AI Coding Tools

Explore the best AI Coding Tools — independent reviews, comparisons, pricing and step-by-step how-to guides, curated by Aizhi.

  • Google Books Ngram Viewer

    Google Books Ngram Viewer

    The Google Books Ngram Viewer is an online search engine that charts the frequencies of any set of search strings using a yearly count of n-grams found in printed sources published between 1500 and 2022 in Google's text corpora in English, Chinese (simplified), French, German, Hebrew, Italian, Russian, or Spanish. There are also some specialized English corpora, such as American English, British English, and English Fiction. The program can search for a word or a phrase. The n-grams are matched with the text within the selected corpus, and if found in 40 or more books, are then displayed as a graph. The program supports searches for parts of speech and wildcards. It is routinely used in research. == History == The Ngram Viewer was created by Google software engineers Will Brockman and Jon Orwant , who teamed up with Harvard researchers Jean-Baptiste Michel and Erez Lieberman Aiden. The service was released on December 16, 2010. Before the release, it was difficult to quantify the rate of linguistic change because of the absence of a database that was designed for this purpose, said Steven Pinker, a well-known linguist who was one of the co-authors of the Science paper published on the same day. The Google Books Ngram Viewer was developed in the hope of opening a new window to quantitative research in the humanities field, and the database contained 500 billion words from 5.2 million books publicly available from the very beginning. The intended audience was scholarly, but the Google Books Ngram Viewer made it possible for anyone with a computer to see a graph that represents the diachronic change of the use of words and phrases with ease. Lieberman said in response to The New York Times that the developers aimed to provide even children with the ability to browse cultural trends throughout history. In the Science paper, Lieberman and his collaborators called the method of high-volume data analysis in digitized texts "culturomics". == Usage == Commas delimit user-entered search terms, where each comma-separated term is searched in the database as an n-gram (for example, "nursery school" is a 2-gram or bigram). The Ngram Viewer then returns a plotted line chart. Due to limitations on the size of the Ngram database, only matches found in at least 40 books are indexed. == Limitations == The data sets of the Ngram Viewer have been criticized for their reliance upon inaccurate optical character recognition (OCR) and for including large numbers of incorrectly dated and categorized texts. Because of these errors, and because they are uncontrolled for bias (such as the increasing amount of scientific literature, which causes other terms to appear to decline in popularity), care must be taken in using the corpora to study language or test theories. Furthermore, the data sets may not reflect general linguistic or cultural change and can only hint at such an effect because they do not involve any metadata like date published, author, length, or genre, to avoid any potential copyright infringements. Systemic errors like the confusion of s and f in pre-19th century texts (due to the use of ſ, the long s, which is similar in appearance to f) can cause systemic bias. Although the Google Books team claims that the results are reliable from 1800 onwards, poor OCR and insufficient data mean that frequencies given for languages such as Chinese may only be accurate from 1970 onward, with earlier parts of the corpus showing no results at all for common terms, and data for some years containing more than 50% noise. Guidelines for doing research with data from Google Ngram have been proposed that try to address some of the issues discussed above.

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  • Promoter based genetic algorithm

    Promoter based genetic algorithm

    The promoter based genetic algorithm (PBGA) is a genetic algorithm for neuroevolution developed by F. Bellas and R.J. Duro in the Integrated Group for Engineering Research (GII) at the University of Coruña, in Spain. It evolves variable size feedforward artificial neural networks (ANN) that are encoded into sequences of genes for constructing a basic ANN unit. Each of these blocks is preceded by a gene promoter acting as an on/off switch that determines if that particular unit will be expressed or not. == PBGA basics == The basic unit in the PBGA is a neuron with all of its inbound connections as represented in the following figure: The genotype of a basic unit is a set of real valued weights followed by the parameters of the neuron and proceeded by an integer valued field that determines the promoter gene value and, consequently, the expression of the unit. By concatenating units of this type we can construct the whole network. With this encoding it is imposed that the information that is not expressed is still carried by the genotype in evolution but it is shielded from direct selective pressure, maintaining this way the diversity in the population, which has been a design premise for this algorithm. Therefore, a clear difference is established between the search space and the solution space, permitting information learned and encoded into the genotypic representation to be preserved by disabling promoter genes. == Results == The PBGA was originally presented within the field of autonomous robotics, in particular in the real time learning of environment models of the robot. It has been used inside the Multilevel Darwinist Brain (MDB) cognitive mechanism developed in the GII for real robots on-line learning. In another paper it is shown how the application of the PBGA together with an external memory that stores the successful obtained world models, is an optimal strategy for adaptation in dynamic environments. Recently, the PBGA has provided results that outperform other neuroevolutionary algorithms in non-stationary problems, where the fitness function varies in time.

