AI Generator Detector

AI Generator Detector — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

• 

  Loebner Prize

  The Loebner Prize was an annual competition in artificial intelligence that awarded prizes to the computer programs considered by the judges to be the most human-like. The format of the competition was that of a standard Turing test. In each round, a human judge simultaneously held textual conversations with a computer program and a human being via computer. Based upon the responses, the judge would attempt to determine which was which. The contest was launched in 1990 by Hugh Loebner in conjunction with the Cambridge Center for Behavioral Studies, Massachusetts, United States. In 2004 and 2005, it was held in Loebner's apartment in New York City. Within the field of artificial intelligence, the Loebner Prize is somewhat controversial; the most prominent critic, Marvin Minsky, called it a publicity stunt that does not help the field along. Beginning in 2014, it was organised by the AISB at Bletchley Park. It has also been associated with Flinders University, Dartmouth College, the Science Museum in London, University of Reading and Ulster University, Magee Campus, Derry, UK City of Culture. For the final 2019 competition, the format changed. There was no panel of judges. Instead, the chatbots were judged by the public and there were to be no human competitors. The prize has been reported as defunct as of 2020. == Prizes == Originally, $2,000 was awarded for the most human-seeming program in the competition. The prize was $3,000 in 2005 and $2,250 in 2006. In 2008, $3,000 was awarded. In addition, there were two one-time-only prizes that have never been awarded. $25,000 is offered for the first program that judges cannot distinguish from a real human and which can convince judges that the human is the computer program. $100,000 is the reward for the first program that judges cannot distinguish from a real human in a Turing test that includes deciphering and understanding text, visual, and auditory input. The competition was planned to end after the achievement of this prize. == Competition rules and restrictions == The rules varied over the years and early competitions featured restricted conversation Turing tests but since 1995 the discussion has been unrestricted. For the three entries in 2007, Robert Medeksza, Noah Duncan and Rollo Carpenter, some basic "screening questions" were used by the sponsor to evaluate the state of the technology. These included simple questions about the time, what round of the contest it is, etc.; general knowledge ("What is a hammer for?"); comparisons ("Which is faster, a train or a plane?"); and questions demonstrating memory for preceding parts of the same conversation. "All nouns, adjectives and verbs will come from a dictionary suitable for children or adolescents under the age of 12." Entries did not need to respond "intelligently" to the questions to be accepted. For the first time in 2008 the sponsor allowed introduction of a preliminary phase to the contest opening up the competition to previously disallowed web-based entries judged by a variety of invited interrogators. The available rules do not state how interrogators are selected or instructed. Interrogators (who judge the systems) have limited time: 5 minutes per entity in the 2003 competition, 20+ per pair in 2004–2007 competitions, 5 minutes to conduct simultaneous conversations with a human and the program in 2008–2009, increased to 25 minutes of simultaneous conversation since 2010. == Criticisms == The prize has long been scorned by experts in the field, for a variety of reasons. It is regarded by many as a publicity stunt. Marvin Minsky scathingly offered a "prize" to anyone who could stop the competition. Loebner responded by jokingly observing that Minsky's offering a prize to stop the competition effectively made him a co-sponsor. The rules of the competition have encouraged poorly qualified judges to make rapid judgements. Interactions between judges and competitors was originally very brief, for example effectively 2.5 mins of questioning, which permitted only a few questions. Questioning was initially restricted to a single topic of the contestant's choice, such as "whimsical conversation", a domain suiting standard chatbot tricks. Competition entrants do not aim at understanding or intelligence but resort to basic ELIZA style tricks, and successful entrants find deception and pretense is rewarded. == Contests == See article history for more details of some earlier contests. A very incomplete listing of a few of the contests: === 2003 === In 2003, the contest was organised by Professor Richard H. R. Harper and Dr. Lynne Hamill from the Digital World Research Centre at the University of Surrey. Although no bot passed the Turing test, the winner was Jabberwock, created by Juergen Pirner. Second was Elbot (Fred Roberts, Artificial Solutions). Third was Jabberwacky, (Rollo Carpenter). === 2006 === In 2006, the contest was organised by Tim Child (CEO of Televirtual) and Huma Shah. On August 30, the four finalists were announced: Rollo Carpenter Richard Churchill and Marie-Claire Jenkins Noah Duncan Robert Medeksza The contest was held on 17 September in the VR theatre, Torrington Place campus of University College London. The judges included the University of Reading's cybernetics professor, Kevin Warwick, a professor of artificial intelligence, John Barnden (specialist in metaphor research at the University of Birmingham), a barrister, Victoria Butler-Cole and a journalist, Graham Duncan-Rowe. The latter's experience of the event can be found in an article in Technology Review. The winner was 'Joan', based on Jabberwacky, both created by Rollo Carpenter. === 2007 === The 2007 competition was held on October 21 in New York City. The judges were: computer science professor Russ Abbott, philosophy professor Hartry Field, psychology assistant professor Clayton Curtis and English lecturer Scott Hutchins. No bot passed the Turing test, but the judges ranked the three contestants as follows: 1st: Robert Medeksza, creator of Ultra Hal 2nd: Noah Duncan, a private entry, creator of Cletus 3rd: Rollo Carpenter from Icogno, creator of Jabberwacky The winner received $2,250 and the annual medal. The runners-up received $250 each. === 2008 === The 2008 competition was organised by professor Kevin Warwick, coordinated by Huma Shah and held on October 12 at the University of Reading, UK. After testing by over one hundred judges during the preliminary phase, in June and July 2008, six finalists were selected from thirteen original entrant artificial conversational entities (ACEs). Five of those invited competed in the finals: Brother Jerome, Peter Cole and Benji Adams Elbot, Fred Roberts / Artificial Solutions Eugene Goostman, Vladimir Veselov, Eugene Demchenko and Sergey Ulasen Jabberwacky, Rollo Carpenter Ultra Hal, Robert Medeksza In the finals, each of the judges was given five minutes to conduct simultaneous, split-screen conversations with two hidden entities. Elbot of Artificial Solutions won the 2008 Loebner Prize bronze award, for most human-like artificial conversational entity, through fooling three of the twelve judges who interrogated it (in the human-parallel comparisons) into believing it was human. This is coming very close to the 30% traditionally required to consider that a program has actually passed the Turing test. Eugene Goostman and Ultra Hal both deceived one judge each that it was the human. Will Pavia, a journalist for The Times, has written about his experience; a Loebner finals' judge, he was deceived by Elbot and Eugene. Kevin Warwick and Huma Shah have reported on the parallel-paired Turing tests. === 2009 === The 2009 Loebner Prize Competition was held September 6, 2009, at the Brighton Centre, Brighton UK in conjunction with the Interspeech 2009 conference. The prize amount for 2009 was $3,000. Entrants were David Levy, Rollo Carpenter, and Mohan Embar, who finished in that order. The writer Brian Christian participated in the 2009 Loebner Prize Competition as a human confederate, and described his experiences at the competition in his book The Most Human Human. === 2010 === The 2010 Loebner Prize Competition was held on October 23 at California State University, Los Angeles. The 2010 competition was the 20th running of the contest. The winner was Bruce Wilcox with Suzette. === 2011 === The 2011 Loebner Prize Competition was held on October 19 at the University of Exeter, Devon, United Kingdom. The prize amount for 2011 was $4,000. The four finalists and their chatterbots were Bruce Wilcox (Rosette), Adeena Mignogna (Zoe), Mohan Embar (Chip Vivant) and Ron Lee (Tutor), who finished in that order. That year there was an addition of a panel of junior judges, namely Georgia-Mae Lindfield, William Dunne, Sam Keat and Kirill Jerdev. The results of the junior contest were markedly different from the main contest, with chatterbots Tutor and Zoe tying for first place and Chip Vivant and Rosette coming in third and fourt

