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TowIt

"TowIt" is a free, global, cross-platform mobile app, website, and Web API that allows civilians to report parking violations and dangerous driving in real-time. The mission is to remove the barriers required to make cities effectively fight and deter bad parking and dangerous driving habits. The company ultimately aims to better existing social controls in order to drive necessary behavioral change through increased education, real-time reporting, optimized enforcement, as well as the resulting reactivity. == User base and adoption == The application has users reporting vehicular infractions in upwards of 30 countries. The top reporting countries are: Portugal, Canada, United States of America and Australia. Users have adopted TowIt for a variety of reasons, usually central to their geographical location and the prominent offences in those specific areas. For instance, the majority of Portuguese reports are cars parked on sidewalks, footpaths and pedestrian crossings, Australian reports are largely focused on the abuse of disabled parking spaces, and in Toronto or San Francisco users generally capture cars parked in bicycle lanes. == Functions == === Data collection === TowIt gathers data on individual parking offences, the prominence of various offence types, as well as recurring offenders. This allows the company to identify trends and hotspots in order to take action against problem vehicles, as well as to help improve urban planning, traffic congestion and gridlock management. Individuals modify or improve an aspect of their behavior in response to their awareness of being observed, theoretically more so when demonstrating selfishness, egocentrism, narcissism and anti-social behavior. The company states that by becoming a user, one can "help TowIt relieve congestion, reduce collisions, open up economies, improve the environment and enhance the lives of urban residents and suburban commuters alike". The company has acknowledged that there are numerous legislative changes that would be required to integrate with governments at any level in many countries. A simple three-step process allows users to take a photo of an offending vehicle and subsequently verifying the offending vehicle's license plate information before submitting by tapping the TowIt (submit) button. Photographical evidence can only be captured with the camera from within the TowIt application. An Internet connection is required. The company has stated that this was purposefully done for quality control and report validation purposes. Users may only submit and view their own report history on either the iOS or Android applications. Globally submitted reports are displayed uncensored and in aggregate only on the Android application and the TowIt website. The "Global Feed" feature was removed from iOS (see iTunes Connect Acceptance Issues). TowIt's back-end automatically geotags the report and compares it to local parking by-law data, including by-law types, locations, times, side(s) of street, etc.- where available. Valid reports are posted to the global feed, to the TowIt website, and passed on to municipalities and police for enforcement (where connected). === Technologies used under license === TowIt currently utilizes the following software or software libraries under license: AngularJS, Apache Cordova, Apple iTunes Store EULA, Chart.js, Google Play Distribution Agreement, Ionic Framework, MongoDB, Moment.js, Python 2.7, Python Flask, and jQuery. == Company history == The TowIt application was conceived by Michael Duncan McArthur on December 5, 2014, as a response to Toronto Mayor John Tory's election mandate to "get this city moving". The application was announced via TowIt's official Twitter page on January 6, 2015. After the initial public announcement, Michael & Gregory were contacted by members of John Tory's staff on January 8, 2015, and invited to demo a prototype at Toronto City Hall on January 12, 2015. The two were also invited to meet with Toronto Councillor Norm Kelly, in his City Hall office, for a subsequent demo of the live Android application on January 28, 2015. A similar meeting and demo took place with members of the Traffic Services department of Toronto Police Service on February 2, 2015. Michael & Gregory teamed up with friends and Toronto-based developers Dae-Seon Moon, Jesse Malone, and Marcus Veres to complete the prototype in time to meet the city's imposed demo deadline and to launch the initial Android version of the application. TowIt officially launched on the Android platform on January 16, 2015. A subsequent iOS launch took place on March 19, 2015. === iTunes connect acceptance issues === The iOS version of the application was delayed for approximately two months, only after significant deliberation with Apple's iTunes Connect review board around (as then stated) rule: "14.1 - Any App that is defamatory, offensive, mean-spirited, or likely to place the targeted individual or group in harm's way will be rejected." The result was having to remove the "Global Feed" feature from the iOS platform, in which civilian users could view all recent reports from within the application. This feature still exists on the Android platform. === Business and legal === TowIt engaged Wildeboer Dellelce, one of Canada's leading business law and transactional corporate finance law firms, on January 17, 2015. The company filed for incorporation as "TowIt Solutions Inc." by both Michael & Gregory in the Canadian province of Ontario on January 22, 2015. TowIt continues to operate under a Freemium business model. The company is 100% bootstrapped and has received no outside investment to date. TowIt was accepted into the MaRS Discovery District's Venture Services program on March 4, 2015. === Lobbyist registration === After receiving initial press coverage in January and February 2015, an unknown entity reported Michael & Gregory's initial communications with city staff to the City of Toronto's Lobbyist Registrar. This complaint resulted in legal threats of fines received on February 10, 2015, for apparently and unknowingly breaking municipal lobbying by-laws. These fines (of up to $100,000) were eventually withdrawn after Michael & Gregory immediately provided all records of communication with city officials and registered as lobbyists in the City of Toronto on the subjects of By-law / Regulation, Parking, and Technology. Their registration was accepted by the Lobbyist Registrar on March 6, 2015. However, communication with Toronto city staff was reduced greatly as a result, which the company believes may have been the desired intent of the original complaint. === Outreach and activism === TowIt encourages its global user base to reach out to their local government representatives to promote the app at the users' own will. This tactic is used not only to demonstrate grassroots support, but also to avoid future lobbying issues. On June 2, 2015, the company officially partnered with Australian campaign "No Permit No Park" who advocate for the creation of inclusive communities. == Reception == The Best Planning Apps for 2016 by Planetizen, 5 Toronto apps you should be using by Indie88, 12 Best Apps Made In Canada by TechVibes.
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Margin-infused relaxed algorithm

Margin-infused relaxed algorithm (MIRA) is a machine learning and online algorithm for multiclass classification problems. It is designed to learn a set of parameters (vector or matrix) by processing all the given training examples one-by-one and updating the parameters according to each training example, so that the current training example is classified correctly with a margin against incorrect classifications at least as large as their loss. The change of the parameters is kept as small as possible. A two-class version called binary MIRA simplifies the algorithm by not requiring the solution of a quadratic programming problem (see below). When used in a one-vs-all configuration, binary MIRA can be extended to a multiclass learner that approximates full MIRA, but may be faster to train. The flow of the algorithm looks as follows: The update step is then formalized as a quadratic programming problem: Find m i n ‖ w ( i + 1 ) − w ( i ) ‖ {\displaystyle min\|w^{(i+1)}-w^{(i)}\|} , so that s c o r e ( x t , y t ) − s c o r e ( x t , y ′ ) ≥ L ( y t , y ′ ) ∀ y ′ {\displaystyle score(x_{t},y_{t})-score(x_{t},y')\geq L(y_{t},y')\ \forall y'} , i.e. the score of the current correct training y {\displaystyle y} must be greater than the score of any other possible y ′ {\displaystyle y'} by at least the loss (number of errors) of that y ′ {\displaystyle y'} in comparison to y {\displaystyle y} .