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  • Genetic programming

    Genetic programming

    Genetic programming (GP) is an evolutionary algorithm, an artificial intelligence technique mimicking natural evolution, which operates on a population of programs. It applies the genetic operators selection according to a predefined fitness measure, mutation and crossover. The crossover operation involves swapping specified parts of selected pairs (parents) to produce new and different offspring that become part of the new generation of programs. Some programs not selected for reproduction are copied from the current generation to the new generation. Mutation involves substitution of some random part of a program with some other random part of a program. Then the selection and other operations are recursively applied to the new generation of programs. Typically, members of each new generation are on average more fit than the members of the previous generation, and the best-of-generation program is often better than the best-of-generation programs from previous generations. Termination of the evolution usually occurs when some individual program reaches a predefined proficiency or fitness level. It may and often does happen that a particular run of the algorithm results in premature convergence to some local maximum that is not a globally optimal or even good solution. Multiple runs (dozens to hundreds) are usually necessary to produce a very good result. It may also be necessary to have a large starting population size and variability of the individuals to avoid pathologies. == History == The first record of the proposal to evolve programs is probably that of Alan Turing in 1950 in "Computing Machinery and Intelligence". There was a gap of 25 years before the publication of John Holland's 'Adaptation in Natural and Artificial Systems' laid out the theoretical and empirical foundations of the science. In 1981, Richard Forsyth demonstrated the successful evolution of small programs, represented as trees, to perform classification of crime scene evidence for the UK Home Office. Although the idea of evolving programs, initially in the computer language Lisp, was current amongst John Holland's students, it was not until they organised the first Genetic Algorithms (GA) conference in Pittsburgh that Nichael Cramer published evolved programs in two specially designed languages, which included the first statement of modern "tree-based" genetic programming (that is, procedural languages organized in tree-based structures and operated on by suitably defined GA-operators). In 1988, John Koza (also a PhD student of John Holland) patented his invention of a GA for program evolution. This was followed by publication in the International Joint Conference on Artificial Intelligence IJCAI-89. Koza followed this with 205 publications on "genetic programming", a term coined by David Goldberg, also a PhD student of John Holland. However, it is the series of 4 books by Koza, starting in 1992 with accompanying videos, that really established GP. Subsequently, there was an enormous expansion of the number of publications with the Genetic Programming Bibliography, surpassing 10,000 entries. In 2010, Koza listed 77 results where genetic programming was human competitive. The departure of GP from the rigid, fixed-length representations typical of early GA models was not entirely without precedent. Early work on variable-length representations laid the groundwork. One notable example is messy genetic algorithms, which introduced irregular, variable-length chromosomes to address building block disruption and positional bias in standard GAs. Another precursor was robot trajectory programming, where genome representations encoded program instructions for robotic movements—structures inherently variable in length. Even earlier, unfixed-length representations were proposed in a doctoral dissertation by Cavicchio, who explored adaptive search using simulated evolution. His work provided foundational ideas for flexible program structures. In 1996, Koza started the annual Genetic Programming conference, which was followed in 1998 by the annual EuroGP conference, and the first book in a GP series edited by Koza. 1998 also saw the first GP textbook. GP continued to flourish, leading to the first specialist GP journal and three years later (2003) the annual Genetic Programming Theory and Practice (GPTP) workshop was established by Rick Riolo. Genetic programming papers continue to be published at a