  Read more →
• 

  Mixture model

  In statistics, a mixture model is a probabilistic model for representing the presence of subpopulations within an overall population, without requiring that an observed data set should identify the sub-population to which an individual observation belongs. Formally a mixture model corresponds to the mixture distribution that represents the probability distribution of observations in the overall population. However, while problems associated with "mixture distributions" relate to deriving the properties of the overall population from those of the sub-populations, "mixture models" are used to make statistical inferences about the properties of the sub-populations given only observations on the pooled population, without sub-population identity information. Mixture models are used for clustering, under the name model-based clustering, and also for density estimation. Mixture models should not be confused with models for compositional data, i.e., data whose components are constrained to sum to a constant value (1, 100%, etc.). However, compositional models can be thought of as mixture models, where members of the population are sampled at random. Conversely, mixture models can be thought of as compositional models, where the total size reading population has been normalized to 1. == Structure == === General mixture model === A typical finite-dimensional mixture model is a hierarchical model consisting of the following components: N random variables that are observed, each distributed according to a mixture of K components, with the components belonging to the same parametric family of distributions (e.g., all normal, all Zipfian, etc.) but with different parameters. However, it is also possible to have a finite mixture model where each component belongs to a different parametric family of distributions, for example, a mixture of a multivariate normal distribution and a generalized hyperbolic distribution. N random latent variables specifying the identity of the mixture component of each observation, each distributed according to a K-dimensional categorical distribution A set of K mixture weights, which are probabilities that sum to 1. A set of K parameters, each specifying the parameter of the corresponding mixture component. In many cases, each "parameter" is actually a set of parameters. For example, if the mixture components are Gaussian distributions, there will be a mean and variance for each component. If the mixture components are categorical distributions (e.g., when each observation is a token from a finite alphabet of size V), there will be a vector of V probabilities summing to 1. In addition, in a Bayesian setting, the mixture weights and parameters will themselves be random variables, and prior distributions will be placed over the variables. In such a case, the weights are typically viewed as a K-dimensional random vector drawn from a Dirichlet distribution (the conjugate prior of the categorical distribution), and the parameters will be distributed according to their respective conjugate priors. Mathematically, a basic parametric mixture model can be described as follows: K = number of mixture components N = number of observations θ i = 1 … K = parameter of distribution of observation associated with component i ϕ i = 1 … K = mixture weight, i.e., prior probability of a particular component i ϕ = K -dimensional vector composed of all the individual ϕ 1 … K ; must sum to 1 z i = 1 … N = component of observation i x i = 1 … N = observation i F ( x | θ ) = probability distribution of an observation, parametrized on θ z i = 1 … N ∼ Categorical ⁡ ( ϕ ) x i = 1 … N | z i = 1 … N ∼ F ( θ z i ) {\displaystyle {\begin{array}{lcl}K&=&{\text{number of mixture components}}\\N&=&{\text{number of observations}}\\\theta _{i=1\dots K}&=&{\text{parameter of distribution of observation associated with component }}i\\\phi _{i=1\dots K}&=&{\text{mixture weight, i.e., prior probability of a particular component }}i\\{\boldsymbol {\phi }}&=&K{\text{-dimensional vector composed of all the individual }}\phi _{1\dots K}{\text{; must sum to 1}}\\z_{i=1\dots N}&=&{\text{component of observation }}i\\x_{i=1\dots N}&=&{\text{observation }}i\\F(x|\theta )&=&{\text{probability distribution of an observation, parametrized on }}\theta \\z_{i=1\dots N}&\sim &\operatorname {Categorical} ({\boldsymbol {\phi }})\\x_{i=1\dots N}|z_{i=1\dots N}&\sim &F(\theta _{z_{i}})\end{array}}} In a Bayesian setting, all parameters are associated with random variables, as follows: K , N = as above θ i = 1 … K , ϕ i = 1 … K , ϕ = as above z i = 1 … N , x i = 1 … N , F ( x | θ ) = as above α = shared hyperparameter for component parameters β = shared hyperparameter for mixture weights H ( θ | α ) = prior probability distribution of component parameters, parametrized on α θ i = 1 … K ∼ H ( θ | α ) ϕ ∼ S y m m e t r i c - D i r i c h l e t K ⁡ ( β ) z i = 1 … N | ϕ ∼ Categorical ⁡ ( ϕ ) x i = 1 … N | z i = 1 … N , θ i = 1 … K ∼ F ( θ z i ) {\displaystyle {\begin{array}{lcl}K,N&=&{\text{as above}}\\\theta _{i=1\dots K},\phi _{i=1\dots K},{\boldsymbol {\phi }}&=&{\text{as above}}\\z_{i=1\dots N},x_{i=1\dots N},F(x|\theta )&=&{\text{as above}}\\\alpha &=&{\text{shared hyperparameter for component parameters}}\\\beta &=&{\text{shared hyperparameter for mixture weights}}\\H(\theta |\alpha )&=&{\text{prior probability distribution of component parameters, parametrized on }}\alpha \\\theta _{i=1\dots K}&\sim &H(\theta |\alpha )\\{\boldsymbol {\phi }}&\sim &\operatorname {Symmetric-Dirichlet} _{K}(\beta )\\z_{i=1\dots N}|{\boldsymbol {\phi }}&\sim &\operatorname {Categorical} ({\boldsymbol {\phi }})\\x_{i=1\dots N}|z_{i=1\dots N},\theta _{i=1\dots K}&\sim &F(\theta _{z_{i}})\end{array}}} This characterization uses F and H to describe arbitrary distributions over observations and parameters, respectively. Typically H will be the conjugate prior of F. The two most common choices of F are Gaussian aka "normal" (for real-valued observations) and categorical (for discrete observations). Other common possibilities for the distribution of the mixture components are: Binomial distribution, for the number of "positive occurrences" (e.g., successes, yes votes, etc.) given a fixed number of total occurrences Multinomial distribution, similar to the binomial distribution, but for counts of multi-way occurrences (e.g., yes/no/maybe in a survey) Negative binomial distribution, for binomial-type observations but where the quantity of interest is the number of failures before a given number of successes occurs Poisson distribution, for the number of occurrences of an event in a given period of time, for an event that is characterized by a fixed rate of occurrence Exponential distribution, for the time before the next event occurs, for an event that is characterized by a fixed rate of occurrence Log-normal distribution, for positive real numbers that are assumed to grow exponentially, such as incomes or prices Multivariate normal distribution (aka multivariate Gaussian distribution), for vectors of correlated outcomes that are individually Gaussian-distributed Multivariate Student's t-distribution, for vectors of heavy-tailed correlated outcomes A vector of Bernoulli-distributed values, corresponding, e.g., to a black-and-white image, with each value representing a pixel; see the handwriting-recognition example below === Specific examples === ==== Gaussian mixture model ==== A typical non-Bayesian Gaussian mixture model looks like this: K , N = as above ϕ i = 1 … K , ϕ = as above z i = 1 … N , x i = 1 … N = as above θ i = 1 … K = { μ i = 1 … K , σ i = 1 … K 2 } μ i = 1 … K = mean of component i σ i = 1 … K 2 = variance of component i z i = 1 … N ∼ Categorical ⁡ ( ϕ ) x i = 1 … N ∼ N ( μ z i , σ z i 2 ) {\displaystyle {\begin{array}{lcl}K,N&=&{\text{as above}}\\\phi _{i=1\dots K},{\boldsymbol {\phi }}&=&{\text{as above}}\\z_{i=1\dots N},x_{i=1\dots N}&=&{\text{as above}}\\\theta _{i=1\dots K}&=&\{\mu _{i=1\dots K},\sigma _{i=1\dots K}^{2}\}\\\mu _{i=1\dots K}&=&{\text{mean of component }}i\\\sigma _{i=1\dots K}^{2}&=&{\text{variance of component }}i\\z_{i=1\dots N}&\sim &\operatorname {Categorical} ({\boldsymbol {\phi }})\\x_{i=1\dots N}&\sim &{\mathcal {N}}(\mu _{z_{i}},\sigma _{z_{i}}^{2})\end{array}}} A Bayesian version of a Gaussian mixture model is as follows: K , N = as above ϕ i = 1 … K , ϕ = as above z i = 1 … N , x i = 1 … N = as above θ i = 1 … K = { μ i = 1 … K , σ i = 1 … K 2 } μ i = 1 … K = mean of component i σ i = 1 … K 2 = variance of component i μ 0 , λ , ν , σ 0 2 = shared hyperparameters μ i = 1 … K ∼ N ( μ 0 , λ σ i 2 ) σ i = 1 … K 2 ∼ I n v e r s e - G a m m a ⁡ ( ν , σ 0 2 ) ϕ ∼ S y m m e t r i c - D i r i c h l e t K ⁡ ( β ) z i = 1 … N ∼ Categorical ⁡ ( ϕ ) x i = 1 … N ∼ N ( μ z i , σ z i 2 ) {\displaystyle {\begin{array}{lcl}K,N&=&{\text{as above}}\\\phi _{i=1\dots K},{\boldsymbol {\phi }}&=&{\text{as above}}\\z_{i=1\dots N},x_{i=1\dots N}&=&{\text{as above}}\\\theta _{i=1\

  Read more →
• 

  Almeida–Pineda recurrent backpropagation

  Almeida–Pineda recurrent backpropagation is an extension to the backpropagation algorithm that is applicable to recurrent neural networks. It is a type of supervised learning. It was described somewhat cryptically in Richard Feynman's senior thesis, and rediscovered independently in the context of artificial neural networks by both Fernando Pineda and Luis B. Almeida. A recurrent neural network for this algorithm consists of some input units, some output units and eventually some hidden units. For a given set of (input, target) states, the network is trained to settle into a stable activation state with the output units in the target state, based on a given input state clamped on the input units.

  Read more →
• 

  Multidimensional analysis

  In statistics, econometrics and related fields, multidimensional analysis (MDA) is a data analysis process that groups data into two categories: data dimensions and measurements. For example, a data set consisting of the number of wins for a single football team at each of several years is a single-dimensional (in this case, longitudinal) data set. A data set consisting of the number of wins for several football teams in a single year is also a single-dimensional (in this case, cross-sectional) data set. A data set consisting of the number of wins for several football teams over several years is a two-dimensional data set. == Higher dimensions == In many disciplines, two-dimensional data sets are also called panel data. While, strictly speaking, two- and higher-dimensional data sets are "multi-dimensional", the term "multidimensional" tends to be applied only to data sets with three or more dimensions. For example, some forecast data sets provide forecasts for multiple target periods, conducted by multiple forecasters, and made at multiple horizons. The three dimensions provide more information than can be gleaned from two-dimensional panel data sets. == Software == Computer software for MDA include Online analytical processing (OLAP) for data in relational databases, pivot tables for data in spreadsheets, and Array DBMSs for general multi-dimensional data (such as raster data) in science, engineering, and business.