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KNIME

KNIME ( ), the Konstanz Information Miner, is a data analytics, reporting and integrating platform. KNIME integrates various components for machine learning and data mining through its modular data pipelining "Building Blocks of Analytics" concept. A graphical user interface and use of Java Database Connectivity (JDBC) allows assembly of nodes blending different data sources, including preprocessing (extract, transform, load, or ETL), for modeling, data analysis and visualization with minimal, or no, programming. It is free and open-source software released under a GNU General Public License. Since 2006, KNIME has been used in pharmaceutical research, and in other areas including customer relationship management (CRM) and data analysis, business intelligence, text mining and financial data analysis. Recently, attempts were made to use KNIME as robotic process automation (RPA) tool. KNIME's headquarters are based in Zurich, with other offices in Konstanz, Berlin, and Austin (USA). == History == Development of KNIME began in January 2004, with a team of software engineers at the University of Konstanz, as an open-source platform. The original team, headed by Michael Berthold, came from a Silicon Valley pharmaceutical industry software company. The initial goal was to create a modular, highly scalable and open data processing platform that allows easy integration of different data loading, processing, transforming, analyzing, and visual exploring modules, without focus on any one application area. The platform was intended for collaborating, research, and for integrating various other data analysis projects. In 2006, the first version of KNIME was released. Several pharmaceutical companies began using KNIME, and several life science software vendors began integrating their tools into the platform. Later that year, after an article in the German magazine c't, users from a number of other areas joined ship. As of 2012, KNIME is in use by over 15,000 actual users (i.e. not counting downloads, but users regularly retrieving updates) in the life sciences and at banks, publishers, car manufacturer, telcos, consulting firms, and various other industries, and a large number of research groups, worldwide. Latest updates to KNIME Server and KNIME Big Data Extensions, provide support for Apache Spark 2.3, Parquet and HDFS-type storage. For the sixth year in a row, KNIME has been placed as a leader for data science and machine learning platforms in Gartner's Magic Quadrant. == Design philosophy, features == These are the design principles and features that KNIME software follows: Visual, Interactive Framework: KNIME Software prioritizes a user-friendly and intuitive approach to data analysis. This is achieved through a visual and interactive framework where data flows can be combined using a drag-and-drop interface. Users can develop customized and interactive applications by creating simple to advanced and highly-automated data pipelines. These may include, for example, access to databases, machine learning libraries, logic for workflow control (e.g., loops, switches, etc.), abstraction (e.g., interactive widgets), invocation, dynamic data apps, integrated deployment, or error handling. Modularity: processing units and data containers should remain independent of each other. This design choice enables easy distribution of computation and allows for the independent development of different algorithms. Data types within KNIME are encapsulated, meaning no types are predefined. This design choice facilitates adding new data types, and integrating them with extant types, while including type-specific renderers and comparators. This principle also enables inspecting results at the end of each single data operation. Extensibility: KNIME Software is designed to be extensible. Adding new processing nodes or views is made simple through a plug-in mechanism. This mechanism ensures that users can distribute their custom functionalities without the need for complicated install or uninstall procedures. Interleaving No-Code with Code: the platform supports integrating both visual programming (no-code) and script-based programming (e.g., Python, R, JavaScript) approaches to data analysis. This design principle is termed low-code. Automation and Scalability: for example, the use of parameterization via flow variables, or the encapsulation of workflow segments in components contribute to reduce manual work and errors in analyses. Further, the scheduling of workflow execution (available in KNIME Business Hub and KNIME Community Hub for Teams) reduces dependency on human resources. In terms of scalability, a few examples include the ability to handle large datasets (millions of rows), execute multiple processes simultaneously out of the box and reuse workflow segments. Full Usability: due to the open source nature, KNIME Analytics Platform provides free full usability with no limited trial periods. == Internals == KNIME allows users to visually create data flows (or pipelines), selectively execute some or all analysis steps, and later inspect the results, models, using interactive widgets and views. KNIME is written in Java and based on Eclipse. It makes use of an extension mechanism to add plug-ins providing added functions. The core version includes hundreds of modules for data integration (file input/output (I/O), database nodes supporting all common database management systems through JDBC or native connectors: SQLite, MS-Access, SQL Server, MySQL, Oracle, PostgreSQL, Vertica and H2), data transformation (filter, converter, splitter, combiner, joiner), and the commonly used methods of statistics, data mining, analysis and text analytics. Visualization is supported with the Report Designer extension. KNIME workflows can be used as data sets to create report templates that can be exported to document formats such as doc, ppt, xls, pdf and others. Other KNIME abilities are: KNIMEs core-architecture allows processing of large data volumes that are only limited by the available hard disk space (not limited to the available RAM). E.g., KNIME allows analyzing 300 million customer addresses, 20 million cell images, and 10 million molecular structures. Added plug-ins allow integrating methods for text mining, image mining, time series analysis, and networking. KNIME integrates various other open-source projects, e.g., machine learning algorithms from Weka, H2O, Keras, Spark, the R project and LIBSVM; plotly, JFreeChart, ImageJ, and the Chemistry Development Kit. KNIME is implemented in Java, allows for wrappers calling other code, in addition to providing nodes that allow it to run Java, Python, R, Ruby and other code fragments. Since 2021, KNIME's Python Integration utilizes Anaconda for Python distribution and environment management. == License == In 2024, KNIME version 5.3 is released under the same GPLv3 license as previous versions. As of version 2.1, KNIME is released under the GPLv3 license, with an exception that allow commercial software vendors to use the well-defined node application programming interface (API) to add proprietary extensions, or wrappers calling their tools from KNIME. == Courses == KNIME allows the performance of data analysis without programming skills. Several free, online courses are provided.
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KXEN Inc.

KXEN was an American software company which existed from 1998 to 2013 when it was acquired by SAP AG. == History == KXEN was founded in June 1998 by Roger Haddad and Michel Bera. It was based in San Francisco, California with offices in Paris and London. On September 10, 2013, SAP AG announced plans to acquire KXEN. On October 1, 2013, a letter to KXEN customers announced the acquisition closed. KXEN primarily marketed predictive analytics software. == Predictive analytics == InfiniteInsight is a predictive modeling suite developed by KXEN that assists analytic professionals, and business executives to extract information from data. Among other functions, InfiniteInsight is used for variable importance, classification, regression, segmentation, time series, product recommendation, as described and expressed by the Java Data Mining interface, and for social network analysis. InfiniteInsight allows prediction of a behavior or a value, the forecast of a time series or the understanding of a group of individuals with similar behavior. Advanced functions include behavioral modeling, exporting the model code into different target environments or building predictive models on top of SAS or SPSS data files. Competitors are SAS Enterprise Miner, IBM SPSS Modeler, and Statistica. Open source predictive tools like the R package or Weka are also competitors, since they provide similar features free of charge.
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Empowerment (artificial intelligence)

Empowerment in the field of artificial intelligence formalises and quantifies (via information theory) the potential an agent perceives that it has to influence its environment. An agent which follows an empowerment maximising policy, acts to maximise future options (typically up to some limited horizon). Empowerment can be used as a (pseudo) utility function that depends only on information gathered from the local environment to guide action, rather than seeking an externally imposed goal, thus is a form of intrinsic motivation. The empowerment formalism depends on a probabilistic model commonly used in artificial intelligence. An autonomous agent operates in the world by taking in sensory information and acting to change its state, or that of the environment, in a cycle of perceiving and acting known as the perception-action loop. Agent state and actions are modelled by random variables ( S : s ∈ S , A : a ∈ A {\displaystyle S:s\in {\mathcal {S}},A:a\in {\mathcal {A}}} ) and time ( t {\displaystyle t} ). The choice of action depends on the current state, and the future state depends on the choice of action, thus the perception-action loop unrolled in time forms a causal bayesian network. == Definition == Empowerment ( E {\displaystyle {\mathfrak {E}}} ) is defined as the channel capacity ( C {\displaystyle C} ) of the actuation channel of the agent, and is formalised as the maximal possible information flow between the actions of the agent and the effect of those actions some time later. Empowerment can be thought of as the future potential of the agent to affect its environment, as measured by its sensors. E := C ( A t ⟶ S t + 1 ) ≡ max p ( a t ) I ( A t ; S t + 1 ) {\displaystyle {\mathfrak {E}}:=C(A_{t}\longrightarrow S_{t+1})\equiv \max _{p(a_{t})}I(A_{t};S_{t+1})} In a discrete time model, Empowerment can be computed for a given number of cycles into the future, which is referred to in the literature as 'n-step' empowerment. E ( A t n ⟶ S t + n ) = max p ( a t , . . . , a t + n − 1 ) I ( A t , . . . , A t + n − 1 ; S t + n ) {\displaystyle {\mathfrak {E}}(A_{t}^{n}\longrightarrow S_{t+n})=\max _{p(a_{t},...,a_{t+n-1})}I(A_{t},...,A_{t+n-1};S_{t+n})} The unit of empowerment depends on the logarithm base. Base 2 is commonly used in which case the unit is bits. === Contextual Empowerment === In general the choice of action (action distribution) that maximises empowerment varies from state to state. Knowing the empowerment of an agent in a specific state is useful, for example to construct an empowerment maximising policy. State-specific empowerment can be found using the more general formalism for 'contextual empowerment'. C {\displaystyle C} is a random variable describing the context (e.g. state). E ( A t n ⟶ S t + n ∣ C ) = ∑ c ∈ C p ( c ) E ( A t n ⟶ S t + n ∣ C = c ) {\displaystyle {\mathfrak {E}}(A_{t}^{n}\longrightarrow S_{t+n}{\mid }C)=\sum _{c{\in }C}p(c){\mathfrak {E}}(A_{t}^{n}\longrightarrow S_{t+n}{\mid }C=c)} == Application == Empowerment maximisation can be used as a pseudo-utility function to enable agents to exhibit intelligent behaviour without requiring the definition of external goals, for example balancing a pole in a cart-pole balancing scenario where no indication of the task is provided to the agent. Empowerment has been applied in studies of collective behaviour and in continuous domains. As is the case with Bayesian methods in general, computation of empowerment becomes computationally expensive as the number of actions and time horizon extends, but approaches to improve efficiency have led to usage in real-time control. Empowerment has been used for intrinsically motivated reinforcement learning agents playing video games, and in the control of underwater vehicles.