diversity of conferences and associated journals. Today there are nineteen GP books including several for students. === Foundational work in GP === Early work that set the stage for current genetic programming research topics and applications is diverse, and includes software synthesis and repair, predictive modeling, data mining, financial modeling, soft sensors, design, and image processing. Applications in some areas, such as design, often make use of intermediate representations, such as Fred Gruau's cellular encoding. Industrial uptake has been significant in several areas including finance, the chemical industry, bioinformatics and the steel industry. == Methods == === Program representation === GP evolves computer programs, traditionally represented in memory as tree structures. Trees can be easily evaluated in a recursive manner. Every internal node has an operator function and every terminal node has an operand, making mathematical expressions easy to evolve and evaluate. Thus traditionally GP favors the use of programming languages that naturally embody tree structures (for example, Lisp; other functional programming languages are also suitable). Non-tree representations have been suggested and successfully implemented, such as linear genetic programming, which perhaps suits the more traditional imperative languages. The commercial GP software Discipulus uses automatic induction of binary machine code ("AIM") to achieve better performance. μGP uses directed multigraphs to generate programs that fully exploit the syntax of a given assembly language. Multi expression programming uses three-address code for encoding solutions. Other program representations on which significant research and development have been conducted include programs for stack-based virtual machines, and sequences of integers that are mapped to arbitrary programming languages via grammars. Cartesian genetic programming is another form of GP, which uses a graph representation instead of the usual tree based representation to encode computer programs. Most representations have structurally noneffective code (introns). Such non-coding genes may seem to be useless because they have no effect on the performance of any one individual. However, they alter the probabilities of generating different offspring under the variation operators, and thus alter the individual's variational properties. Experiments seem to show faster convergence when using program representations that allow such non-coding genes, compared to program representations that do not have any non-coding genes. Instantiations may have both trees with introns and those without; the latter are called canonical trees. Special canonical crossover operators are introduced that maintain the canonical structure of parents in their children. === Initialisation === The methods for creation of the initial population include: Grow creates the individuals sequentially. Every GP tree is created starting from the root, creating functional nodes with children as well as terminal nodes up to a certain depth. Full is similar to the Grow. The difference is that all brunches in a tree are of same predetermined depth. Ramped half-and-half creates a population consisting of m d − 1 {\displaystyle md-1} parts and a maximum depth of m d {\displaystyle md} for its trees. The first part has a maximum depth of 2, second of 3 and so on up to the m d − 1 {\displaystyle md-1} -th part with maximum depth m d {\displaystyle md} . Half of every part is created by Grow, while the other part is created by Full. === Selection === Selection is a process whereby certain individuals are selected from the current generation that would serve as parents for the next generation. The individuals are selected probabilistically such that the better performing individuals have a higher chance of getting selected. The most commonly used selection method in GP is tournament selection, although other methods such as fitness proportionate selection, lexicase selection, and others have been demonstrated to perform better for many GP problems. Elitism, which involves seeding the next generation with the best individual (or best n individuals) from the current generation, is a technique sometimes employed to avoid regression. === Crossover === In genetic programming two fit individuals are chosen from the population to be parents for one or two children. In tree genetic programming, these parents are represented as inverted lisp like trees, with their root nodes at the top. In subtree cro