  Read more →
• 

  Tradeshift

  Tradeshift is a cloud based business network and platform for purchase-to-pay automation, supply chain payments, marketplaces, virtual cards and supply chain financing. Its 2018 round of funding, led by Goldman Sachs, raised US$250 million at a valuation of $1.1 billion, giving the company unicorn status. Tradeshift is headquartered in San Francisco, California and has offices in London, Copenhagen, Bucharest and Kuala Lumpur. Tradeshift has reprocessed over $1 trillion USD through transactions on its network. == History == Tradeshift was founded in 2010 by Christian Lanng, Mikkel Hippe Brun, and Gert Sylvest. Inspiration for Tradeshift came after they created the world's first large scale peer-to-peer infrastructure for an e-business called NemHandel. The founders also had leading roles (Governing board member, Technical Director) in the European Commission project PEPPOL inside the European Union. In 2010, the Tradeshift platform launched in May in Copenhagen. Tradeshift won the European Startup Awards in the category of "Best Business or Enterprise Startup." In 2011, Tradeshift made its app marketplace available. In 2012, Tradeshift moved their headquarters from Copenhagen to San Francisco. In 2013, Tradeshift opened an R&D center in Suzhou, China. Tradeshift opened an additional office in London. And LATAM e-invoicing capabilities were added through partnership with Invoiceware. In 2014, Tradeshift expanded with offices in Tokyo, Paris, and Munich. The EU Commission officially approved the Universal Business Language (UBL) data format – a format Tradeshift supports – as eligible for referencing in tenders from public administrations. In 2015, Tradeshift won the Circulars "Digital Disruptor" Award at the WEF conference in Davos, Switzerland. Tradeshift also acquired product information management company Merchantry, and launched e-procurement and supplier risk management solutions. In 2016, Tradeshift acquired Hyper Travel and secured a $75 million series-D round funding. In 2017, Tradeshift acquired IBX Business Network and launches Tradeshift Ada. In 2018, Tradeshift secured a $250 million series-E round funding. and launched Blockchain Payments, the latter as part of Tradeshift Pay. In December 2018 Tradeshift acquired Babelway, an online B2B integration platform. The acquisition added three new office locations to Tradeshift (Salt Lake City, Louvain-la-neuve, Belgium, Cairo Egypt). In Q3 2018, Tradeshift reported year-over-year revenue growth of 400%, new bookings growth of 284%, and gross merchandise volume (GMV) growth of 262%. New total contract value also grew by US$47 million. Additionally, it added 27 new customers including Hertz, Shiseido, ECU and multiple Fortune 500 companies. In July 2023, HSBC and Tradeshift announced an agreement to launch a new, jointly owned business focused on the development of embedded finance solutions and financial services apps. As part of the agreement, HSBC made a $35 million investment into Tradeshift and joined its board. The agreement was part of a funding round which is expected to raise a minimum of $70 million from HSBC and other investors. The new joint venture will allow HSBC and Tradeshift to deploy a range of digital solutions across Tradeshift and other platforms. This includes payment and fintech services embedded into trade, e-commerce and marketplace experiences. In September 2023, CEO Lanng was fired for "gross misconduct on multiple grounds," including "allegations of sexual assault and harassment." Tradeshift was alleged to have fired his accuser after she complained to the company's human resources department, its co-founders and members of its board of directors about his abuse. == Financials == The company's valuation as of May 2018 was $1.1 billion. Tradeshift is now considered a unicorn, and, according to Bloomberg, will not need any further funding. Jan 14, 2020, Tradeshift announced that they had raised $240 million in Series F finance. == Acquisitions == In 2015, Tradeshift acquired product information management company Merchantry. Merchantry is a retail product information management (PIM) software for multi-vendor ecommerce retailers. In 2016, Tradeshift acquired Hyper Travel. Hyper Travel is a travel management service that allows customers to access travel agents via its native messaging apps, SMS, and email. In 2017, Tradeshift acquired IBX Group. In 2018, Tradeshift acquired Babelway, an online B2B integration platform.

  Read more →
• 

  Yooreeka

  Yooreeka is a library for data mining, machine learning, soft computing, and mathematical analysis. The project started with the code of the book "Algorithms of the Intelligent Web". Although the term "Web" prevailed in the title, in essence, the algorithms are valuable in any software application. It covers all major algorithms and provides many examples. Yooreeka 2.x is licensed under the Apache License rather than the somewhat more restrictive LGPL (which was the license of v1.x). The library is written 100% in the Java language. == Algorithms == The following algorithms are covered: Clustering Hierarchical—Agglomerative (e.g. MST single link; ROCK) and Divisive Partitional (e.g. k-means) Classification Bayesian Decision trees Neural Networks Rule based (via Drools) Recommendations Collaborative filtering Content based Search PageRank DocRank Personalization

  Read more →
• 

  Bootstrap aggregating

  Bootstrap aggregating, also called bagging (from bootstrap aggregating) or bootstrapping, is a machine learning (ML) ensemble meta-algorithm designed to improve the stability and accuracy of ML classification and regression algorithms. It also reduces variance and overfitting. Although it is usually applied to decision tree methods, it can be used with any type of method. Bagging is a special case of the ensemble averaging approach. == Description of the technique == Given a standard training set D {\displaystyle D} of size n {\displaystyle n} , bagging generates m {\displaystyle m} new training sets D i {\displaystyle D_{i}} , each of size n ′ {\displaystyle n'} , by sampling from D {\displaystyle D} uniformly and with replacement. By sampling with replacement, some observations may be repeated in each D i {\displaystyle D_{i}} . If n ′ = n {\displaystyle n'=n} , then for large n {\displaystyle n} the set D i {\displaystyle D_{i}} is expected to have the fraction (1 - 1/e) (~63.2%) of the unique samples of D {\displaystyle D} , the rest being duplicates. This kind of sample is known as a bootstrap sample. Sampling with replacement ensures each bootstrap is independent from its peers, as it does not depend on previous chosen samples when sampling. Then, m {\displaystyle m} models are fitted using the above bootstrap samples and combined by averaging the output (for regression) or voting (for classification). Bagging leads to "improvements for unstable procedures", which include, for example, artificial neural networks, classification and regression trees, and subset selection in linear regression. Bagging was shown to improve preimage learning. On the other hand, it can mildly degrade the performance of stable methods such as k-nearest neighbors. == Process of the algorithm == === Key Terms === There are three types of datasets in bootstrap aggregating. These are the original, bootstrap, and out-of-bag datasets. Each section below will explain how each dataset is made except for the original dataset. The original dataset is whatever information is given. === Creating the bootstrap dataset === The bootstrap dataset is made by randomly picking objects from the original dataset. Also, it must be the same size as the original dataset. However, the difference is that the bootstrap dataset can have duplicate objects. Here is a simple example to demonstrate how it works along with the illustration below: Suppose the original dataset is a group of 12 people. Their names are Emily, Jessie, George, Constantine, Lexi, Theodore, John, James, Rachel, Anthony, Ellie, and Jamal. By randomly picking a group of names, let us say our bootstrap dataset had James, Ellie, Constantine, Lexi, John, Constantine, Theodore, Constantine, Anthony, Lexi, Constantine, and Theodore. In this case, the bootstrap sample contained four duplicates for Constantine, and two duplicates for Lexi, and Theodore. === Creating the out-of-bag dataset === The out-of-bag dataset represents the remaining people who were not in the bootstrap dataset. It can be calculated by taking the difference between the original and the bootstrap datasets. In this case, the remaining samples who were not selected are Emily, Jessie, George, Rachel, and Jamal. Keep in mind that since both datasets are sets, when taking the difference the duplicate names are ignored in the bootstrap dataset. The illustration below shows how the math is done: === Application === Creating the bootstrap and out-of-bag datasets is crucial since it is used to test the accuracy of ensemble learning algorithms like random forest. For example, a model that produces 50 trees using the bootstrap/out-of-bag datasets will have a better accuracy than if it produced 10 trees. Since the algorithm generates multiple trees and therefore multiple datasets the chance that an object is left out of the bootstrap dataset is low. The next few sections talk about how the random forest algorithm works in more detail. === Creation of Decision Trees === The next step of the algorithm involves the generation of decision trees from the bootstrapped dataset. To achieve this, the process examines each gene/feature and determines for how many samples the feature's presence or absence yields a positive or negative result. This information is then used to compute a confusion matrix, which lists the true positives, false positives, true negatives, and false negatives of the feature when used as a classifier. These features are then ranked according to various classification metrics based on their confusion matrices. Some common metrics include estimate of positive correctness (calculated by subtracting false positives from true positives), measure of "goodness", and information gain. These features are then used to partition the samples into two sets: those that possess the top feature, and those that do not. The diagram below shows a decision tree of depth two being used to classify data. For example, a data point that exhibits Feature 1, but not Feature 2, will be given a "No". Another point that does not exhibit Feature 1, but does exhibit Feature 3, will be given a "Yes". This process is repeated recursively for successive levels of the tree until the desired depth is reached. At the very bottom of the tree, samples that test positive for the final feature are generally classified as positive, while those that lack the feature are classified as negative. These trees are then used as predictors to classify new data. === Random Forests === The next part of the algorithm involves introducing yet another element of variability amongst the bootstrapped trees. In addition to each tree only examining a bootstrapped set of samples, only a small but consistent number of unique features are considered when ranking them as classifiers. This means that each tree only knows about the data pertaining to a small constant number of features, and a variable number of samples that is less than or equal to that of the original dataset. Consequently, the trees are more likely to return a wider array of answers, derived from more diverse knowledge. This results in a random forest, which possesses numerous benefits over a single decision tree generated without randomness. In a random forest, each tree "votes" on whether or not to classify a sample as positive based on its features. The sample is then classified based on majority vote. An example of this is given in the diagram below, where the four trees in a random forest vote on whether or not a patient with mutations A, B, F, and G has cancer. Since three out of four trees vote yes, the patient is then classified as cancer positive. Because of their properties, random forests are considered one of the most accurate data mining algorithms, are less likely to overfit their data, and run quickly and efficiently even for large datasets. They are primarily useful for classification as opposed to regression, which attempts to draw observed connections between statistical variables in a dataset. This makes random forests particularly useful in such fields as banking, healthcare, the stock market, and e-commerce where it is important to be able to predict future results based on past data. One of their applications would be as a useful tool for predicting cancer based on genetic factors, as seen in the above example. There are several important factors to consider when designing a random forest. If the trees in the random forests are too deep, overfitting can still occur due to over-specificity. If the forest is too large, the algorithm may become less efficient due to an increased runtime. Random forests also do not generally perform well when given sparse data with little variability. However, they still have numerous advantages over similar data classification algorithms such as neural networks, as they are much easier to interpret and generally require less data for training. As an integral component of random forests, bootstrap aggregating is very important to classification algorithms, and provides a critical element of variability that allows for increased accuracy when analyzing new data, as discussed below. == Improving Random Forests and Bagging == While the techniques described above utilize random forests and bagging (otherwise known as bootstrapping), there are certain techniques that can be used in order to improve their execution and voting time, their prediction accuracy, and their overall performance. The following are key steps in creating an efficient random forest: Specify the maximum depth of trees: Instead of allowing the random forest to continue until all nodes are pure, it is better to cut it off at a certain point in order to further decrease chances of overfitting. Prune the dataset: Using an extremely large dataset may create results that are less indicative of the data provided than a smaller set that more accurately represents what is being focused on. Continue pruning the data at each