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Sufficient dimension reduction

In statistics, sufficient dimension reduction (SDR) is a paradigm for analyzing data that combines the ideas of dimension reduction with the concept of sufficiency. Dimension reduction has long been a primary goal of regression analysis. Given a response variable y and a p-dimensional predictor vector x {\displaystyle {\textbf {x}}} , regression analysis aims to study the distribution of y ∣ x {\displaystyle y\mid {\textbf {x}}} , the conditional distribution of y {\displaystyle y} given x {\displaystyle {\textbf {x}}} . A dimension reduction is a function R ( x ) {\displaystyle R({\textbf {x}})} that maps x {\displaystyle {\textbf {x}}} to a subset of R k {\displaystyle \mathbb {R} ^{k}} , k < p, thereby reducing the dimension of x {\displaystyle {\textbf {x}}} . For example, R ( x ) {\displaystyle R({\textbf {x}})} may be one or more linear combinations of x {\displaystyle {\textbf {x}}} . A dimension reduction R ( x ) {\displaystyle R({\textbf {x}})} is said to be sufficient if the distribution of y ∣ R ( x ) {\displaystyle y\mid R({\textbf {x}})} is the same as that of y ∣ x {\displaystyle y\mid {\textbf {x}}} . In other words, no information about the regression is lost in reducing the dimension of x {\displaystyle {\textbf {x}}} if the reduction is sufficient. == Graphical motivation == In a regression setting, it is often useful to summarize the distribution of y ∣ x {\displaystyle y\mid {\textbf {x}}} graphically. For instance, one may consider a scatterplot of y {\displaystyle y} versus one or more of the predictors or a linear combination of the predictors. A scatterplot that contains all available regression information is called a sufficient summary plot. When x {\displaystyle {\textbf {x}}} is high-dimensional, particularly when p ≥ 3 {\displaystyle p\geq 3} , it becomes increasingly challenging to construct and visually interpret sufficiency summary plots without reducing the data. Even three-dimensional scatter plots must be viewed via a computer program, and the third dimension can only be visualized by rotating the coordinate axes. However, if there exists a sufficient dimension reduction R ( x ) {\displaystyle R({\textbf {x}})} with small enough dimension, a sufficient summary plot of y {\displaystyle y} versus R ( x ) {\displaystyle R({\textbf {x}})} may be constructed and visually interpreted with relative ease. Hence sufficient dimension reduction allows for graphical intuition about the distribution of y ∣ x {\displaystyle y\mid {\textbf {x}}} , which might not have otherwise been available for high-dimensional data. Most graphical methodology focuses primarily on dimension reduction involving linear combinations of x {\displaystyle {\textbf {x}}} . The rest of this article deals only with such reductions. == Dimension reduction subspace == Suppose R ( x ) = A T x {\displaystyle R({\textbf {x}})=A^{T}{\textbf {x}}} is a sufficient dimension reduction, where A {\displaystyle A} is a p × k {\displaystyle p\times k} matrix with rank k ≤ p {\displaystyle k\leq p} . Then the regression information for y ∣ x {\displaystyle y\mid {\textbf {x}}} can be inferred by studying the distribution of y ∣ A T x {\displaystyle y\mid A^{T}{\textbf {x}}} , and the plot of y {\displaystyle y} versus A T x {\displaystyle A^{T}{\textbf {x}}} is a sufficient summary plot. Without loss of generality, only the space spanned by the columns of A {\displaystyle A} need be considered. Let η {\displaystyle \eta } be a basis for the column space of A {\displaystyle A} , and let the space spanned by η {\displaystyle \eta } be denoted by S ( η ) {\displaystyle {\mathcal {S}}(\eta )} . It follows from the definition of a sufficient dimension reduction that F y ∣ x = F y ∣ η T x , {\displaystyle F_{y\mid x}=F_{y\mid \eta ^{T}x},} where F {\displaystyle F} denotes the appropriate distribution function. Another way to express this property is y ⊥ ⊥ x ∣ η T x , {\displaystyle y\perp \!\!\!\perp {\textbf {x}}\mid \eta ^{T}{\textbf {x}},} or y {\displaystyle y} is conditionally independent of x {\displaystyle {\textbf {x}}} , given η T x {\displaystyle \eta ^{T}{\textbf {x}}} . Then the subspace S ( η ) {\displaystyle {\mathcal {S}}(\eta )} is defined to be a dimension reduction subspace (DRS). === Structural dimensionality === For a regression y ∣ x {\displaystyle y\mid {\textbf {x}}} , the structural dimension, d {\displaystyle d} , is the smallest number of distinct linear combinations of x {\displaystyle {\textbf {x}}} necessary to preserve the conditional distribution of y ∣ x {\displaystyle y\mid {\textbf {x}}} . In other words, the smallest dimension reduction that is still sufficient maps x {\displaystyle {\textbf {x}}} to a subset of R d {\displaystyle \mathbb {R} ^{d}} . The corresponding DRS will be d-dimensional. === Minimum dimension reduction subspace === A subspace S {\displaystyle {\mathcal {S}}} is said to be a minimum DRS for y ∣ x {\displaystyle y\mid {\textbf {x}}} if it is a DRS and its dimension is less than or equal to that of all other DRSs for y ∣ x {\displaystyle y\mid {\textbf {x}}} . A minimum DRS S {\displaystyle {\mathcal {S}}} is not necessarily unique, but its dimension is equal to the structural dimension d {\displaystyle d} of y ∣ x {\displaystyle y\mid {\textbf {x}}} , by definition. If S {\displaystyle {\mathcal {S}}} has basis η {\displaystyle \eta } and is a minimum DRS, then a plot of y versus η T x {\displaystyle \eta ^{T}{\textbf {x}}} is a minimal sufficient summary plot, and it is (d + 1)-dimensional. == Central subspace == If a subspace S {\displaystyle {\mathcal {S}}} is a DRS for y ∣ x {\displaystyle y\mid {\textbf {x}}} , and if S ⊂ S drs {\displaystyle {\mathcal {S}}\subset {\mathcal {S}}_{\text{drs}}} for all other DRSs S drs {\displaystyle {\mathcal {S}}_{\text{drs}}} , then it is a central dimension reduction subspace, or simply a central subspace, and it is denoted by S y ∣ x {\displaystyle {\mathcal {S}}_{y\mid x}} . In other words, a central subspace for y ∣ x {\displaystyle y\mid {\textbf {x}}} exists if and only if the intersection ⋂ S drs {\textstyle \bigcap {\mathcal {S}}_{\text{drs}}} of all dimension reduction subspaces is also a dimension reduction subspace, and that intersection is the central subspace S y ∣ x {\displaystyle {\mathcal {S}}_{y\mid x}} . The central subspace S y ∣ x {\displaystyle {\mathcal {S}}_{y\mid x}} does not necessarily exist because the intersection ⋂ S drs {\textstyle \bigcap {\mathcal {S}}_{\text{drs}}} is not necessarily a DRS. However, if S y ∣ x {\displaystyle {\mathcal {S}}_{y\mid x}} does exist, then it is also the unique minimum dimension reduction subspace. === Existence of the central subspace === While the existence of the central subspace S y ∣ x {\displaystyle {\mathcal {S}}_{y\mid x}} is not guaranteed in every regression situation, there are some rather broad conditions under which its existence follows directly. For example, consider the following proposition from Cook (1998): Let S 1 {\displaystyle {\mathcal {S}}_{1}} and S 2 {\displaystyle {\mathcal {S}}_{2}} be dimension reduction subspaces for y ∣ x {\displaystyle y\mid {\textbf {x}}} . If x {\displaystyle {\textbf {x}}} has density f ( a ) > 0 {\displaystyle f(a)>0} for all a ∈ Ω x {\displaystyle a\in \Omega _{x}} and f ( a ) = 0 {\displaystyle f(a)=0} everywhere else, where Ω x {\displaystyle \Omega _{x}} is convex, then the intersection S 1 ∩ S 2 {\displaystyle {\mathcal {S}}_{1}\cap {\mathcal {S}}_{2}} is also a dimension reduction subspace. It follows from this proposition that the central subspace S y ∣ x {\displaystyle {\mathcal {S}}_{y\mid x}} exists for such