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  • Jubatus

    Jubatus

    Jubatus is an open-source online machine learning and distributed computing framework developed at Nippon Telegraph and Telephone and Preferred Infrastructure. Its features include classification, recommendation, regression, anomaly detection and graph mining. It supports many client languages, including C++, Java, Ruby and Python. It uses Iterative Parameter Mixture for distributed machine learning. == Notable Features == Jubatus supports: Multi-classification algorithms: Perceptron Passive Aggressive Confidence Weighted Adaptive Regularization of Weight Vectors Normal Herd Recommendation algorithms using: Inverted index Minhash Locality-sensitive hashing Regression algorithms: Passive Aggressive feature extraction method for natural language: n-gram Text segmentation

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  • Graphics

    Graphics

    Graphics (from Ancient Greek γραφικός (graphikós) 'pertaining to drawing, painting, writing, etc.') are visual images or designs on some surface, such as a wall, canvas, screen, paper, or stone, to inform, illustrate, or entertain. In contemporary usage, it includes a pictorial representation of data, as in design and manufacture, in typesetting and the graphic arts, and in educational and recreational software. Images that are generated by a computer are called computer graphics. Examples are photographs, drawings, line art, mathematical graphs, line graphs, charts, diagrams, typography, numbers, symbols, geometric designs, maps, engineering drawings, or other images. Graphics often combine text, illustration, and color. Graphic design may consist of the deliberate selection, creation, or arrangement of typography alone, as in a brochure, flyer, poster, web site, or book without any other element. The objective can be clarity or effective communication, association with other cultural elements, or merely the creation of a distinctive style. Graphics can be functional or artistic. The latter can be a recorded version, such as a photograph, or an interpretation by a scientist to highlight essential features, or an artist, in which case the distinction with imaginary graphics may become blurred. It can also be used for architecture. == History == The earliest graphics known to anthropologists studying prehistoric periods are cave paintings and markings on boulders, bone, ivory, and antlers, which were created during the Upper Palaeolithic period from 40,000 to 10,000 B.C. or earlier. Many of these were found to record astronomical, seasonal, and chronological details. Some of the earliest graphics and drawings are known to the modern world, from almost 6,000 years ago, are that of engraved stone tablets and ceramic cylinder seals, marking the beginning of the historical periods and the keeping of records for accounting and inventory purposes. Records from Egypt predate these and papyrus was used by the Egyptians as a material on which to plan the building of pyramids; they also used slabs of limestone and wood. From 600 to 250 BC, the Greeks played a major role in geometry. They used graphics to represent their mathematical theories such as the Circle Theorem and the Pythagorean theorem. In art, "graphics" is often used to distinguish work in a monotone and made up of lines, as opposed to painting. === Drawing === Drawing generally involves making marks on a surface by applying pressure from a tool or moving a tool across a surface. In which a tool is always used as if there were no tools it would be art. Graphical drawing is an instrumental guided drawing. === Printmaking === Woodblock printing, including images is first seen in China after paper was invented (about A.D. 105). In the West, the main techniques have been woodcut, engraving and etching, but there are many others. ==== Etching ==== Etching is an intaglio method of printmaking in which the image is incised into the surface of a metal plate using an acid. The acid eats the metal, leaving behind roughened areas, or, if the surface exposed to the acid is very thin, burning a line into the plate. The use of the process in printmaking is believed to have been invented by Daniel Hopfer (c. 1470–1536) of Augsburg, Germany, who decorated armour in this way. Etching is also used in the manufacturing of printed circuit boards and semiconductor devices. === Line art === Line art is a rather non-specific term sometimes used for any image that consists of distinct straight and curved lines placed against a (usually plain) background, without gradations in shade (darkness) or hue (color) to represent two-dimensional or three-dimensional objects. Line art is usually monochromatic, although lines may be of different colors. === Illustration === An illustration is a visual representation such as a drawing, painting, photograph or other work of art that stresses the subject more than form. The aim of an