  Read more →
• 

  Yooreeka

  Yooreeka is a library for data mining, machine learning, soft computing, and mathematical analysis. The project started with the code of the book "Algorithms of the Intelligent Web". Although the term "Web" prevailed in the title, in essence, the algorithms are valuable in any software application. It covers all major algorithms and provides many examples. Yooreeka 2.x is licensed under the Apache License rather than the somewhat more restrictive LGPL (which was the license of v1.x). The library is written 100% in the Java language. == Algorithms == The following algorithms are covered: Clustering Hierarchical—Agglomerative (e.g. MST single link; ROCK) and Divisive Partitional (e.g. k-means) Classification Bayesian Decision trees Neural Networks Rule based (via Drools) Recommendations Collaborative filtering Content based Search PageRank DocRank Personalization

  Read more →
• 

  Apertus (LLM)

  Apertus is a public large language model, developed by the Swiss AI Initiative (a collaboration between EPFL, ETH Zurich, and the Swiss National Supercomputing Centre). It was released on September 2, 2025, under the free and open-source Apache 2.0 license. Designed initially for business and research use cases around the world, Apertus was trained on over 1800 languages, and comes in 8 billion or 70 billion parameter versions and is available on Hugging Face for download. The model was developed aiming to adhere to European copyright law, and is one of the first examples of AI as a public good in the vein of AI Sovereignty. It is also the first large model to comply with the European Union's Artificial Intelligence Act. At its launch, the model creators emphasized multilinguality, transparency, and auditability as priorities in contrast to commercial frontier model. While international reception was largely positive, the first iteration was significantly behind the capabilities of frontier models and needs adaptation for many use cases with chatbots being a secondary but not a primary use case. As of late 2025, it was considered the largest and most capable fully open model. The capability of future models will depend in part on how much more funding can be secured.

  Read more →
• 

  Receptron

  The receptron (short for "reservoir perceptron") is a neuromorphic data processing model — specifically neuromorphic computing — that generalizes the traditional perceptron, by incorporating non-linear interactions between inputs. Unlike classical perceptron, which rely on linearly independent weights, the receptron leverages complexity in physical substrates, such as the electric conduction properties of nanostructured materials or optical speckle fields, to perform classification tasks. The receptron bridges unconventional computing and neural network principles, enabling solutions that do not require the training approaches typical of artificial neural networks based on the perceptron model. == Algorithm == The receptron is an algorithm for supervised learning of binary classifiers, so a classification algorithm that makes its predictions based on a predictor function, combining a set of weights with the feature vector. The mathematical model is based on the sum of inputs with non-linear interactions: S = ∑ k = 1 n x j w ~ j ( x → ) | S ∈ R {\displaystyle S=\sum _{k=1}^{n}x_{j}{\widetilde {w}}_{j}({\vec {x}})|S\in R} (1) where j ∈ [ 1 , n ] {\displaystyle j\in [1,n]} and w ~ j {\displaystyle {\widetilde {w}}_{j}} are non-linear weight functions depending on the inputs, x → {\displaystyle {\vec {x}}} . Nonlinearity will typically make the system extremely complex, and allowing for the solution of problems not solvable through the simpler rules of a linear system, such as the perceptron or McCulloch Pitts neurons, which is based on the sum of linearly independent weights: S = ∑ k = 1 n x j w j p {\displaystyle S=\sum _{k=1}^{n}x_{j}w_{j}^{p}} (2) where w j {\displaystyle w_{j}} are constant real values. A consequence of this simplicity is the limitation to linearly separable functions, which necessitates multi-layer architectures and training algorithms like backpropagation As in the perceptron case, the summation in Eq. 1 origins the activation of the receptron output through the thresholding process, Y ( x 1 , . . . , x n ) = { 1 if S > th 0 if S ≤ th {\displaystyle Y(x_{1},...,x_{n})={\begin{cases}1&{\text{if }}S>{\text{th}}\\0&{\text{if }}S\leq {\text{th}}\end{cases}}} (3) where th is a constant threshold parameter. Equation 3 can be written by using the Heaviside step function. The weight functions w ~ ( x → ) {\displaystyle {\widetilde {w}}({\vec {x}})} can be written with a finite number of parameters w j 1 . . . j n {\displaystyle w_{j_{1}...j_{n}}} , simplifying the model representation. One can Taylor-expand w ~ ( x → ) {\displaystyle {\widetilde {w}}({\vec {x}})} and use the idempotency of Boolean variables ( x j ) q = x j ∀ q ≥ 1 {\displaystyle (x_{j})^{q}=x_{j}\forall q\geq 1} such that S ′ = b + ∑ k = 1 n x j w ~ j ( x → ) {\displaystyle S'=b+\sum _{k=1}^{n}x_{j}{\widetilde {w}}_{j}({\vec {x}})} can be written as S ′ ( x → ) = b + ∑ j w j x j + ∑ j < k w j k x j x k + ∑ j < k < l w j k l x j x k x l + . . . {\displaystyle S'({\vec {x}})=b+\sum _{j}w_{j}x_{j}+\sum _{j Read more →

