewise linear network. == Subset selection == Subset selection evaluates a subset of features as a group for suitability. Subset selection algorithms can be broken up into wrappers, filters, and embedded methods. Wrappers use a search algorithm to search through the space of possible features and evaluate each subset by running a model on the subset. Wrappers can be computationally expensive and have a risk of over fitting to the model. Filters are similar to wrappers in the search approach, but instead of evaluating against a model, a simpler filter is evaluated. Embedded techniques are embedded in, and specific to, a model. Many popular search approaches use greedy hill climbing, which iteratively evaluates a candidate subset of features, then modifies the subset and evaluates if the new subset is an improvement over the old. Evaluation of the subsets requires a scoring metric that grades a subset of features. Exhaustive search is generally impractical, so at some implementor (or operator) defined stopping point, the subset of features with the highest score discovered up to that point is selected as the satisfactory feature subset. The stopping criterion varies by algorithm; possible criteria include: a subset score exceeds a threshold, a program's maximum allowed run time has been surpassed, etc. Alternative search-based techniques are based on targeted projection pursuit which finds low-dimensional projections of the data that score highly: the features that have the largest projections in the lower-dimensional space are then selected. Search approaches include: Exhaustive Best first Simulated annealing Genetic algorithm Greedy forward selection Greedy backward elimination Particle swarm optimization Targeted projection pursuit Scatter search Variable neighborhood search Two popular filter metrics for classification problems are correlation and mutual information, although neither are true metrics or 'distance measures' in the mathematical sense, since they fail to obey the triangle inequality and thus do not compute any actual 'distance' – they should rather be regarded as 'scores'. These scores are computed between a candidate feature (or set of features) and the desired output category. There are, however, true metrics that are a simple function of the mutual information; see here. Other available filter metrics include: Class separability Error probability Inter-class distance Probabilistic distance Entropy Consistency-based feature selection Correlation-based feature selection == Optimality criteria == The choice of optimality criteria is difficult as there are multiple objectives in a feature selection task. Many common criteria incorporate a measure of accuracy, penalised by the number of features selected. Examples include Akaike information criterion (AIC) and Mallows's Cp, which have a penalty of 2 for each added feature. AIC is based on information theory, and is effectively derived via the maximum entropy principle. Other criteria are Bayesian information criterion (BIC), which uses a penalty of log ⁡ n {\displaystyle {\sqrt {\log {n}}}} for each added feature, minimum description length (MDL) which asymptotically uses log ⁡ n {\displaystyle {\sqrt {\log {n}}}} , Bonferroni / RIC which use 2 log ⁡ p {\displaystyle {\sqrt {2\log {p}}}} , maximum dependency feature selection, and a variety of new criteria that are motivated by false discovery rate (FDR), which use something close to 2 log ⁡ p q {\displaystyle {\sqrt {2\log {\frac {p}{q}}}}} . A maximum entropy rate criterion may also be used to select the most relevant subset of features. == Structure learning == Filter feature selection is a specific case of a more general paradigm called structure learning. Feature selection finds the relevant feature set for a specific target variable whereas structure learning finds the relationships between all the variables, usually by expressing these relationships as a graph. The most common structure learning algorithms