illustration is to elucidate or decorate a story, poem or piece of textual information (such as a newspaper article), traditionally by providing a visual representation of something described in the text. The editorial cartoon, also known as a political cartoon, is an illustration containing a political or social message. Illustrations can be used to display a wide range of subject matter and serve a variety of functions, such as: giving faces to characters in a story displaying a number of examples of an item described in an academic textbook (e.g. A Typology) visualizing step-wise sets of instructions in a technical manual communicating subtle thematic tone in a narrative linking brands to the ideas of human expression, individuality, and creativity making a reader laugh or smile for fun (to make laugh) funny === Graphs === A graph or chart is a graphic that represents tabular or numeric data. Charts are often used to make it easier to understand large quantities of data and the relationships between different parts of the data. === Diagrams === A diagram is a simplified and structured visual representation of concepts, ideas, constructions, relations, statistical data, etc., used to visualize and clarify the topic. === Symbols === A symbol, in its basic sense, is a representation of a concept or quantity; i.e., an idea, object, concept, quality, etc. In more psychological and philosophical terms, all concepts are symbolic in nature, and representations for these concepts are simply token artifacts that are allegorical to (but do not directly codify) a symbolic meaning, or symbolism. === Maps === A map is a simplified depiction of a space, a navigational aid which highlights relations between objects within that space. Usually, a map is a two-dimensional, geometrically accurate representation of a three-dimensional space. One of the first 'modern' maps was made by Waldseemüller. === Photography === One difference between photography and other forms of graphics is that a photographer, in principle, just records a single moment in reality, with seemingly no interpretation. However, a photographer can choose the field of view and angle, and may also use other techniques, such as various lenses to choose the view or filters to change the colors. In recent times, digital photography has opened the way to an infinite number of fast, but strong, manipulations. Even in the early days of photography, there was controversy over photographs of enacted scenes that were presented as 'real life' (especially in war photography, where it can be very difficult to record the original events). Shifting the viewer's eyes ever so slightly with simple pinpricks in the negative could have a dramatic effect. The choice of the field of view can have a strong effect, effectively 'censoring out' other parts of the scene, accomplished by cropping them out or simply not including them in the photograph. This even touches on the philosophical question of what reality is. The human brain processes information based on previous experience, making us see what we want to see or what we were taught to see. Photography does the same, although the photographer interprets the scene for their viewer. === Engineering drawings === An engineering drawing is a type of drawing and is technical in nature, used to fully and clearly define requirements for engineered items. It is usually created in accordance with standardized conventions for layout, nomenclature, interpretation, appearance (such as typefaces and line styles), size, etc. === Computer graphics === There are two types of computer graphics: raster graphics, where each pixel is separately defined (as in a digital photograph), and vector graphics, where mathematical formulas are used to draw lines and shapes, which are then interpreted at the viewer's end to produce the graphic. Using vectors results in infinitely sharp graphics and often smaller files, but, when complex, like vectors take time to render and may have larger file sizes than a raster equivalent. In 1950, the first computer-driven display was attached to MIT's Whirlwind I computer to generate simple pictures. This was followed by MIT's TX-0 and TX-2, interactive computing which increased interest in computer graphics during the late 1950s. In 1962, Ivan Sutherland invented Sketchpad, an innovative program that influenced alternative forms of interaction with computers. In the mid-1960s, large computer graphics research projects were begun at MIT, General Motors, Bell Labs, and Lockheed Corporation. Douglas T. Ross of MIT developed an advanced compiler language for graphics programming. S.A.Coons, also at MIT, and J. C. Ferguson at Boeing, began work in sculptured surfaces. GM developed their DAC-1 system, and other companies, such as Douglas, Lockheed, and McDonnell, also made significant developments. In 1968, ray tracing was first described by Arthur Appel of the IBM Research Center, Yorktown Heights, N