• 

  Least-squares support vector machine

  Least-squares support-vector machines (LS-SVM) for statistics and in statistical modeling, are least-squares versions of support-vector machines (SVM), which are a set of related supervised learning methods that analyze data and recognize patterns, and which are used for classification and regression analysis. In this version one finds the solution by solving a set of linear equations instead of a convex quadratic programming (QP) problem for classical SVMs. Least-squares SVM classifiers were proposed by Johan Suykens and Joos Vandewalle. LS-SVMs are a class of kernel-based learning methods. == From support-vector machine to least-squares support-vector machine == Given a training set { x i , y i } i = 1 N {\displaystyle \{x_{i},y_{i}\}_{i=1}^{N}} with input data x i ∈ R n {\displaystyle x_{i}\in \mathbb {R} ^{n}} and corresponding binary class labels y i ∈ { − 1 , + 1 } {\displaystyle y_{i}\in \{-1,+1\}} , the SVM classifier, according to Vapnik's original formulation, satisfies the following conditions: { w T ϕ ( x i ) + b ≥ 1 , if y i = + 1 , w T ϕ ( x i ) + b ≤ − 1 , if y i = − 1 , {\displaystyle {\begin{cases}w^{T}\phi (x_{i})+b\geq 1,&{\text{if }}\quad y_{i}=+1,\\w^{T}\phi (x_{i})+b\leq -1,&{\text{if }}\quad y_{i}=-1,\end{cases}}} which is equivalent to y i [ w T ϕ ( x i ) + b ] ≥ 1 , i = 1 , … , N , {\displaystyle y_{i}\left[{w^{T}\phi (x_{i})+b}\right]\geq 1,\quad i=1,\ldots ,N,} where ϕ ( x ) {\displaystyle \phi (x)} is the nonlinear map from original space to the high- or infinite-dimensional space. === Inseparable data === In case such a separating hyperplane does not exist, we introduce so-called slack variables ξ i {\displaystyle \xi _{i}} such that { y i [ w T ϕ ( x i ) + b ] ≥ 1 − ξ i , i = 1 , … , N , ξ i ≥ 0 , i = 1 , … , N . {\displaystyle {\begin{cases}y_{i}\left[{w^{T}\phi (x_{i})+b}\right]\geq 1-\xi _{i},&i=1,\ldots ,N,\\\xi _{i}\geq 0,&i=1,\ldots ,N.\end{cases}}} According to the structural risk minimization principle, the risk bound is minimized by the following minimization problem: min J 1 ( w , ξ ) = 1 2 w T w + c ∑ i = 1 N ξ i , {\displaystyle \min J_{1}(w,\xi )={\frac {1}{2}}w^{T}w+c\sum \limits _{i=1}^{N}\xi _{i},} Subject to { y i [ w T ϕ ( x i ) + b ] ≥ 1 − ξ i , i = 1 , … , N , ξ i ≥ 0 , i = 1 , … , N , {\displaystyle {\text{Subject to }}{\begin{cases}y_{i}\left[{w^{T}\phi (x_{i})+b}\right]\geq 1-\xi _{i},&i=1,\ldots ,N,\\\xi _{i}\geq 0,&i=1,\ldots ,N,\end{cases}}} To solve this problem, we could construct the Lagrangian function: L 1 ( w , b , ξ , α , β ) = 1 2 w T w + c ∑ i = 1 N ξ i − ∑ i = 1 N α i { y i [ w T ϕ ( x i ) + b ] − 1 + ξ i } − ∑ i = 1 N β i ξ i , {\displaystyle L_{1}(w,b,\xi ,\alpha ,\beta )={\frac {1}{2}}w^{T}w+c\sum \limits _{i=1}^{N}{\xi _{i}}-\sum \limits _{i=1}^{N}\alpha _{i}\left\{y_{i}\left[{w^{T}\phi (x_{i})+b}\right]-1+\xi _{i}\right\}-\sum \limits _{i=1}^{N}\beta _{i}\xi _{i},} where α i ≥ 0 , β i ≥ 0 ( i = 1 , … , N ) {\displaystyle \alpha _{i}\geq 0,\ \beta _{i}\geq 0\ (i=1,\ldots ,N)} are the Lagrangian multipliers. The optimal point will be in the saddle point of the Lagrangian function, and then we obtain By substituting w {\displaystyle w} by its expression in the Lagrangian formed from the appropriate objective and constraints, we will get the following quadratic programming problem: max Q 1 ( α ) = − 1 2 ∑ i , j = 1 N α i α j y i y j K ( x i , x j ) + ∑ i = 1 N α i , {\displaystyle \max Q_{1}(\alpha )=-{\frac {1}{2}}\sum \limits _{i,j=1}^{N}{\alpha _{i}\alpha _{j}y_{i}y_{j}K(x_{i},x_{j})}+\sum \limits _{i=1}^{N}\alpha _{i},} where K ( x i , x j ) = ⟨ ϕ ( x i ) , ϕ ( x j ) ⟩ {\displaystyle K(x_{i},x_{j})=\left\langle \phi (x_{i}),\phi (x_{j})\right\rangle } is called the kernel function. Solving this QP problem subject to constraints in (1), we will get the hyperplane in the high-dimensional space and hence the classifier in the original space. === Least-squares SVM formulation === The least-squares version of the SVM classifier is obtained by reformulating the minimization problem as min J 2 ( w , b , e ) = μ 2 w T w + ζ 2 ∑ i = 1 N e i 2 , {\displaystyle \min J_{2}(w,b,e)={\frac {\mu }{2}}w^{T}w+{\frac {\zeta }{2}}\sum \limits _{i=1}^{N}e_{i}^{2},} subject to the equality constraints y i [ w T ϕ ( x i ) + b ] = 1 − e i , i = 1 , … , N . {\displaystyle y_{i}\left[{w^{T}\phi (x_{i})+b}\right]=1-e_{i},\quad i=1,\ldots ,N.} The least-squares SVM (LS-SVM) classifier formulation above implicitly corresponds to a regression interpretation with binary targets y i = ± 1 {\displaystyle y_{i}=\pm 1} . Using y i 2 = 1 {\displaystyle y_{i}^{2}=1} , we have ∑ i = 1 N e i 2 = ∑ i = 1 N ( y i e i ) 2 = ∑ i = 1 N e i 2 = ∑ i = 1 N ( y i − ( w T ϕ ( x i ) + b ) ) 2 , {\displaystyle \sum \limits _{i=1}^{N}e_{i}^{2}=\sum \limits _{i=1}^{N}(y_{i}e_{i})^{2}=\sum \limits _{i=1}^{N}e_{i}^{2}=\sum \limits _{i=1}^{N}\left(y_{i}-(w^{T}\phi (x_{i})+b)\right)^{2},} with e i = y i − ( w T ϕ ( x i ) + b ) . {\displaystyle e_{i}=y_{i}-(w^{T}\phi (x_{i})+b).} Notice, that this error would also make sense for least-squares data fitting, so that the same end results holds for the regression case. Hence the LS-SVM classifier formulation is equivalent to J 2 ( w , b , e ) = μ E W + ζ E D {\displaystyle J_{2}(w,b,e)=\mu E_{W}+\zeta E_{D}} with E W = 1 2 w T w {\displaystyle E_{W}={\frac {1}{2}}w^{T}w} and E D = 1 2 ∑ i = 1 N e i 2 = 1 2 ∑ i = 1 N ( y i − ( w T ϕ ( x i ) + b ) ) 2 . {\displaystyle E_{D}={\frac {1}{2}}\sum \limits _{i=1}^{N}e_{i}^{2}={\frac {1}{2}}\sum \limits _{i=1}^{N}\left(y_{i}-(w^{T}\phi (x_{i})+b)\right)^{2}.} Both μ {\displaystyle \mu } and ζ {\displaystyle \zeta } should be considered as hyperparameters to tune the amount of regularization versus the sum squared error. The solution does only depend on the ratio γ = ζ / μ {\displaystyle \gamma =\zeta /\mu } , therefore the original formulation uses only γ {\displaystyle \gamma } as tuning parameter. We use both μ {\displaystyle \mu } and ζ {\displaystyle \zeta } as parameters in order to provide a Bayesian interpretation to LS-SVM. The solution of LS-SVM regressor will be obtained after we construct the Lagrangian function: { L 2 ( w , b , e , α ) = J 2 ( w , e ) − ∑ i = 1 N α i { [ w T ϕ ( x i ) + b ] + e i − y i } , = 1 2 w T w + γ 2 ∑ i = 1 N e i 2 − ∑ i = 1 N α i { [ w T ϕ ( x i ) + b ] + e i − y i } , {\displaystyle {\begin{cases}L_{2}(w,b,e,\alpha )\;=J_{2}(w,e)-\sum \limits _{i=1}^{N}\alpha _{i}\left\{{\left[{w^{T}\phi (x_{i})+b}\right]+e_{i}-y_{i}}\right\},\\\quad \quad \quad \quad \quad \;={\frac {1}{2}}w^{T}w+{\frac {\gamma }{2}}\sum \limits _{i=1}^{N}e_{i}^{2}-\sum \limits _{i=1}^{N}\alpha _{i}\left\{\left[w^{T}\phi (x_{i})+b\right]+e_{i}-y_{i}\right\},\end{cases}}} where α i ∈ R {\displaystyle \alpha _{i}\in \mathbb {R} } are the Lagrange multipliers. The conditions for optimality are { ∂ L 2 ∂ w = 0 → w = ∑ i = 1 N α i ϕ ( x i ) , ∂ L 2 ∂ b = 0 → ∑ i = 1 N α i = 0 , ∂ L 2 ∂ e i = 0 → α i = γ e i , i = 1 , … , N , ∂ L 2 ∂ α i = 0 → y i = w T ϕ ( x i ) + b + e i , i = 1 , … , N . {\displaystyle {\begin{cases}{\frac {\partial L_{2}}{\partial w}}=0\quad \to \quad w=\sum \limits _{i=1}^{N}\alpha _{i}\phi (x_{i}),\\{\frac {\partial L_{2}}{\partial b}}=0\quad \to \quad \sum \limits _{i=1}^{N}\alpha _{i}=0,\\{\frac {\partial L_{2}}{\partial e_{i}}}=0\quad \to \quad \alpha _{i}=\gamma e_{i},\;i=1,\ldots ,N,\\{\frac {\partial L_{2}}{\partial \alpha _{i}}}=0\quad \to \quad y_{i}=w^{T}\phi (x_{i})+b+e_{i},\,i=1,\ldots ,N.\end{cases}}} Elimination of w {\displaystyle w} and e {\displaystyle e} will yield a linear system instead of a quadratic programming problem: [ 0 1 N T 1 N Ω + γ − 1 I N ] [ b α ] = [ 0 Y ] , {\displaystyle \left[{\begin{matrix}0&1_{N}^{T}\\1_{N}&\Omega +\gamma ^{-1}I_{N}\end{matrix}}\right]\left[{\begin{matrix}b\\\alpha \end{matrix}}\right]=\left[{\begin{matrix}0\\Y\end{matrix}}\right],} with Y = [ y 1 , … , y N ] T {\displaystyle Y=[y_{1},\ldots ,y_{N}]^{T}} , 1 N = [ 1 , … , 1 ] T {\displaystyle 1_{N}=[1,\ldots ,1]^{T}} and α = [ α 1 , … , α N ] T {\displaystyle \alpha =[\alpha _{1},\ldots ,\alpha _{N}]^{T}} . Here, I N {\displaystyle I_{N}} is an N × N {\displaystyle N\times N} identity matrix, and Ω ∈ R N × N {\displaystyle \Omega \in \mathbb {R} ^{N\times N}} is the kernel matrix defined by Ω i j = ϕ ( x i ) T ϕ ( x j ) = K ( x i , x j ) {\displaystyle \Omega _{ij}=\phi (x_{i})^{T}\phi (x_{j})=K(x_{i},x_{j})} . === Kernel function K === For the kernel function K(•, •) one typically has the following choices: Linear kernel : K ( x , x i ) = x i T x , {\displaystyle K(x,x_{i})=x_{i}^{T}x,} Polynomial kernel of degree d {\displaystyle d} : K ( x , x i ) = ( 1 + x i T x / c ) d , {\displaystyle K(x,x_{i})=\left({1+x_{i}^{T}x/c}\right)^{d},} Radial basis function RBF kernel : K ( x , x i ) = exp ⁡ ( − ‖ x − x i ‖ 2 / σ 2 ) , {\displaystyle K(x,x_{i})=\exp \left({-\left\|{x-x_{i}}\right\|^{2}/\sigma ^{2}}\right),} MLP kernel : K ( x , x i ) = tanh ⁡ ( k x i T x + θ ) , {\displaystyle K(x,x_{i})=\tanh \left({k