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  Modern Hopfield network

  Modern Hopfield networks (also known as Dense Associative Memories) are generalizations of the classical Hopfield networks that break the linear scaling relationship between the number of input features and the number of stored memories. This is achieved by introducing stronger non-linearities (either in the energy function or neurons’ activation functions) leading to super-linear (even an exponential) memory storage capacity as a function of the number of feature neurons. The network still requires a sufficient number of hidden neurons. The key theoretical idea behind the modern Hopfield networks is to use an energy function and an update rule that is more sharply peaked around the stored memories in the space of neuron’s configurations compared to the classical Hopfield network. == Classical Hopfield networks == Hopfield networks are recurrent neural networks with dynamical trajectories converging to fixed point attractor states and described by an energy function. The state of each model neuron i {\textstyle i} is defined by a time-dependent variable V i {\displaystyle V_{i}} , which can be chosen to be either discrete or continuous. A complete model describes the mathematics of how the future state of activity of each neuron depends on the known present or previous activity of all the neurons. In the original Hopfield model of associative memory, the variables were binary, and the dynamics were described by a one-at-a-time update of the state of the neurons. An energy function quadratic in the V i {\displaystyle V_{i}} was defined, and the dynamics consisted of changing the activity of each single neuron i {\displaystyle i} only if doing so would lower the total energy of the system. This same idea was extended to the case of V i {\displaystyle V_{i}} being a continuous variable representing the output of neuron i {\displaystyle i} , and V i {\displaystyle V_{i}} being a monotonic function of an input current. The dynamics became expressed as a set of first-order differential equations for which the "energy" of the system always decreased. The energy in the continuous case has one term which is quadratic in the V i {\displaystyle V_{i}} (as in the binary model), and a second term which depends on the gain function (neuron's activation function). While having many desirable properties of associative memory, both of these classical systems suffer from a small memory storage capacity, which scales linearly with the number of input features. == Discrete variables == A simple example of the Modern Hopfield network can be written in terms of binary variables V i {\displaystyle V_{i}} that represent the active V i = + 1 {\displaystyle V_{i}=+1} and inactive V i = − 1 {\displaystyle V_{i}=-1} state of the model neuron i {\displaystyle i} . E = − ∑ μ = 1 N mem F ( ∑ i = 1 N f ξ μ i V i ) {\displaystyle E=-\sum \limits _{\mu =1}^{N_{\text{mem}}}F{\Big (}\sum \limits _{i=1}^{N_{f}}\xi _{\mu i}V_{i}{\Big )}} In this formula the weights ξ μ i {\textstyle \xi _{\mu i}} represent the matrix of memory vectors (index μ = 1... N mem {\displaystyle \mu =1...N_{\text{mem}}} enumerates different memories, and index i = 1... N f {\displaystyle i=1...N_{f}} enumerates the content of each memory corresponding to the i {\displaystyle i} -th feature neuron), and the function F ( x ) {\displaystyle F(x)} is a rapidly growing non-linear function. The update rule for individual neurons (in the asynchronous case) can be written in the following form V i ( t + 1 ) = sign ⁡ [ ∑ μ = 1 N mem ( F ( ξ μ i + ∑ j ≠ i ξ μ j V j ( t ) ) − F ( − ξ μ i + ∑ j ≠ i ξ μ j V j ( t ) ) ) ] {\displaystyle V_{i}^{(t+1)}=\operatorname {sign} {\bigg [}\sum \limits _{\mu =1}^{N_{\text{mem}}}{\bigg (}F{\Big (}\xi _{\mu i}+\sum \limits _{j\neq i}\xi _{\mu j}V_{j}^{(t)}{\Big )}-F{\Big (}-\xi _{\mu i}+\sum \limits _{j\neq i}\xi _{\mu j}V_{j}^{(t)}{\Big )}{\bigg )}{\bigg ]}} which states that in order to calculate the updated state of the i {\textstyle i} -th neuron the network compares two energies: the energy of the network with the i {\displaystyle i} -th neuron in the ON state and the energy of the network with the i {\displaystyle i} -th neuron in the OFF state, given the states of the remaining neuron. The updated state of the i {\displaystyle i} -th neuron selects the state that has the lowest of the two energies. In the limiting case when the non-linear energy function is quadratic F ( x ) = x 2 {\displaystyle F(x)=x^{2}} these equations reduce to the familiar energy function and the update rule for the classical binary Hopfield network. The memory storage capacity of these networks can be calculated for random binary patterns. For the power energy function F ( x ) = x n {\displaystyle F(x)=x^{n}} the maximal number of memories that can be stored and retrieved from this network without errors is given by N mem max ≈ 1 2 ( 2 n − 3 ) ! ! N f n − 1 ln ⁡ ( N f ) {\displaystyle N_{\text{mem}}^{\max }\approx {\frac {1}{2(2n-3)!!