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  • Multispectral pattern recognition

    Multispectral pattern recognition

    Multispectral remote sensing is the collection and analysis of reflected, emitted, or back-scattered energy from an object or an area of interest in multiple bands of regions of the electromagnetic spectrum (Jensen, 2005). Subcategories of multispectral remote sensing include hyperspectral, in which hundreds of bands are collected and analyzed, and ultraspectral remote sensing where many hundreds of bands are used (Logicon, 1997). The main purpose of multispectral imaging is the potential to classify the image using multispectral classification. This is a much faster method of image analysis than is possible by human interpretation. == Multispectral remote sensing systems == Remote sensing systems gather data via instruments typically carried on satellites in orbit around the Earth. The remote sensing scanner detects the energy that radiates from the object or area of interest. This energy is recorded as an analog electrical signal and converted into a digital value though an A-to-D conversion. There are several multispectral remote sensing systems that can be categorized in the following way: === Multispectral imaging using discrete detectors and scanning mirrors === Landsat Multispectral Scanner (MSS) Landsat Thematic Mapper (TM) NOAA Geostationary Operational Environmental Satellite (GOES) NOAA Advanced Very High Resolution Radiometer (AVHRR) NASA and ORBIMAGE, Inc., Sea-viewing Wide field-of-view Sensor (SeaWiFS) Daedalus, Inc., Aircraft Multispectral Scanner (AMS) NASA Airborne Terrestrial Applications Sensor (ATLAS) === Multispectral imaging using linear arrays === SPOT 1, 2, and 3 High Resolution Visible (HRV) sensors and Spot 4 and 5 High Resolution Visible Infrared (HRVIR) and vegetation sensor Indian Remote Sensing System (IRS) Linear Imaging Self-scanning Sensor (LISS) Space Imaging, Inc. (IKONOS) Digital Globe, Inc. (QuickBird) ORBIMAGE, Inc. (OrbView-3) ImageSat International, Inc. (EROS A1) NASA Terra Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) NASA Terra Multiangle Imaging Spectroradiometer (MISR) === Imaging spectrometry using linear and area arrays === NASA Jet Propulsion Laboratory Airborne Visible/Infrared Imaging Spectrometer (AVIRIS) Compact Airborne Spectrographic Imager 3 (CASI 3) NASA Terra Moderate Resolution Imaging Spectrometer (MODIS) NASA Earth Observer (EO-1) Advanced Land Imager (ALI), Hyperion, and LEISA Atmospheric Corrector (LAC) === Satellite analog and digital photographic systems === Russian SPIN-2 TK-350, and KVR-1000 NASA Space Shuttle and International Space Station Imagery == Multispectral classification methods == A variety of methods can be used for the multispectral classification of images: Algorithms based on parametric and nonparametric statistics that use ratio-and interval-scaled data and nonmetric methods that can also incorporate nominal scale data (Duda et al., 2001), Supervised or unsupervised classification logic, Hard or soft (fuzzy) set classification logic to create hard or fuzzy thematic output products, Per-pixel or object-oriented classification logic, and Hybrid approaches == Supervised classification == In this classification method, the identity and location of some of the land-cover types are obtained beforehand from a combination of fieldwork, interpretation of aerial photography, map analysis, and personal experience. The analyst would locate sites that have similar characteristics to the known land-cover types. These areas are known as training sites because the known characteristics of these sites are used to train the classification algorithm for eventual land-cover mapping of the remainder of the image. Multivariate statistical parameters (means, standard deviations, covariance matrices, correlation matrices, etc.) are calculated for each training site. All pixels inside and outside of the training sites are evaluated and allocated to the class with the more similar characteristics. === Classification scheme === The first step in the supervised classification method is to identify the land-cover and land-use classes to be used. Land-cover refers to the type of material present on the site (e.g. water, crops, forest, wet land, asphalt, and concrete). Land-use refers to the modifications made by people to the land cover (e.g. agriculture, commerce, settlement). All classes should be selected and defined carefully to properly classify remotely sensed data into the