  Read more →
• 

  Sparse PCA

  Sparse principal component analysis (SPCA or sparse PCA) is a technique used in statistical analysis and, in particular, in the analysis of multivariate data sets. It extends the classic method of principal component analysis (PCA) for the reduction of dimensionality of data by introducing sparsity structures to the input variables. A particular disadvantage of ordinary PCA is that the principal components are usually linear combinations of all input variables. SPCA overcomes this disadvantage by finding components that are linear combinations of just a few input variables (SPCs). This means that some of the coefficients of the linear combinations defining the SPCs, called loadings, are equal to zero. The number of nonzero loadings is called the cardinality of the SPC. == Mathematical formulation == Consider a data matrix, X {\displaystyle X} , where each of the p {\displaystyle p} columns represent an input variable, and each of the n {\displaystyle n} rows represents an independent sample from data population. One assumes each column of X {\displaystyle X} has mean zero, otherwise one can subtract column-wise mean from each element of X {\displaystyle X} . Let Σ = 1 n − 1 X ⊤ X {\displaystyle \Sigma ={\frac {1}{n-1}}X^{\top }X} be the empirical covariance matrix of X {\displaystyle X} , which has dimension p × p {\displaystyle p\times p} . Given an integer k {\displaystyle k} with 1 ≤ k ≤ p {\displaystyle 1\leq k\leq p} , the sparse PCA problem can be formulated as maximizing the variance along a direction represented by vector v ∈ R p {\displaystyle v\in \mathbb {R} ^{p}} while constraining its cardinality: max v T Σ v subject to ‖ v ‖ 2 = 1 ‖ v ‖ 0 ≤ k . {\displaystyle {\begin{aligned}\max \quad &v^{T}\Sigma v\\{\text{subject to}}\quad &\left\Vert v\right\Vert _{2}=1\\&\left\Vert v\right\Vert _{0}\leq k.\end{aligned}}} Eq. 1 The first constraint specifies that v is a unit vector. In the second constraint, ‖ v ‖ 0 {\displaystyle \left\Vert v\right\Vert _{0}} represents the ℓ 0 {\displaystyle \ell _{0}} pseudo-norm of v, which is defined as the number of its non-zero components. So the second constraint specifies that the number of non-zero components in v is less than or equal to k, which is typically an integer that is much smaller than dimension p. The optimal value of Eq. 1 is known as the k-sparse largest eigenvalue. If one takes k=p, the problem reduces to the ordinary PCA, and the optimal value becomes the largest eigenvalue of covariance matrix Σ. After finding the optimal solution v, one deflates Σ to obtain a new matrix Σ 1 = Σ − ( v T Σ v ) v v T , {\displaystyle \Sigma _{1}=\Sigma -(v^{T}\Sigma v)vv^{T},} and iterate this process to obtain further principal components. However, unlike PCA, sparse PCA cannot guarantee that different principal components are orthogonal. In order to achieve orthogonality, additional constraints must be enforced. The following equivalent definition is in matrix form. Let V {\displaystyle V} be a p×p symmetric matrix, one can rewrite the sparse PCA problem as max T r ( Σ V ) subject to T r ( V ) = 1 ‖ V ‖ 0 ≤ k 2 R a n k ( V ) = 1 , V ⪰ 0. {\displaystyle {\begin{aligned}\max \quad &Tr(\Sigma V)\\{\text{subject to}}\quad &Tr(V)=1\\&\Vert V\Vert _{0}\leq k^{2}\\&Rank(V)=1,V\succeq 0.\end{aligned}}} Eq. 2 Tr is the matrix trace, and ‖ V ‖ 0 {\displaystyle \Vert V\Vert _{0}} represents the non-zero elements in matrix V. The last line specifies that V has matrix rank one and is positive semidefinite. The last line means that one has V = v v T {\displaystyle V=vv^{T}} , so Eq. 2 is equivalent to Eq. 1. Moreover, the rank constraint in this formulation is actually redundant, and therefore sparse PCA can be cast as the following mixed-integer semidefinite program max T r ( Σ V ) subject to T r ( V ) = 1 | V i , i | ≤ z i , ∀ i ∈ { 1 , . . . , p } , | V i , j | ≤ 1 2 z i , ∀ i , j ∈ { 1 , . . . , p } : i ≠ j , V ⪰ 0 , z ∈ { 0 , 1 } p , ∑ i z i ≤ k {\displaystyle {\begin{aligned}\max \quad &Tr(\Sigma V)\\{\text{subject to}}\quad &Tr(V)=1\\&\vert V_{i,i}\vert \leq z_{i},\forall i\in \{1,...,p\},\vert V_{i,j}\vert \leq {\frac {1}{2}}z_{i},\forall i,j\in \{1,...,p\}:i\neq j,\\&V\succeq 0,z\in \{0,1\}^{p},\sum _{i}z_{i}\leq k\end{aligned}}} Eq. 3 Because of the cardinality constraint, the maximization problem is hard to solve exactly, especially when dimension p is high. In fact, the sparse PCA problem in Eq. 1 is NP-hard in the strong sense. == Computational considerations == As most sparse problems, variable selection in SPCA is a computationally intractable non-convex NP-hard problem, therefore greedy sub-optimal algorithms are often employed to find solutions. Note also that SPCA introduces hyperparameters quantifying in what capacity large parameter values are penalized. These might need tuning to achieve satisfactory performance, thereby adding to the total computational cost. == Algorithms for SPCA == Several alternative approaches (of Eq. 1) have been proposed, including a regression framework, a penalized matrix decomposition framework, a convex relaxation/semidefinite programming framework, a generalized power method framework an alternating maximization framework forward-backward greedy search and exact methods using branch-and-bound techniques, a certifiably optimal branch-and-bound approach Bayesian formulation framework. A certifiably optimal mixed-integer semidefinite branch-and-cut approach The methodological and theoretical developments of Sparse PCA as well as its applications in scientific studies are recently reviewed in a survey paper. === Notes on Semidefinite Programming Relaxation === It has been proposed that sparse PCA can be approximated by semidefinite programming (SDP). If one drops the rank constraint and relaxes the cardinality constraint by a 1-norm convex constraint, one gets a semidefinite programming relaxation, which can be solved efficiently in polynomial time: max T r ( Σ V ) subject to T r ( V ) = 1 1 T | V | 1 ≤ k V ⪰ 0. {\displaystyle {\begin{aligned}\max \quad &Tr(\Sigma V)\\{\text{subject to}}\quad &Tr(V)=1\\&\mathbf {1} ^{T}|V|\mathbf {1} \leq k\\&V\succeq 0.\end{aligned}}} Eq. 3 In the second constraint, 1 {\displaystyle \mathbf {1} } is a p×1 vector of ones, and |V| is the matrix whose elements are the absolute values of the elements of V. The optimal solution V {\displaystyle V} to the relaxed problem Eq. 3 is not guaranteed to have rank one. In that case, V {\displaystyle V} can be truncated to retain only the dominant eigenvector. While the semidefinite program does not scale beyond n=300 covariates, it has been shown that a second-order cone relaxation of the semidefinite relaxation is almost as tight and successfully solves problems with n=1000s of covariates == Applications == === Financial Data Analysis === Suppose ordinary PCA is applied to a dataset where each input variable represents a different asset, it may generate principal components that are weighted combination of all the assets. In contrast, sparse PCA would produce principal components that are weighted combination of only a few input assets, so one can easily interpret its meaning. Furthermore, if one uses a trading strategy based on these principal components, fewer assets imply less transaction costs. === Biology === Consider a dataset where each input variable corresponds to a specific gene. Sparse PCA can produce a principal component that involves only a few genes, so researchers can focus on these specific genes for further analysis. === High-dimensional Hypothesis Testing === Contemporary datasets often have the number of input variables ( p {\displaystyle p} ) comparable with or even much larger than the number of samples ( n {\displaystyle n} ). It has been shown that if p / n {\displaystyle p/n} does not converge to zero, the classical PCA is not consistent. In other words, if we let k = p {\displaystyle k=p} in Eq. 1, then the optimal value does not converge to the largest eigenvalue of data population when the sample size n → ∞ {\displaystyle n\rightarrow \infty } , and the optimal solution does not converge to the direction of maximum variance. But sparse PCA can retain consistency even if p ≫ n . {\displaystyle p\gg n.} The k-sparse largest eigenvalue (the optimal value of Eq. 1) can be used to discriminate an isometric model, where every direction has the same variance, from a spiked covariance model in high-dimensional setting. Consider a hypothesis test where the null hypothesis specifies that data X {\displaystyle X} are generated from a multivariate normal distribution with mean 0 and covariance equal to an identity matrix, and the alternative hypothesis specifies that data X {\displaystyle X} is generated from a spiked model with signal strength θ {\displaystyle \theta } : H 0 : X ∼ N ( 0 , I p ) , H 1 : X ∼ N ( 0 , I p + θ v v T ) , {\displaystyle H_{0}:X\sim N(0,I_{p}),\quad H_{1}:X\sim N(0,I_{p}+\theta vv^{T}),} where v ∈ R p {\displaystyle v\in \mathbb {R} ^{p}