x {\displaystyle {\textbf {x}}} . == Methods for dimension reduction == There are many existing methods for dimension reduction, both graphical and numeric. For example, sliced inverse regression (SIR) and sliced average variance estimation (SAVE) were introduced in the 1990s and continue to be widely used. Although SIR was originally designed to estimate an effective dimension reducing subspace, it is now understood that it estimates only the central subspace, which is generally different. More recent methods for dimension reduction include likelihood-based sufficient dimension reduction, estimating the central subspace based on the inverse third moment (or kth moment), estimating the central solution space, graphical regression, envelope model, and the principal support vector machine. For more details on these and other methods, consult the statistical literature. Principal components analysis (PCA) and similar methods for dimension reduction are not based on the sufficiency principle. === Example: linear regression === Consider the regression model y = α + β T x + ε , where ε ⊥ ⊥ x . {\displaystyle y=\alpha +\beta ^{T}{\textbf {x}}+\varepsilon ,{\text{ where }}\varepsilon \perp \!\!\!\perp {\textbf {x}}.} Note that the distribution of y ∣ x {\displaystyle y\mid {\textbf {x}}} is the same as the distribution of y ∣ β T x {\displ
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Count sketch

Count sketch is a type of dimensionality reduction that is particularly efficient in statistics, machine learning and algorithms. It was invented by Moses Charikar, Kevin Chen and Martin Farach-Colton in an effort to speed up the AMS Sketch by Alon, Matias and Szegedy for approximating the frequency moments of streams (these calculations require counting of the number of occurrences for the distinct elements of the stream). The sketch is nearly identical to the Feature hashing algorithm by John Moody, but differs in its use of hash functions with low dependence, which makes it more practical. In order to still have a high probability of success, the median trick is used to aggregate multiple count sketches, rather than the mean. These properties allow use for explicit kernel methods, bilinear pooling in neural networks and is a cornerstone in many numerical linear algebra algorithms. == Intuitive explanation == The inventors of this data structure offer the following iterative explanation of its operation: at the simplest level, the output of a single hash function s mapping stream elements q into {+1, -1} is feeding a single up/down counter C. After a single pass over the data, the frequency n ( q ) {\displaystyle n(q)} of a stream element q can be approximated, although extremely poorly, by the expected value E [ C ⋅ s ( q ) ] {\displaystyle {\mathbf {E}}[C\cdot s(q)]} ; a straightforward way to improve the variance of the previous estimate is to use an array of different hash functions s i {\displaystyle s_{i}} , each connected to its own counter C i {\displaystyle C_{i}} . For each i, the E [ C i ⋅ s i ( q ) ] = n ( q ) {\displaystyle {\mathbf {E}}[C_{i}\cdot s_{i}(q)]=n(q)} still holds, so averaging across the i range will tighten the approximation; the previous construct still has a major deficiency: if a lower-frequency-but-still-important output element a exhibits a hash collision with a high-frequency element even for one of the s i {\displaystyle s_{i}} hashes, n ( a ) {\displaystyle n(a)} estimate can be significantly affected. Avoiding this requires reducing the frequency of collision counter updates between any two distinct elements. This is achieved by replacing each C i {\displaystyle C_{i}} in the previous construct with an array of m counters (making the counter set into a two-dimensional matrix C i , j {\displaystyle C_{i,j}} ), with index j of a particular counter to be incremented/decremented selected via another set of hash functions h i {\displaystyle h_{i}} that map element q into the range {1..m}. Since E [ C i , h i ( q ) ⋅ s i ( q ) ] = n ( q ) {\displaystyle {\mathbf {E}}[C_{i,h_{i}(q)}\cdot s_{i}(q)]=n(q)} , averaging across all values of i will work. == Mathematical definition == 1. For constants w {\displaystyle w} and t {\displaystyle t} (to be defined later) independently choose d = 2 t + 1 {\displaystyle d=2t+1} random hash functions h 1 , … , h d {\displaystyle h_{1},\dots ,h_{d}} and s 1 , … , s d {\displaystyle s_{1},\dots ,s_{d}} such that h i : [ n ] → [ w ] {\displaystyle h_{i}:[n]\to [w]} and s i : [ n ] → { ± 1 } {\displaystyle s_{i}:[n]\to \{\pm 1\}} . It is necessary that the hash families from which h i {\displaystyle h_{i}} and s i {\displaystyle s_{i}} are chosen be pairwise independent. 2. For each item q i {\displaystyle q_{i}} in the stream, add s j ( q i ) {\displaystyle s_{j}(q_{i})} to the h j ( q i ) {\displaystyle h_{j}(q_{i})} th bucket of the j {\displaystyle j} th hash. At the end of this process, one has w d {\displaystyle wd} sums ( C i j ) {\displaystyle (C_{ij})} where C i , j = ∑ h i ( k ) = j s i ( k ) . {\displaystyle C_{i,j}=\sum _{h_{i}(k)=j}s_{i}(k).} To estimate the count of q {\displaystyle q} s one computes the following value: r q = median i = 1 d s i ( q ) ⋅ C i , h i ( q ) . {\displaystyle r_{q}={\text{median}}_{i=1}^{d}\,s_{i}(q)\cdot C_{i,h_{i}(q)}.} The values s i ( q ) ⋅ C i , h i ( q ) {\displaystyle s_{i}(q)\cdot C_{i,h_{i}(q)}} are unbiased estimates of how many times q {\displaystyle q} has appeared in the stream. The estimate r q {\displaystyle r_{q}} has variance O ( m i n { m 1 2 / w 2 , m 2 2 / w } ) {\displaystyle O(\mathrm {min} \{m_{1}^{2}/w^{2},m_{2}^{2}/w\})} , where m 1 {\displaystyle m_{1}} is the length of the stream and m 2 2 {\displaystyle m_{2}^{2}} is ∑ q ( ∑ i [ q i = q ] ) 2 {\displaystyle \sum _{q}(\sum _{i}[q_{i}=q])^{2}} . Furthermore, r q {\displaystyle r_{q}} is guaranteed to never be more than 2 m 2 / w {\displaystyle 2m_{2}/{\sqrt {w}}} off from the true value, with probability 1 − e − O ( t ) {\displaystyle 1-e^{-O(t)}} . === Vector formulation === Alternatively Count-Sketch can be seen as a linear mapping with a non-linear reconstruction function. Let M ( i ∈ [ d ] ) ∈ { − 1 , 0 , 1 } w × n {\displaystyle M^{(i\in [d])}\in \{-1,0,1\}^{w\times n}} , be a collection of d = 2 t + 1 {\displaystyle d=2t+1} matrices, defined by M h i ( j ) , j ( i ) = s i ( j ) {\displaystyle M_{h_{i}(j),j}^{(i)}=s_{i}(j)} for j ∈ [ w ] {\displaystyle j\in [w]} and 0 everywhere else. Then a vector v ∈ R n {\displaystyle v\in \mathbb {R} ^{n}} is sketched by C ( i ) = M ( i ) v ∈ R w {\displaystyle C^{(i)}=M^{(i)}v\in \mathbb {R} ^{w}} . To reconstruct v {\displaystyle v} we take v j ∗ = median i C j ( i ) s i ( j ) {\displaystyle v_{j}^{}={\text{median}}_{i}C_{j}^{(i)}s_{i}(j)} . This gives the same guarantees as stated above, if we take m 1 = ‖ v ‖ 1 {\displaystyle m_{1}=\|v\|_{1}} and m 2 = ‖ v ‖ 2 {\displaystyle m_{2}=\|v\|_{2}} . == Relation to Tensor sketch == The count sketch projection of the outer product of two vectors is equivalent to the convolution of two component count sketches. The count sketch computes a vector convolution C ( 1 ) x ∗ C ( 2 ) x T {\displaystyle C^{(1)}x\ast C^{(2)}x^{T}} , where C ( 1 ) {\displaystyle C^{(1)}} and C ( 2 ) {\displaystyle C^{(2)}} are independent count sketch matrices. Pham and Pagh show that this equals C ( x ⊗ x T ) {\displaystyle C(x\otimes x^{T})} – a count sketch C {\displaystyle C} of the outer product of vectors, where ⊗ {\displaystyle \otimes } denotes Kronecker product. The fast Fourier transform can be used to do fast convolution of count sketches. By using the face-splitting product such structures can be computed much faster than normal matrices.