correct land-use and/or land-cover information. To achieve this purpose, it is necessary to use a classification system that contains taxonomically correct definitions of classes. If a hard classification is desired, the following classes should be used: Mutually exclusive: there is not any taxonomic overlap of any classes (i.e., rain forest and evergreen forest are distinct classes). Exhaustive: all land-covers in the area have been included. Hierarchical: sub-level classes (e.g., single-family residential, multiple-family residential) are created, allowing that these classes can be included in a higher category (e.g., residential). Some examples of hard classification schemes are: American Planning Association Land-Based Classification System United States Geological Survey Land-use/Land-cover Classification System for Use with Remote Sensor Data U.S. Department of the Interior Fish and Wildlife Service U.S. National Vegetation and Classification System International Geosphere-Biosphere Program IGBP Land Cover Classification System === Training sites === Once the classification scheme is adopted, the image analyst may select training sites in the image that are representative of the land-cover or land-use of interest. If the environment where the data was collected is relatively homogeneous, the training data can be used. If different conditions are found in the site, it would not be possible to extend the remote sensing training data to the site. To solve this problem, a geographical stratification should be done during the preliminary stages of the project. All differences should be recorded (e.g. soil type, water turbidity, crop species, etc.). These differences should be recorded on the imagery and the selection training sites made based on the geographical stratification of this data. The final classification map would be a composite of the individual stratum classifications. After the data are organized in different training sites, a measurement vector is created. This vector would contain the brightness values for each pixel in each band in each training class. The mean, standard deviation, variance-covariance matrix, and correlation matrix are calculated from the measurement vectors. Once the statistics from each training site are determined, the most effective bands for each class should be selected. The objective of this discrimination is to eliminate the bands that can provide redundant information. Graphical and statistical methods can be used to achieve this objective. Some of the graphic methods are: Bar graph spectral plots Cospectral mean vector plots Feature space plots Cospectral parallelepiped or ellipse plots === Classification algorithm === The last step in supervised classification is selecting an appropriate algorithm. The choice of a specific algorithm depends on the input data and the desired output. Parametric algorithms are based on the fact that the data is normally distributed. If the data is not normally distributed, nonparametric algorithms should be used. The more common nonparametric algorithms are: One-dimensional density slicing Parallelipiped Minimum distance Nearest-neighbor Expert system analysis Convolutional neural network == Unsupervised classification == Unsupervised classification (also known as clustering) is a method of partitioning remote sensor image data in multispectral feature space and extracting land-cover information. Unsupervised classification require less input information from the analyst compared to supervised classification because clustering does not require training data. This process consists in a series of numerical operations to search for the spectral properties of pixels. From this process, a map with m spectral classes is obtained. Using the map, the analyst tries to assign or transform the spectral classes into thematic information of interest (i.e. forest, agriculture, urban). This process may not be easy because some spectral clusters represent mixed classes of surface materials and may not be useful. The analyst has to understand the spectral characteristics of the terrain to be able to label clusters as a specific information class. There are hundreds of clustering algorithms. Two of the most conceptually simple algorithms are the chain method and the ISODATA method. === Chain method === The algorithm used in this method operates in a two-pass mode (it passes through the multispectral dataset two times. In the first pass, the program reads through the dataset and sequentially builds clusters (groups of p