  Read more →
• 

  Automated penetration testing

  Automated penetration testing (also known as autonomous penetration testing or automated offensive security) is the application of software-driven workflows and orchestration to simulate cyberattack techniques. These methods are used to identify, validate, and exploit security vulnerabilities in IT assets such as networks, applications, and cloud infrastructure. Automated penetration testing is the use of software to simulate cyberattacks in order to rapidly identify exploitable vulnerabilities across systems without relying solely on human testers. In technical literature, the term describes a spectrum of activities ranging from scripted exploit orchestration to experimental systems designed for fully autonomous attack planning. Automated Penetration Testing falls short of testing using manual experts in terms of discovery of deep complex vulnerabilities and contextual business logic vulnerabilities. == Terminology and scope == The label “automated penetration testing” appears frequently in vendor and practitioner writing but lacks a single, neutral, standards-based definition. In the literature the term’s scope varies: some authors use it to mean automation of specific penetration-testing tasks (scanning, exploitation attempts, evidence collection), others to describe integrated, repeatable assessment pipelines, and a smaller body of work investigates autonomous decision-making agents that select attack steps algorithmically. To avoid implying consensus, this article describes common techniques and architectures reported in the literature and industry, and it notes where claims are primarily found in practitioner publications or early-stage research. Its important to note the differences between automated penetration testing and traditional penetration testing using human skill. The most important difference is scope and speed. Automated penetration testing generally fails at discovering exposures and weakness associated with business logic due to a lack of contextual understanding. The benefit of Automated Penetration testing is speed at which it can be conducted. Traditional penetration testing also is expected to be accurate and contain no false positives. This is due to the human validation aspect of the test. Automated approaches are expected to contain mistakes and false positives which need to be validated upon completion of the test. == History == Automated offensive techniques build on decades of tools and scripting that aided vulnerability discovery and exploitation. Early vulnerability scanners and community scripting in the 1990s and 2000s created the first layers of automation. Later, modular exploitation frameworks (notably Metasploit) integrated scanning and exploitation modules and made automated proof-of-concept attacks more accessible. Over the 2010s–2020s, as cloud platforms, APIs and continuous delivery practices increased the need for frequent validation, academic and industry interest in formalizing automated approaches also grew. == Methodologies and architectures == Descriptions in the literature and technical reports cluster automated capabilities into several overlapping models: Scripted/engineered playbooks (task automation): Predefined workflows or playbooks encode common attack paths (for example, web application exploit sequences or lateral-movement chains). These playbooks are designed to reproduce known techniques in a controlled way to validate exploitability and reduce manual repetition. Exploit-oriented orchestration: Automation orchestrates exploitation modules from established frameworks to perform controlled proof-of-concept attacks that confirm exploitability rather than simply flagging potential weaknesses. This approach can reduce false positives versus passive scanning when tests are run in an appropriately controlled environment. Orchestrated multi-tool pipelines: A coordinated toolchain integrates reconnaissance, vulnerability scanning, credential testing, exploitation modules and reporting. Data and state persist across stages so that multi-step workflows (e.g., discover → escalate → pivot) can be executed repeatably, approximating manual penetration-test methodologies at larger scale. Continuous / CI-integrated testing: Automation embedded in build or deployment pipelines (CI/CD) triggers assessments automatically on new builds, configuration changes, or on a schedule, supporting frequent, repeatable validation aligned with DevOps practices. Academic theses and experimental work describe CI/CD-integrated proof-of-concept systems for web applications and internal networks. Research on autonomous planning and learning: Recent academic work explores machine learning and reinforcement-learning approaches to select or prioritise attack steps, generate attack sequences, or optimize the testing path; these approaches are largely experimental and raise distinct validation and safety questions. == Tools and vendors == Automated penetration testing is provided by a mix of open-source projects, commercial platforms, and professional services. These often follow the penetration testing as a service (PTaaS) model, which integrates automated scanning with manual validation by security analysts. Examples of widely known tools and vendors in the space include exploitation frameworks such as Metasploit, commercial automated platforms and PTaaS providers, and specialist vendors that offer breach-and-attack simulation (BAS) or continuous testing capabilities. == Applications and deployment models == In industry practice, some organizations deploy automated techniques through dedicated security validation platforms rather than bespoke toolchains. These platforms are typically used for continuous or scheduled validation in pre-production or controlled environments and are often positioned alongside, rather than in place of, human-led penetration testing. Examples discussed in secondary literature include platforms such as Pentera, which are commonly classified under breach-and-attack simulation or automated security validation rather than as standalone penetration-testing methodologies.