}}{\frac {N_{f}^{n-1}}{\ln(N_{f})}}} For an exponential energy function F ( x ) = e x {\textstyle F(x)=e^{x}} the memory storage capacity is exponential in the number of feature neurons N mem max ≈ 2 N f / 2 {\displaystyle N_{\text{mem}}^{\max }\approx 2^{N_{f}/2}} == Continuous variables == Modern Hopfield networks or Dense Associative Memories can be best understood in continuous variables and continuous time. Consider the network architecture, shown in Fig.1, and the equations for the neurons' state evolutionwhere the currents of the feature neurons are denoted by x i {\textstyle x_{i}} , and the currents of the memory neurons are denoted by h μ {\displaystyle h_{\mu }} ( h {\displaystyle h} stands for hidden neurons). There are no synaptic connections among the feature neurons or the memory neurons. A matrix ξ μ i {\displaystyle \xi _{\mu i}} denotes the strength of synapses from a feature neuron i {\displaystyle i} to the memory neuron μ {\displaystyle \mu } . The synapses are assumed to be symmetric, so that the same value characterizes a different physical synapse from the memory neuron μ {\displaystyle \mu } to the feature neuron i {\displaystyle i} . The outputs of the memory neurons and the feature neurons are denoted by f μ {\displaystyle f_{\mu }} and g i {\displaystyle g_{i}} , which are non-linear functions of the corresponding currents. In general these outputs can depend on the currents of all the neurons in that layer so that f μ = f ( { h μ } ) {\displaystyle f_{\mu }=f(\{h_{\mu }\})} and g i = g ( { x i } ) {\textstyle g_{i}=g(\{x_{i}\})} . It is convenient to define these activation function as derivatives of the Lagrangian functions for the two groups of neuronsThis way the specific form of the equations for neuron's states is completely defined once the Lagrangian functions are specified. Finally, the time constants for the two groups of neurons are denoted by τ f {\displaystyle \tau _{f}} and τ h {\displaystyle \tau _{h}} , I i {\displaystyle I_{i}} is the input current to the network that can be driven by the presented data. General systems of non-linear differential equations can have many complicated behaviors that can depend on the choice of the non-linearities and the initial conditions. For Hopfield networks, however, this is not the case - the dynamical trajectories always converge to a fixed point attractor state. This property is achieved because these equations are specifically engineered so that they have an underlying energy function The terms grouped into square brackets represent a Legendre transform of the Lagrangian function with respect to the states of the neurons. If the Hessian matrices of the Lagrangian functions are positive semi-definite, the energy function is guaranteed to decrease on the dynamical trajectory This property makes it possible to prove that the system of dynamical equations describing temporal evolution of neurons' activities will eventually reach a fixed point attractor state. In certain situations one can assume that the dynamics of hidden neurons equilibrates at a much faster time scale compared to the feature neurons, τ h ≪ τ f {\textstyle \tau _{h}\ll \tau _{f}} . In this case the steady state solution of the second equation in the system (1) can be used to express the currents of the hidden units through the outputs of the feature neurons. This makes it possible to reduce the general theory (1) to an effective theory for feature neurons only. The resulting effective update rules and the energies for various common choices of the Lagrangian functions are shown in Fig.2. In the case of log-sum-exponential Lagrangian function the update rule (if applied once) for the states of the feature neurons is the attention mechanism commonly used in many modern AI systems (see Ref. for the derivation of this result from the continuous time formulation). == Relationship to classical Hopfield network with continuous variables == Classical formulation of continuous Hopfield networks can be understood as a