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Dispersive flies optimisation

Dispersive flies optimisation (DFO) is a bare-bones swarm intelligence algorithm which is inspired by the swarming behaviour of flies hovering over food sources. DFO is a simple optimiser which works by iteratively trying to improve a candidate solution with regard to a numerical measure that is calculated by a fitness function. Each member of the population, a fly or an agent, holds a candidate solution whose suitability can be evaluated by their fitness value. Optimisation problems are often formulated as either minimisation or maximisation problems. DFO was introduced with the intention of analysing a simplified swarm intelligence algorithm with the fewest tunable parameters and components. In the first work on DFO, this algorithm was compared against a few other existing swarm intelligence techniques using error, efficiency and diversity measures. It is shown that despite the simplicity of the algorithm, which only uses agents’ position vectors at time t to generate the position vectors for time t + 1, it exhibits a competitive performance. Since its inception, DFO has been used in a variety of applications including medical imaging and image analysis as well as data mining and machine learning. == Algorithm == DFO bears many similarities with other existing continuous, population-based optimisers (e.g. particle swarm optimization and differential evolution). In that, the swarming behaviour of the individuals consists of two tightly connected mechanisms, one is the formation of the swarm and the other is its breaking or weakening. DFO works by facilitating the information exchange between the members of the population (the swarming flies). Each fly x {\displaystyle \mathbf {x} } represents a position in a d-dimensional search space: x = ( x 1 , x 2 , … , x d ) {\displaystyle \mathbf {x} =(x_{1},x_{2},\ldots ,x_{d})} , and the fitness of each fly is calculated by the fitness function f ( x ) {\displaystyle f(\mathbf {x} )} , which takes into account the flies' d dimensions: f ( x ) = f ( x 1 , x 2 , … , x d ) {\displaystyle f(\mathbf {x} )=f(x_{1},x_{2},\ldots ,x_{d})} . The pseudocode below represents one iteration of the algorithm: for i = 1 : N flies x i . fitness = f ( x i ) {\displaystyle \mathbf {x_{i}} .{\text{fitness}}=f(\mathbf {x} _{i})} end for i x s {\displaystyle \mathbf {x} _{s}} = arg min [ f ( x i ) ] , i ∈ { 1 , … , N } {\textstyle [f(\mathbf {x} _{i})],\;i\in \{1,\ldots ,N\}} for i = 1 : N and i ≠ s {\displaystyle i\neq s} for d = 1 : D dimensions if U ( 0 , 1 ) < Δ {\displaystyle U(0,1)<\Delta } x i d t + 1 = U ( x min , d , x max , d ) {\displaystyle x_{id}^{t+1}=U(x_{\min ,d},x_{\max ,d})} else x i d t + 1 = x i n d t + U ( 0 , 1 ) ( x s d t − x i d t ) {\displaystyle x_{id}^{t+1}=x_{i_{nd}}^{t}+U(0,1)(x_{sd}^{t}-x_{id}^{t})} end if end for d end for i In the algorithm above, x i d t + 1 {\displaystyle x_{id}^{t+1}} represents fly i {\displaystyle i} at dimension d {\displaystyle d} and time t + 1 {\displaystyle t+1} ; x i n d t {\displaystyle x_{i_{nd}}^{t}} presents x i {\displaystyle x_{i}} 's best neighbouring fly in ring topology (left or right, using flies indexes), at dimension d {\displaystyle d} and time t {\displaystyle t} ; and x s d t {\displaystyle x_{sd}^{t}} is the swarm's best fly. Using this update equation, the swarm's population update depends on each fly's best neighbour (which is used as the focus μ {\displaystyle \mu } , and the difference between the current fly and the best in swarm represents the spread of movement, σ {\displaystyle \sigma } ). Other than the population size N {\displaystyle N} , the only tunable parameter is the disturbance threshold Δ {\displaystyle \Delta } , which controls the dimension-wise restart in each fly vector. This mechanism is proposed to control the diversity of the swarm. Other notable minimalist swarm algorithm is Bare bones particle swarms (BB-PSO), which is based on particle swarm optimisation, along with bare bones differential evolution (BBDE) which is a hybrid of the bare bones particle swarm optimiser and differential evolution, aiming to reduce the number of parameters. Alhakbani in her PhD thesis covers many aspects of the algorithms including several DFO applications in feature selection as well as parameter tuning. == Applications == Some of the recent applications of DFO are listed below: Optimising support vector machine kernel to classify imbalanced data Quantifying symmetrical complexity in computational aesthetics Analysing computational autopoiesis and computational creativity Identifying calcifications in medical images Building non-identical organic structures for game's space development Deep Neuroevolution: Training Deep Neural Networks for False Alarm Detection in Intensive Care Units Identification of animation key points from 2D-medialness maps
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Language Server Protocol

The Language Server Protocol (LSP) is an open, JSON-RPC-based protocol for use between source-code editors or integrated development environments (IDEs) and servers that provide "language intelligence tools": programming language-specific features like code completion, syntax highlighting and marking of warnings and errors, as well as refactoring routines. The goal of the protocol is to allow programming language support to be implemented and distributed independently of any given editor or IDE. In the early 2020s, LSP quickly became a "norm" for language intelligence tools providers. == History == LSP was originally developed for Microsoft Visual Studio Code and is now an open standard. On June 27, 2016, Microsoft announced a collaboration with Red Hat and Codenvy to standardize the protocol's specification. Its specification is hosted and developed on GitHub. == Background == Modern IDEs provide programmers with sophisticated features like code completion, refactoring, navigating to a symbol's definition, syntax highlighting, and error and warning markers. For example, in a text-based programming language, a programmer might want to rename a method read. The programmer could either manually edit the respective source code files and change the appropriate occurrences of the old method name into the new name, or instead use an IDE's refactoring capabilities to make all the necessary changes automatically. To be able to support this style of refactoring, an IDE needs a sophisticated understanding of the programming language that the program's source is written in. A programming tool without such an understanding—for example, one that performs a naive search-and-replace instead—could introduce errors. When renaming a read method, for example, the tool should not replace the partial match in a variable that might be called readyState, nor should it replace the portion of a code comment containing the word "already". Neither should renaming a local variable read, for example, end up altering identically-named variables in other scopes. Conventional compilers or interpreters for a specific programming language are typically unable to provide these language services, because they are written with the goal of either transforming the source code into object code or immediately executing the code. Additionally, language services must be able to handle source code that is not well-formed, e.g. because the programmer is in the middle of editing and has not yet finished typing a statement, procedure, or other construct. Additionally, small changes to a source code file which are done during typing usually change the semantics of the program. In order to provide instant feedback to the user, the editing tool must be able to very quickly evaluate the syntactical and semantical consequences of a specific modification. Compilers and interpreters therefore provide a poor candidate for producing the information needed for an editing tool to consume. Prior to the design and implementation of the Language Server Protocol for the development of Visual Studio Code, most language services were generally tied to a given IDE or other editor. In the absence of the Language Server Protocol, language services are typically implemented by using a tool-specific extension API. Providing the same language service to another editing tool requires effort to adapt the existing code so that the service may target the second editor's extension interfaces. The Language Server Protocol allows for decoupling language services from the editor so that the services may be contained within a general-purpose language server. Any editor can inherit sophisticated support for many different languages by making use of existing language servers. Similarly, a programmer involved with the development of a new programming language can make services for that language available to existing editing tools. Making use of language servers via the Language Server Protocol thus also reduces the burden on vendors of editing tools, because vendors do not need to develop language services of their own for the languages the vendor intends to support, as long as the language servers have already been implemented. The Language Server Protocol also enables the distribution and development of servers contributed by an interested third party, such as end users, without additional involvement by either the vendor of the compiler for the programming language in use or the vendor of the editor to which the language support is being added. LSP is not restricted to programming languages. It can be used for any kind of text-based language, like specifications or domain-specific languages (DSL). == Technical overview == When a user edits one or more source code files using a language server protocol-enabled tool, the tool acts as a client that consumes the language services provided by a language server. The tool may be a text editor or IDE and the language services could be refactoring, code completion, etc. The client informs the server about what the user is doing, e.g., opening a file or inserting a character at a specific text position. The client can also request the server to perform a language service, e.g. to format a specified range in the text document. The server answers a client's request with an appropriate response. For example, the formatting request is answered either by a response that transfers the formatted text to the client or by an error response containing details about the error. The Language Server Protocol defines the messages to be exchanged between client and language server. They are JSON-RPC preceded by headers similar to HTTP. Messages may originate from the server or client. The protocol does not make any provisions about how requests, responses and notifications are transferred between client and server. For example, client and server could be components within the same process exchanging JSON strings via method calls. They could also be different processes on the same or on different machines communicating via network sockets. == Registry == There are lists of LSP-compatible implementations, maintained by the community-driven Langserver.org or Microsoft.