  Read more →
• 

  Loss function

  In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost" associated with the event. An optimization problem seeks to minimize a loss function. An objective function is either a loss function or its opposite (in specific domains, variously called a reward function, a profit function, a utility function, a fitness function, etc.), in which case it is to be maximized. The loss function could include terms from several levels of the hierarchy. In statistics, typically a loss function is used for parameter estimation, and the event in question is some function of the difference between estimated and true values for an instance of data. The concept, as old as Laplace, was reintroduced in statistics by Abraham Wald in the middle of the 20th century. In the context of economics, for example, this is usually economic cost or regret. In classification, it is the penalty for an incorrect classification of an example. In actuarial science, it is used in an insurance context to model benefits paid over premiums, particularly since the works of Harald Cramér in the 1920s. In optimal control, the loss is the penalty for failing to achieve a desired value. In financial risk management, the function is mapped to a monetary loss. == Examples == === Regret === Leonard J. Savage argued that using non-Bayesian methods such as minimax, the loss function should be based on the idea of regret, i.e., the loss associated with a decision should be the difference between the consequences of the best decision that could have been made under circumstances will be known and the decision that was in fact taken before they were known. === Quadratic loss function === The use of a quadratic loss function is common, for example when using least squares techniques. It is often more mathematically tractable than other loss functions because of the properties of variances, as well as being symmetric: an error above the target causes the same loss as the same magnitude of error below the target. If the target is t {\displaystyle t} , then a quadratic loss function is λ ( x ) = C ( t − x ) 2 {\displaystyle \lambda (x)=C(t-x)^{2}\;} for some constant C {\displaystyle C} ; the value of the constant makes no difference to a decision, and can be ignored by setting it equal to 1. This is also known as the squared error loss (SEL). Many common statistics, including t-tests, regression models, design of experiments, and much else, use least squares methods applied using linear regression theory, which is based on the quadratic loss function. The quadratic loss function is also used in linear-quadratic optimal control problems. In these problems, even in the absence of uncertainty, it may not be possible to achieve the desired values of all target variables. Often loss is expressed as a quadratic form in the deviations of the variables of interest from their desired values; this approach is tractable because it results in linear first-order conditions. In the context of stochastic control, the expected value of the quadratic form is used. The quadratic loss assigns more importance to outliers than to the true data due to its square nature, so alternatives like the Huber, log-cosh and SMAE losses are used when the data has many large outliers. === 0-1 loss function === In statistics and decision theory, a frequently used loss function is the 0-1 loss function L ( y ^ , y ) = { 0 if y = y ^ 1 if y ≠ y ^ {\displaystyle L({\hat {y}},y)={\begin{cases}0&{\text{if }}y={\hat {y}}\\1&{\text{if }}y\neq {\hat {y}}\end{cases}}} In information theory, this loss function is known as Hamming distortion. == Constructing loss and objective functions == In many applications, objective functions, including loss functions as a particular case, are determined by the problem formulation. In other situations, the decision maker’s preference must be elicited and represented by a scalar-valued function (called also utility function) in a form suitable for optimization — the problem that Ragnar Frisch has highlighted in his Nobel Prize lecture. The existing methods for constructing objective functions are collected in the proceedings of two dedicated conferences. In particular, Andranik Tangian showed that the most usable objective functions — quadratic and additive — are determined by a few indifference points. He used this property in the models for constructing these objective functions from either ordinal or cardinal data that were elicited through computer-assisted interviews with decision makers. Among other things, he constructed objective functions to optimally distribute budgets for 16 Westfalian universities and the European subsidies for equalizing unemployment rates among 271 German regions. == Expected loss == In some contexts, the value of the loss function itself is a random quantity because it depends on the outcome of a random variable X {\displaystyle X} . === Statistics === Both frequentist and Bayesian statistical theory involve making a decision based on the expected value of the loss function; however, this quantity is defined differently under the two paradigms. ==== Frequentist expected loss ==== We first define the expected loss in the frequentist context. It is obtained by taking the expected value with respect to the probability distribution, P θ {\displaystyle P_{\theta }} , of the observed data, X {\displaystyle X} . This is also referred to as the risk function of the decision rule δ {\displaystyle \delta } and the parameter θ {\displaystyle \theta } . Here the decision rule depends on the outcome of X {\displaystyle X} . The risk function is given by: R ( θ , δ ) = E θ ⁡ L ( θ , δ ( X ) ) = ∫ X L ( θ , δ ( x ) ) d P θ ( x ) . {\displaystyle R(\theta ,\delta )=\operatorname {E} _{\theta }L{\big (}\theta ,\delta (X){\big )}=\int _{X}L{\big (}\theta ,\delta (x){\big )}\,\mathrm {d} P_{\theta }(x).} Here, θ {\displaystyle \theta } is a fixed but possibly unknown state of nature, X {\displaystyle X} is a vector of observations stochastically drawn from a population, E θ {\displaystyle \operatorname {E} _{\theta }} is the expectation over all population values of X {\displaystyle X} , d P θ {\displaystyle \mathrm {d} P_{\theta }} is a probability measure over the event space of X {\displaystyle X} (parametrized by θ {\displaystyle \theta } ) and the integral is evaluated over the entire support of X {\displaystyle X} . ==== Bayes Risk ==== In a Bayesian approach, the expectation is calculated using the prior distribution π ∗ {\displaystyle \pi ^{}} of the parameter θ {\displaystyle \theta } : ρ ( π ∗ , a ) = ∫ Θ ∫ X L ( θ , a ( x ) ) d P ( x | θ ) d π ∗ ( θ ) = ∫ X ∫ Θ L ( θ , a ( x ) ) d π ∗ ( θ | x ) d M ( x ) {\displaystyle \rho (\pi ^{},a)=\int _{\Theta }\int _{\mathbf {X}}L(\theta ,a({\mathbf {x}}))\,\mathrm {d} P({\mathbf {x}}\vert \theta )\,\mathrm {d} \pi ^{}(\theta )=\int _{\mathbf {X}}\int _{\Theta }L(\theta ,a({\mathbf {x}}))\,\mathrm {d} \pi ^{}(\theta \vert {\mathbf {x}})\,\mathrm {d} M({\mathbf {x}})} where M ( x ) {\displaystyle M(\mathbf {x} )} is known as the predictive likelihood wherein θ {\displaystyle \theta } has been "integrated out," π ∗ ( θ | x ) {\displaystyle \pi ^{}(\theta |\mathbf {x} )} is the posterior distribution, and the order of integration has been changed. One then should choose the action a ∗ {\displaystyle a^{}} which minimises this expected loss, which is referred to as Bayes Risk. In the latter equation, the integrand inside d x {\displaystyle \mathrm {d} x} is known as the Posterior Risk, and minimising it with respect to decision a {\displaystyle a} also minimizes the overall Bayes Risk. This optimal decision, a ∗ {\displaystyle a^{}} is known as the Bayes (decision) Rule - it minimises the average loss over all possible states of nature θ {\displaystyle \theta } , over all possible (probability-weighted) data outcomes. One advantage of the Bayesian approach is to that one need only choose the optimal action under the actual observed data to obtain a uniformly optimal one, whereas choosing the actual frequentist optimal decision rule as a function of all possible observations, is a much more difficult problem. Of equal importance though, the Bayes Rule reflects consideration of loss outcomes under different states of nature, θ {\displaystyle \theta } . ==== Examples in statistics ==== For a scalar parameter θ {\displaystyle \theta } , a decision function whose output θ ^ {\displaystyle {\hat {\theta }}} is an estimate of θ {\displaystyle \theta } , and a quadratic loss function (squared error loss) L ( θ , θ ^ ) = ( θ − θ ^ ) 2 , {\displaystyle L(\theta ,{\hat {\theta }})=(\theta -{\hat {\theta }})^{2},} the risk function becomes the mean squared error of the estimate, R ( θ , θ ^ ) = E θ ⁡ [ ( θ − θ ^ ) 2 ] . {\displaystyle R(\theta ,{\hat {\thet

  Read more →
• 

  Markov blanket

  In statistics and machine learning, a Markov blanket of a random variable is a set of variables that renders the variable conditionally independent of all other variables in the system. This concept is central in probabilistic graphical models and feature selection. If a Markov blanket is minimal—meaning that no variable in it can be removed without losing this conditional independence—it is called a Markov boundary. Identifying a Markov blanket or boundary allows for efficient inference and helps isolate relevant variables for prediction or causal reasoning. The terms Markov blanket and Markov boundary were coined by Judea Pearl in 1988. A Markov blanket may be derived from the structure of a probabilistic graphical model such as a Bayesian network or Markov random field. == Definition == A Markov blanket of a random variable Y {\displaystyle Y} in a random variable set S = { X 1 , … , X n } {\displaystyle {\mathcal {S}}=\{X_{1},\ldots ,X_{n}\}} is any subset S 1 {\displaystyle {\mathcal {S}}_{1}} of S {\displaystyle {\mathcal {S}}} , conditioned on which other variables are independent with Y {\displaystyle Y} : Y ⊥ ⊥ S ∖ S 1 ∣ S 1 {\displaystyle Y\perp \!\!\!\perp {\mathcal {S}}\smallsetminus {\mathcal {S}}_{1}\mid {\mathcal {S}}_{1}} It means that S 1 {\displaystyle {\mathcal {S}}_{1}} contains at least all the information one needs to infer Y {\displaystyle Y} , where the variables in S ∖ S 1 {\displaystyle {\mathcal {S}}\smallsetminus {\mathcal {S}}_{1}} are redundant. In general, a given Markov blanket is not unique. Any set in S {\displaystyle {\mathcal {S}}} that contains a Markov blanket is also a Markov blanket itself. Specifically, S {\displaystyle {\mathcal {S}}} is a Markov blanket of Y {\displaystyle Y} in S {\displaystyle {\mathcal {S}}} . === Example === In a Bayesian network, the Markov blanket of a node consists of its parents, its children, and its children's other parents (i.e., co-parents). Knowing the values of these nodes makes the target node conditionally independent of the rest of the network. In a Markov random field, the Markov blanket of a node is simply its immediate neighbors. == Markov condition == The concept of a Markov blanket is rooted in the Markov condition, which states that in a probabilistic graphical model, each variable is conditionally independent of its non-descendants given its parents. This condition implies the existence of a minimal separating set — the Markov blanket — that shields a variable from the rest of the network. For instance, when a person holds an object stationary against gravity, the object’s acceleration is fully determined by its direct causes—namely, the upward force from the hand and the downward gravitational pull. Other variables such as air pressure or temperature are causally irrelevant. == Markov boundary == A Markov boundary of Y {\displaystyle Y} in S {\displaystyle {\mathcal {S}}} is a subset S 2 {\displaystyle {\mathcal {S}}_{2}} of S {\displaystyle {\mathcal {S}}} , such that S 2 {\displaystyle {\mathcal {S}}_{2}} itself is a Markov blanket of Y {\displaystyle Y} , but any proper subset of S 2 {\displaystyle {\mathcal {S}}_{2}} is not a Markov blanket of Y {\displaystyle Y} . In other words, a Markov boundary is a minimal Markov blanket. The Markov boundary of a node A {\displaystyle A} in a Bayesian network is the set of nodes composed of A {\displaystyle A} 's parents, A {\displaystyle A} 's children, and A {\displaystyle A} 's children's other parents. In a Markov random field, the Markov boundary for a node is the set of its neighboring nodes. In a dependency network, the Markov boundary for a node is the set of its parents. === Uniqueness of Markov boundary === The Markov boundary always exists. Under some mild conditions, the Markov boundary is unique. However, for most practical and theoretical scenarios multiple Markov boundaries may provide alternative solutions. When there are multiple Markov boundaries, quantities measuring causal effect could fail. == In cognitive science == In the study of consciousness, brain function, and complex adaptive systems, Markov blankets are proposed as a mathematical mechanism which delimits the extent of cognitive entities, whether it be physical or causal.

  Read more →