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  JasPer

  JasPer is a computer software project to create a reference implementation of the codec specified in the JPEG-2000 Part-1 standard (i.e. ISO/IEC 15444-1) - started in 1997 at Image Power Inc. and at the University of British Columbia. It consists of a C library and some sample applications useful for testing the codec. The copyright owner began licensing the code to the public under an MIT License-style license in 2004 in response to requests from the open-source community. As of 2011 JasPer operated as a component of many software projects, both free and proprietary, including (but not limited to) netpbm (as of release 10.12), ImageMagick and KDE (as of version 3.2). As of 22 June 2010 the GEGL graphics library supported JasPer in its latest Git versions. In a series of objective JPEG-2000-compression quality tests conducted in 2004, "JasPer was the best codec, closely followed by IrfanView and Kakadu". However, Jasper remains one of the slowest implementations of the JPEG-2000 codec, as it was designed for reference, not performance. == Etymology == The name "JasPer" has simultaneous connotations with Canada's Jasper National Park, with the semi-precious gemstone, jasper, and with "JP" as an abbreviation of the JPEG-2000 standard.

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  Dynamic Bayesian network

  A dynamic Bayesian network (DBN) is a Bayesian network (BN) which relates variables to each other over adjacent time steps. == History == A dynamic Bayesian network (DBN) is often called a "two-timeslice" BN (2TBN) because it says that at any point in time T, the value of a variable can be calculated from the internal regressors and the immediate prior value (time T-1). DBNs were developed by Paul Dagum in the early 1990s at Stanford University's Section on Medical Informatics. Dagum developed DBNs to unify and extend traditional linear state-space models such as Kalman filters, linear and normal forecasting models such as ARMA and simple dependency models such as hidden Markov models into a general probabilistic representation and inference mechanism for arbitrary nonlinear and non-normal time-dependent domains. Today, DBNs are common in robotics, and have shown potential for a wide range of data mining applications. For example, they have been used in speech recognition, digital forensics, protein sequencing, and bioinformatics. DBN is a generalization of hidden Markov models and Kalman filters. DBNs are conceptually related to probabilistic Boolean networks and can, similarly, be used to model dynamical systems at steady-state.