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Gaussian process emulator

In statistics, Gaussian process emulator is one name for a general type of statistical model that has been used in contexts where the problem is to make maximum use of the outputs of a complicated (often non-random) computer-based simulation model. Each run of the simulation model is computationally expensive and each run is based on many different controlling inputs. The variation of the outputs of the simulation model is expected to vary reasonably smoothly with the inputs, but in an unknown way. The overall analysis involves two models: the simulation model, or "simulator", and the statistical model, or "emulator", which notionally emulates the unknown outputs from the simulator. The Gaussian process emulator model treats the problem from the viewpoint of Bayesian statistics. In this approach, even though the output of the simulation model is fixed for any given set of inputs, the actual outputs are unknown unless the computer model is run and hence can be made the subject of a Bayesian analysis. The main element of the Gaussian process emulator model is that it models the outputs as a Gaussian process on a space that is defined by the model inputs. The model includes a description of the correlation or covariance of the outputs, which enables the model to encompass the idea that differences in the output will be small if there are only small differences in the inputs.
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Cellular evolutionary algorithm

A cellular evolutionary algorithm (cEA) is a kind of evolutionary algorithm (EA) in which individuals cannot mate arbitrarily, but every one interacts with its closer neighbors on which a basic EA is applied (selection, variation, replacement). The cellular model simulates natural evolution from the point of view of the individual, which encodes a tentative optimization, learning, or search problem solution. The essential idea of this model is to provide the EA population with a special structure defined as a connected graph, in which each vertex is an individual who communicates with his nearest neighbors. Particularly, individuals are conceptually set in a toroidal mesh, and are only allowed to recombine with close individuals. This leads to a kind of locality known as "isolation by distance". The set of potential mates of an individual is called its "neighborhood". It is known that, in this kind of algorithm, similar individuals tend to cluster creating niches, and these groups operate as if they were separate sub-populations (islands). There is no clear borderline between adjacent groups, and close niches could be easily colonized by competitive niches and potentially merge solution contents during the process. Simultaneously, farther niches can be affected more slowly. == Introduction == A cellular evolutionary algorithm (cEA) usually evolves a structured bidimensional grid of individuals, although other topologies are also possible. In this grid, clusters of similar individuals are naturally created during evolution, promoting exploration in their boundaries, while exploitation is mainly performed by direct competition and merging inside them. The grid is usually 2D toroidal structure, although the number of dimensions can be easily extended (to 3D) or reduced (to 1D, e.g. a ring). The neighborhood of a particular point of the grid (where an individual is placed) is defined in terms of the Manhattan distance from it to others in the population. Each point of the grid has a neighborhood that overlaps the neighborhoods of nearby individuals. In the basic algorithm, all the neighborhoods have the same size and identical shapes. The two most commonly used neighborhoods are L5, also called the Von Neumann or NEWS (North, East, West and South) neighborhood, and C9, also known as the Moore neighborhood. Here, L stands for "linear" while C stands for "compact". In cEAs, the individuals can only interact with their neighbors in the reproductive cycle where the variation operators are applied. This reproductive cycle is executed inside the neighborhood of each individual and, generally, consists in selecting two parents among its neighbors according to a certain criterion, applying the variation operators to them (recombination and mutation for example), and replacing the considered individual by the recently created offspring following a given criterion, for instance, replace if the offspring represents a better solution than the considered individual. == Synchronous versus asynchronous == In a regular synchronous cEA, the algorithm proceeds from the very first top left individual to the right and then to the several rows by using the information in the population to create a new temporary population. After finishing with the bottom-right last individual the temporary population is full with the newly computed individuals, and the replacement step starts. In it, the old population is completely and synchronously replaced with the newly computed one according to some criterion. Usually, the replacement keeps the best individual in the same position of both populations, that is, elitism is used. According to the update policy of the population used, an asynchronous cEA may also be defined and is a well-known issue in cellular automata. In asynchronous cEAs the order in which the individuals in the grid are update changes depending on the choice of criterion: line sweep, fixed random sweep, new random sweep, and uniform choice. All four proceed using the newly computed individual (or the original if better) for the computations of its neighbors. The overlap of the neighborhoods provides an implicit mechanism of solution migration to the cEA. Since the best solutions spread smoothly through the whole population, genetic diversity in the population is preserved longer than in non structured EAs. This soft dispersion of the best solutions through the population is one of the main issues of the good tradeoff between exploration and exploitation that cEAs perform during the search. This tradeoff can be tuned (and by extension the genetic diversity level along the evolution) by modifying (for instance) the size of the neighborhood used, as the overlap degree between the neighborhoods grows according to the size of the neighborhood. A cEA can be seen as a cellular automaton (CA) with probabilistic rewritable rules, where the alphabet of the CA is equivalent to the potential number of solutions of the problem. Hence, knowledge from research in CAs can be applied to cEAs. == Parallelism == Cellular EAs are very amenable to parallelism, thus usually found in the literature of parallel metaheuristics. In particular, fine grain parallelism can be used to assign independent threads of execution to every individual, thus allowing the whole cEA to run on a concurrent or actually parallel hardware platform. In this way, large time reductions can be obtained when running cEAs on FPGAs or GPUs. However, it is important to stress that cEAs are a model of search, in many senses different from traditional EAs. Also, they can be run in sequential and parallel platforms, reinforcing the fact that the model and the implementation are two different concepts. See here for a complete description on the fundamentals for the understanding, design, and application of cEAs.
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Ni1000

The Ni1000 is an artificial neural network chip developed by Nestor Corporation and Intel, developed in the 1990s. It is Intel's second-generation neural network chip, but the first all-digital chip. The chip is aimed at image analysis applications– containing more than 3 million transistors – and can analyze 40,000 patterns per second. Prototypes running Nestor's OCR software in 1994 were capable of recognizing around 100 handwritten characters per second. The development was funded with money from DARPA and Office of Naval Research.