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  Rectified linear unit

  In the context of artificial neural networks, the rectifier or ReLU (rectified linear unit) activation function is an activation function defined as the non-negative part of its argument, i.e., the ramp function: ReLU ⁡ ( x ) = x + = max ( 0 , x ) = x + | x | 2 = { x if x > 0 , 0 x ≤ 0 {\displaystyle \operatorname {ReLU} (x)=x^{+}=\max(0,x)={\frac {x+|x|}{2}}={\begin{cases}x&{\text{if }}x>0,\\0&x\leq 0\end{cases}}} where x {\displaystyle x} is the input to a neuron. This is analogous to half-wave rectification in electrical engineering. ReLU is one of the most popular activation functions for artificial neural networks, and finds application in computer vision and speech recognition using deep neural nets and computational neuroscience. == History == The ReLU was first used by Alston Householder in 1941 as a mathematical abstraction of biological neural networks. Kunihiko Fukushima in 1969 used ReLU in the context of visual feature extraction in hierarchical neural networks. In 1998, Gregory Woodbury demonstrated that the rectified linear function could account for a broad range of emergent properties in the visual cortex. His work showed that a single unified model could drive the joint development of refined retinotopic maps, ocular dominance columns, and orientation selectivity. By utilizing the rectifier's "cutoff" property, Woodbury achieved a close quantitative fit to biological data, matching the spatial periodicities and topographic refinement patterns observed in macaque and cat cortical maps. Furthermore, he extended this framework to adult plasticity, accurately replicating the spatial and temporal dynamics of lesion-induced cortical reorganization. This research established that the rectified linear response was a necessary mechanism for the stable self-organisation and maintenance of complex, multi-feature neural maps. In 2000, Hahnloser et al. argued that ReLU approximates the biological relationship between neural firing rates and input current, in addition to enabling recurrent neural network dynamics to stabilise under weaker criteria. Prior to 2010, most activation functions used were the logistic sigmoid (which is inspired by probability theory; see logistic regression) and its more numerically efficient counterpart, the hyperbolic tangent. Around 2010, the use of ReLU became common again. Jarrett et al. (2009) noted that rectification by either absolute or ReLU (which they called "positive part") was critical for object recognition in convolutional neural networks (CNNs), specifically because it allows average pooling without neighboring filter outputs cancelling each other out. They hypothesized that the use of sigmoid or tanh was responsible for poor performance in previous CNNs. Nair and Hinton (2010) made a theoretical argument that the softplus activation function should be used, in that the softplus function numerically approximates the sum of an exponential number of linear models that share parameters. They then proposed ReLU as a good approximation to it. Specifically, they began by considering a single binary neuron in a Boltzmann machine that takes x {\displaystyle x} as input, and produces 1 as output with probability σ ( x ) = 1 1 + e − x {\displaystyle \sigma (x)={\frac {1}{1+e^{-x}}}} . They then considered extending its range of output by making infinitely many copies of it X 1 , X 2 , X 3 , … {\displaystyle X_{1},X_{2},X_{3},\dots } , that all take the same input, offset by an amount 0.5 , 1.5 , 2.5 , … {\displaystyle 0.5,1.5,2.5,\dots } , then their outputs are added together as ∑ i = 1 ∞ X i {\displaystyle \sum _{i=1}^{\infty }X_{i}} . They then demonstrated that ∑ i = 1 ∞ X i {\displaystyle \sum _{i=1}^{\infty }X_{i}} is approximately equal to N ( log ⁡ ( 1 + e x ) , σ ( x ) ) {\displaystyle {\mathcal {N}}(\log(1+e^{x}),\sigma (x))} , which is also approximately equal to ReLU ⁡ ( N ( x , σ ( x ) ) ) {\displaystyle \operatorname {ReLU} ({\mathcal {N}}(x,\sigma (x)))} , where N {\displaystyle {\mathcal {N}}} stands for the gaussian distribution. They also argued for another reason for using ReLU: that it allows "intensity equivariance" in image recognition. That is, multiplying input image by a constant k {\displaystyle k} multiplies the output also. In contrast, this is false for other activation functions like sigmoid or tanh. They found that ReLU activation allowed good empirical performance in restricted Boltzmann machines. Glorot et al (2011) argued that ReLU has the following advantages over sigmoid or tanh: ReLU is more similar to biological neurons' responses in their main operating regime. ReLU avoids vanishing gradients. ReLU is cheaper to compute. ReLU creates sparse representation naturally, because many hidden units output exactly zero for a given input. They also found empirically that deep networks trained with ReLU can achieve strong performance without unsupervised pre-training, especially on large, purely supervised tasks. In 2017, the rectified linear function became a central component of the transformer architecture introduced in the Vaswani et al paper "Attention Is All You Need". Within every transformer layer, ReLU is utilized in the position-wise feed-forward networks (FFN), defined by Equation 2 of their paper: FFN ⁡ ( x ) = max ( 0 , x W 1 + b 1 ) W 2 + b 2 {\displaystyle \operatorname {FFN} (x)=\max(0,xW_{1}+b_{1})W_{2}+b_{2}} This equation is foundational to the model's capacity; while the attention mechanism determines the relationships between tokens, the ReLU-based FFN performs the majority of the numerical computation and houses the bulk of the model's parameters. The efficiency and scalability of this rectified framework triggered a global technological revolution, enabling the development of Large Language Models that have had a profound economic impact. The industrial response to this architecture—including the massive expansion of AI-specific hardware and the birth of the generative AI sector—has positioned the Transformer as a cornerstone of 21st-century infrastructure. During the post 2017 period of rapid AI advancement, the rectified linear unit function has been key to achieving increased model performance and scaling due to the fact that it zeros out responses that are immaterial for a given stimuli, preventing them from accumulating in massive scale models. It is the complete silencing of the parts of the model found to be stimuli-irrelevant during learning that allows for scaling. As the stimuli-irrelevant proportion of the model becomes more massive, these highly numerous connections within the model would inevitably accumulate during scaling no matter how small each individual response is. Therefore, the rectified linear unit function, with its absolute zeroing property, enabled the scaling to hundred billion parameter models and beyond. Early Transformer scaling giants like GPT-3 (2020) and Falcon-180B (2023) relied on the rectified linear unit function explicitly, while successors such as GPT-4 (2023) and Llama 3 (2024) utilized smoother variants like GELU or SwiGLU. These variants were used to improve training stability while fundamentally preserving the rectified principle of zeroing low responses. At the centre of modern artificial intelligence ReLU and its variants maintain absolute zero response across the bulk of the model at any one time, while maintaining approximately linear reponses for stimuli-relevant connections enabling high performance on each specific cognitive task. This feature of activation sparsity has been critical for massive scaling and performance gains of AI models right up to the present day. == Advantages == Advantages of ReLU include: Sparse activation: for example, in a randomly initialized network, only about 50% of hidden units are activated (i.e. have a non-zero output). Better gradient propagation: fewer vanishing gradient problems compared to sigmoidal activation functions that saturate in both directions. Efficiency: only requires comparison and addition. Scale-invariant (homogeneous, or "intensity equivariance"): max ( 0 , a x ) = a max ( 0 , x ) for a ≥ 0 {\displaystyle \max(0,ax)=a\max(0,x){\text{ for }}a\geq 0} . == Potential problems == Possible downsides can include: Non-differentiability at zero (however, it is differentiable anywhere else, and the value of the derivative at zero can be chosen to be 0 or 1 arbitrarily). Not zero-centered: ReLU outputs are always non-negative. This can make it harder for the network to learn during backpropagation, because gradient updates tend to push weights in one direction (positive or negative). Batch normalization can help address this. ReLU is unbounded. Redundancy of the parametrization: Because ReLU is scale-invariant, the network computes the exact same function by scaling the weights and biases in front of a ReLU activation by k {\displaystyle k} , and the weights after by 1 / k {\displaystyle 1/k} . Dying ReLU: ReLU neurons can sometimes be pushed into states

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