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Hallucination (artificial intelligence)

In the field of artificial intelligence (AI), a hallucination or artificial hallucination (also called bullshitting, confabulation, or delusion) is a response generated by AI that contains false or misleading information presented as fact. This term draws a loose analogy with human psychology, where a hallucination typically involves false percepts. For example, a chatbot powered by large language models (LLMs), like ChatGPT, may embed plausible-sounding random falsehoods within its generated content. Detecting and mitigating errors and hallucinations pose significant challenges for practical deployment and reliability of LLMs in high-stakes scenarios, such as chip design, supply chain logistics, and medical diagnostics. Some software engineers and statisticians have criticized the specific term "AI hallucination" for unreasonably anthropomorphizing computers. Symbolic artificial intelligence models generally do not produce hallucinations, unlike large language models. == Term == === Origin === Since the 1980s, the term "hallucination" has been used in computer vision with a positive connotation to describe the process of adding detail to an image. For example, the task of generating high-resolution face images from low-resolution inputs is called face hallucination. The first documented use of the term "hallucination" in this sense is in the PhD thesis of Eric Mjolsness in 1986. A notable work is the face hallucination algorithm by Simon Baker and Takeo Kanade published in 1999. In the 2000s, hallucinations were described in statistical machine translation as a failure mode. Since the 2010s, the term has undergone a semantic shift to signify the generation of factually incorrect or misleading outputs by AI systems in tasks like machine translation and object detection. In 2015, hallucinations were identified in visual semantic role labeling tasks by Saurabh Gupta and Jitendra Malik. In 2015, computer scientist Andrej Karpathy used the term "hallucinated" in a blog post to describe his recurrent neural network (RNN) language model generating an incorrect citation link. In 2017, Google researchers used the term to describe the responses generated by neural machine translation (NMT) models when they are not related to the source text, and in 2018, the term was used in computer vision to describe instances where non-existent objects are erroneously detected because of adversarial attacks. In July 2021, Meta warned during its release of BlenderBot 2 that the system is prone to "hallucinations", which Meta defined as "confident statements that are not true". Following OpenAI's ChatGPT release in beta version in November 2022, some users complained that such chatbots often seem to pointlessly embed plausible-sounding random falsehoods within their generated content. Many news outlets, including The New York Times, started to use the term "hallucinations" to describe these models' frequently incorrect or inconsistent responses. In 2023, the Cambridge dictionary updated its definition of hallucination to include this new sense specific to the field of AI. Some researchers have highlighted a lack of consistency in how the term is used, but also identified several alternative terms in the literature, such as confabulations, fabrications, and factual errors. === Definitions and alternatives === Uses, definitions and characterizations of the term "hallucination" in the context of LLMs include: "a tendency to invent facts in moments of uncertainty" (OpenAI, May 2023) "a model's logical mistakes" (OpenAI, May 2023) "fabricating information entirely, but behaving as if spouting facts" (CNBC, May 2023) "making up information" (The Verge, February 2023) "probability distributions" (in scientific contexts) Journalist Benj Edwards, in Ars Technica, writes that the term "hallucination" is controversial, but that some form of metaphor remains necessary; Edwards suggests "confabulation" as an analogy for processes that involve "creative gap-filling". In July 2024, a White House report on fostering public trust in AI research mentioned hallucinations only in the context of reducing them. Notably, when acknowledging David Baker's Nobel Prize-winning work with AI-generated proteins, the Nobel committee avoided the term entirely, instead referring to "imaginative protein creation". Hicks, Humphries, and Slater, in their article in Ethics and Information Technology, argue that the output of LLMs is "bullshit" under Harry Frankfurt's definition of the term, and that the models are "in an important way indifferent to the truth of their outputs", with true statements only accidentally true, and false ones accidentally false. Some researchers also use the derogatory term "botshit", often referring to uncritical use of AI. === Criticism === In the scientific community, some researchers avoid the term "hallucination", seeing it as potentially misleading. It has been criticized by Usama Fayyad, executive director of the Institute for Experimental Artificial Intelligence at Northeastern University, on the grounds that it misleadingly personifies large language models and is vague. Mary Shaw said, "The current fashion for calling generative AI's errors 'hallucinations' is appalling. It anthropomorphizes the software, and it spins actual errors as somehow being idiosyncratic quirks of the system even when they're objectively incorrect." In Salon, statistician Gary Smith argues that LLMs "do not understand what words mean" and consequently that the term "hallucination" unreasonably anthropomorphizes the machine. Murray Shanahan argues that anthropomorphic framing of LLM capabilities, including terms like "hallucination", encourages users and researchers to attribute cognitive processes to systems that operate through statistical pattern completion, and advocates for more careful linguistic practices when discussing LLM behavior. Kristina Šekrst argues that applying psychological vocabulary to LLM outputs obscures the difference between the appearance of mental properties and their genuine presence. Förster & Skop assert that tech companies use the hallucination metaphor to anthropomorphize models and deflect responsibility for non-factual outputs. Some see the AI outputs not as illusory but as prospective—that is, having some chance of being true, similar to early-stage scientific conjectures. The term has also been criticized for its association with psychedelic drug experiences. == In natural language generation == In natural language generation, there are several reasons why natural language models hallucinate: === Hallucination from data === Hallucinations can stem from incomplete, inaccurate or unrepresentative data sets. === Modeling-related causes === The pre-training of generative pretrained transformers (GPT) involves predicting the next word. It incentivizes GPT models to "give a guess" about what the next word is, even when they lack information. Some researchers take an anthropomorphic perspective and posit that hallucinations arise from a tension between novelty and usefulness. For instance, Amabile and Pratt define human creativity as the production of novel and useful ideas. By extension, a focus on novelty in machine creativity can lead to the production of original but inaccurate responses—that is, falsehoods—whereas a focus on usefulness may result in memorized content lacking originality. By 2022, newspapers such as The New York Times expressed concern that, as the adoption of bots based on large language models continued to grow, unwarranted user confidence in bot output could lead to problems. === Interpretability research === In 2025, interpretability research by Anthropic on the LLM Claude identified internal circuits that cause it to decline to answer questions unless it knows the answer. By default, the circuit is active and the LLM doesn't answer. When the LLM has sufficient information, these circuits are inhibited and the LLM answers the question. Hallucinations were found to occur when this inhibition happens incorrectly, such as when Claude recognizes a name but lacks sufficient information about that person, causing it to generate plausible but untrue responses. === Examples === On 15 November 2022, researchers from Meta AI published Galactica, designed to "store, combine and reason about scientific knowledge". Content generated by Galactica came with the warning: "Outputs may be unreliable! Language Models are prone to hallucinate text." In one case, when asked to draft a paper on creating avatars, Galactica cited a fictitious paper from a real author who works in the relevant area. Meta withdrew Galactica on 17 November due to offensiveness and inaccuracy. OpenAI's ChatGPT, released in beta version to the public on November 30, 2022, was based on the foundation model GPT-3.5 (a revision of GPT-3). Professor Ethan Mollick of Wharton called it an "omniscient, eager-to-please intern who sometimes lies to you". Data scientist Teresa Kuba
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LPBoost

Linear Programming Boosting (LPBoost) is a supervised classifier from the boosting family of classifiers. LPBoost maximizes a margin between training samples of different classes, and thus also belongs to the class of margin classifier algorithms. Consider a classification function f : X → { − 1 , 1 } , {\displaystyle f:{\mathcal {X}}\to \{-1,1\},} which classifies samples from a space X {\displaystyle {\mathcal {X}}} into one of two classes, labelled 1 and -1, respectively. LPBoost is an algorithm for learning such a classification function, given a set of training examples with known class labels. LPBoost is a machine learning technique especially suited for joint classification and feature selection in structured domains. == LPBoost overview == As in all boosting classifiers, the final classification function is of the form f ( x ) = ∑ j = 1 J α j h j ( x ) , {\displaystyle f({\boldsymbol {x}})=\sum _{j=1}^{J}\alpha _{j}h_{j}({\boldsymbol {x}}),} where α j {\displaystyle \alpha _{j}} are non-negative weightings for weak classifiers h j : X → { − 1 , 1 } {\displaystyle h_{j}:{\mathcal {X}}\to \{-1,1\}} . Each individual weak classifier h j {\displaystyle h_{j}} may be just a little bit better than random, but the resulting linear combination of many weak classifiers can perform very well. LPBoost constructs f {\displaystyle f} by starting with an empty set of weak classifiers. Iteratively, a single weak classifier to add to the set of considered weak classifiers is selected, added and all the weights α {\displaystyle {\boldsymbol {\alpha }}} for the current set of weak classifiers are adjusted. This is repeated until no weak classifiers to add remain. The property that all classifier weights are adjusted in each iteration is known as totally-corrective property. Early boosting methods, such as AdaBoost do not have this property and converge slower. == Linear program == More generally, let H = { h ( ⋅ ; ω ) | ω ∈ Ω } {\displaystyle {\mathcal {H}}=\{h(\cdot ;\omega )|\omega \in \Omega \}} be the possibly infinite set of weak classifiers, also termed hypotheses. One way to write down the problem LPBoost solves is as a linear program with infinitely many variables. The primal linear program of LPBoost, optimizing over the non-negative weight vector α {\displaystyle {\boldsymbol {\alpha }}} , the non-negative vector ξ {\displaystyle {\boldsymbol {\xi }}} of slack variables and the margin ρ {\displaystyle \rho } is the following. min α , ξ , ρ − ρ + D ∑ n = 1 ℓ ξ n sb.t. ∑ ω ∈ Ω y n α ω h ( x n ; ω ) + ξ n ≥ ρ , n = 1 , … , ℓ , ∑ ω ∈ Ω α ω = 1 , ξ n ≥ 0 , n = 1 , … , ℓ , α ω ≥ 0 , ω ∈ Ω , ρ ∈ R . {\displaystyle {\begin{array}{cl}{\underset {{\boldsymbol {\alpha }},{\boldsymbol {\xi }},\rho }{\min }}&-\rho +D\sum _{n=1}^{\ell }\xi _{n}\\{\textrm {sb.t.}}&\sum _{\omega \in \Omega }y_{n}\alpha _{\omega }h({\boldsymbol {x}}_{n};\omega )+\xi _{n}\geq \rho ,\qquad n=1,\dots ,\ell ,\\&\sum _{\omega \in \Omega }\alpha _{\omega }=1,\\&\xi _{n}\geq 0,\qquad n=1,\dots ,\ell ,\\&\alpha _{\omega }\geq 0,\qquad \omega \in \Omega ,\\&\rho \in {\mathbb {R} }.\end{array}}} Note the effects of slack variables ξ ≥ 0 {\displaystyle {\boldsymbol {\xi }}\geq 0} : their one-norm is penalized in the objective function by a constant factor D {\displaystyle D} , which—if small enough—always leads to a primal feasible linear program. Here we adopted the notation of a parameter space Ω {\displaystyle \Omega } , such that for a choice ω ∈ Ω {\displaystyle \omega \in \Omega } the weak classifier h ( ⋅ ; ω ) : X → { − 1 , 1 } {\displaystyle h(\cdot ;\omega ):{\mathcal {X}}\to \{-1,1\}} is uniquely defined. When the above linear program was first written down in early publications about boosting methods it was disregarded as intractable due to the large number of variables α {\displaystyle {\boldsymbol {\alpha }}} . Only later it was discovered that such linear programs can indeed be solved efficiently using the classic technique of column generation. === Column generation for LPBoost === In a linear program a column corresponds to a primal variable. Column generation is a technique to solve large linear programs. It typically works in a restricted problem, dealing only with a subset of variables. By generating primal variables iteratively and on-demand, eventually the original unrestricted problem with all variables is recovered. By cleverly choosing the columns to generate the problem can be solved such that while still guaranteeing the obtained solution to be optimal for the original full problem, only a small fraction of columns has to be created. ==== LPBoost dual problem ==== Columns in the primal linear program corresponds to rows in the dual linear program. The equivalent dual linear program of LPBoost is the following linear program. max λ , γ γ sb.t. ∑ n = 1 ℓ y n h ( x n ; ω ) λ n + γ ≤ 0 , ω ∈ Ω , 0 ≤ λ n ≤ D , n = 1 , … , ℓ , ∑ n = 1 ℓ λ n = 1 , γ ∈ R . {\displaystyle {\begin{array}{cl}{\underset {{\boldsymbol {\lambda }},\gamma }{\max }}&\gamma \\{\textrm {sb.t.}}&\sum _{n=1}^{\ell }y_{n}h({\boldsymbol {x}}_{n};\omega )\lambda _{n}+\gamma \leq 0,\qquad \omega \in \Omega ,\\&0\leq \lambda _{n}\leq D,\qquad n=1,\dots ,\ell ,\\&\sum _{n=1}^{\ell }\lambda _{n}=1,\\&\gamma \in \mathbb {R} .\end{array}}} For linear programs the optimal value of the primal and dual problem are equal. For the above primal and dual problems, the optimal value is equal to the negative 'soft margin'. The soft margin is the size of the margin separating positive from negative training instances minus positive slack variables that carry penalties for margin-violating samples. Thus, the soft margin may be positive although not all samples are linearly separated by the classification function. The latter is called the 'hard margin' or 'realized margin'. ==== Convergence criterion ==== Consider a subset of the satisfied constraints in the dual problem. For any finite subset we can solve the linear program and thus satisfy all constraints. If we could prove that of all the constraints which we did not add to the dual problem no single constraint is violated, we would have proven that solving our restricted problem is equivalent to solving the original problem. More formally, let γ ∗ {\displaystyle \gamma ^{}} be the optimal objective function value for any restricted instance. Then, we can formulate a search problem for the 'most violated constraint' in the original problem space, namely finding ω ∗ ∈ Ω {\displaystyle \omega ^{}\in \Omega } as ω ∗ = argmax ω ∈ Ω ∑ n = 1 ℓ y n h ( x n ; ω ) λ n . {\displaystyle \omega ^{}={\underset {\omega \in \Omega }{\textrm {argmax}}}\sum _{n=1}^{\ell }y_{n}h({\boldsymbol {x}}_{n};\omega )\lambda _{n}.} That is, we search the space H {\displaystyle {\mathcal {H}}} for a single decision stump h ( ⋅ ; ω ∗ ) {\displaystyle h(\cdot ;\omega ^{})} maximizing the left hand side of the dual constraint. If the constraint cannot be violated by any choice of decision stump, none of the corresponding constraint can be active in the original problem and the restricted problem is equivalent. ==== Penalization constant ==== D {\displaystyle D} The positive value of penalization constant D {\displaystyle D} has to be found using model selection techniques. However, if we choose D = 1 ℓ ν {\displaystyle D={\frac {1}{\ell \nu }}} , where ℓ {\displaystyle \ell } is the number of training samples and 0 < ν < 1 {\displaystyle 0<\nu <1} , then the new parameter ν {\displaystyle \nu } has the following properties. ν {\displaystyle \nu } is an upper bound on the fraction of training errors; that is, if k {\displaystyle k} denotes the number of misclassified training samples, then k ℓ ≤ ν {\displaystyle {\frac {k}{\ell }}\leq \nu } . ν {\displaystyle \nu } is a lower bound on the fraction of training samples outside or on the margin. == Algorithm == Input: Training set X = { x 1 , … , x ℓ } {\displaystyle X=\{{\boldsymbol {x}}_{1},\dots ,{\boldsymbol {x}}_{\ell }\}} , x i ∈ X {\displaystyle {\boldsymbol {x}}_{i}\in {\mathcal {X}}} Training labels Y = { y 1 , … , y ℓ } {\displaystyle Y=\{y_{1},\dots ,y_{\ell }\}} , y i ∈ { − 1 , 1 } {\displaystyle y_{i}\in \{-1,1\}} Convergence threshold θ ≥ 0 {\displaystyle \theta \geq 0} Output: Classification function f : X → { − 1 , 1 } {\displaystyle f:{\mathcal {X}}\to \{-1,1\}} Initialization Weights, uniform λ n ← 1 ℓ , n = 1 , … , ℓ {\displaystyle \lambda _{n}\leftarrow {\frac {1}{\ell }},\quad n=1,\dots ,\ell } Edge γ ← 0 {\displaystyle \gamma \leftarrow 0} Hypothesis count J ← 1 {\displaystyle J\leftarrow 1} Iterate h ^ ← argmax ω ∈ Ω ∑ n = 1 ℓ y n h ( x n ; ω ) λ n {\displaystyle {\hat {h}}\leftarrow {\underset {\omega \in \Omega }{\textrm {argmax}}}\sum _{n=1}^{\ell }y_{n}h({\boldsymbol {x}}_{n};\omega )\lambda _{n}} if ∑ n = 1 ℓ y n h ^ ( x n ) λ n + γ ≤ θ {\displaystyle \sum _{n=1}^{\ell }y_{n}{\hat {h}}({\boldsymbol {x}}_{n})\lambda _{n}+\gamma \leq \theta } then break h J ← h ^ {\displaystyle h_{J}\leftarrow {\hat {h}}} J
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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.
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