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  • Sparse PCA

    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}

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  • Oja's rule

    Oja's rule

    Oja's learning rule, or simply Oja's rule, named after Finnish computer scientist Erkki Oja (Finnish pronunciation: [ˈojɑ], AW-yuh), is a model of how neurons in the brain or in artificial neural networks change connection strength, or learn, over time. It is a modification of the standard Hebb's Rule that, through multiplicative normalization, solves all stability problems and generates an algorithm for principal components analysis. This is a computational form of an effect which is believed to happen in biological neurons. == Theory == Oja's rule requires a number of simplifications to derive, but in its final form it is demonstrably stable, unlike Hebb's rule. It is a single-neuron special case of the Generalized Hebbian Algorithm. However, Oja's rule can also be generalized in other ways to varying degrees of stability and success. === Formula === Consider a simplified model of a neuron y {\displaystyle y} that returns a linear combination of its inputs x using presynaptic weights w: y ( x ) = ∑ j = 1 m x j w j {\displaystyle \,y(\mathbf {x} )~=~\sum _{j=1}^{m}x_{j}w_{j}} Oja's rule defines the change in presynaptic weights w given the output response y {\displaystyle y} of a neuron to its inputs x to be Δ w = w n + 1 − w n = η y n ( x n − y n w n ) , {\displaystyle \,\Delta \mathbf {w} ~=~\mathbf {w} _{n+1}-\mathbf {w} _{n}~=~\eta \,y_{n}(\mathbf {x} _{n}-y_{n}\mathbf {w} _{n}),} where η is the learning rate which can also change with time. Note that the bold symbols are vectors and n defines a discrete time iteration. The rule can also be made for continuous iterations as d w d t = η y ( t ) ( x ( t ) − y ( t ) w ( t ) ) . {\displaystyle \,{\frac {d\mathbf {w} }{dt}}~=~\eta \,y(t)(\mathbf {x} (t)-y(t)\mathbf {w} (t)).} === Derivation === The simplest learning rule known is Hebb's rule, which states in conceptual terms that neurons that fire together, wire together. In component form as a difference equation, it is written Δ w = η y ( x n ) x n {\displaystyle \,\Delta \mathbf {w} ~=~\eta \,y(\mathbf {x} _{n})\mathbf {x} _{n}} , or in scalar form with implicit n-dependence, w i ( n + 1 ) = w i ( n ) + η y ( x ) x i {\displaystyle \,w_{i}(n+1)~=~w_{i}(n)+\eta \,y(\mathbf {x} )x_{i}} , where y(xn) is again the output, this time explicitly dependent on its input vector x. Hebb's rule has synaptic weights approaching infinity with a positive learning rate. We can stop this by normalizing the weights so that each weight's magnitude is restricted between 0, corresponding to no weight, and 1, corresponding to being the only input neuron with any weight. We do this by normalizing the weight vector to be of length one: w i ( n + 1 ) = w i ( n ) + η y ( x ) x i ( ∑ j = 1 m [ w j ( n ) + η y ( x ) x j ] p ) 1 / p {\displaystyle \,w_{i}(n+1)~=~{\frac {w_{i}(n)+\eta \,y(\mathbf {x} )x_{i}}{\left(\sum _{j=1}^{m}[w_{j}(n)+\eta \,y(\mathbf {x} )x_{j}]^{p}\right)^{1/p}}}} . Note that in Oja's original paper, p=2, corresponding to quadrature (root sum of squares), which is the familiar Cartesian normalization rule. However, any type of normalization, even linear, will give the same result without loss of generality. For a small learning rate | η | ≪ 1 {\displaystyle |\eta |\ll 1} the equation can be expanded as a Power series in η {\displaystyle \eta } . w i ( n + 1 ) = w i ( n ) ( ∑ j w j p ( n ) ) 1 / p + η ( y x i ( ∑ j w j p ( n ) ) 1 / p − w i ( n ) ∑ j y x j w j p − 1 ( n ) ( ∑ j w j p ( n ) ) ( 1 + 1 / p ) ) + O ( η 2 ) {\displaystyle \,w_{i}(n+1)~=~{\frac {w_{i}(n)}{\left(\sum _{j}w_{j}^{p}(n)\right)^{1/p}}}~+~\eta \left({\frac {yx_{i}}{\left(\sum _{j}w_{j}^{p}(n)\right)^{1/p}}}-{\frac {w_{i}(n)\sum _{j}yx_{j}w_{j}^{p-1}(n)}{\left(\sum _{j}w_{j}^{p}(n)\right)^{(1+1/p)}}}\right)~+~O(\eta ^{2})} . For small η, our higher-order terms O(η2) go to zero. We again make the specification of a linear neuron, that is, the output of the neuron is equal to the sum of the product of each input and its synaptic weight to the power of p-1, which in the case of p=2 is synaptic weight itself, or y ( x ) = ∑ j = 1 m x j w j p − 1 {\displaystyle \,y(\mathbf {x} )~=~\sum _{j=1}^{m}x_{j}w_{j}^{p-1}} . We also specify that our weights normalize to 1, which will be a necessary condition for stability, so | w | = ( ∑ j = 1 m w j p ) 1 / p = 1 {\displaystyle \,|\mathbf {w} |~=~\left(\sum _{j=1}^{m}w_{j}^{p}\right)^{1/p}~=~1} , which, when substituted into our expansion, gives Oja's rule, or w i ( n + 1 ) = w i ( n ) + η y ( x i − w i ( n ) y ) {\displaystyle \,w_{i}(n+1)~=~w_{i}(n)+\eta \,y(x_{i}-w_{i}(n)y)} . === Stability and PCA === In analyzing the convergence of a single neuron evolving by Oja's rule, one extracts the first principal component, or feature, of a data set. Furthermore, with extensions using the Generalized Hebbian Algorithm, one can create a multi-Oja neural network that can extract as many features as desired, allowing for principal components analysis. A principal component aj is extracted from a dataset x through some associated vector qj, or aj = qj⋅x, and we can restore our original dataset by taking x = ∑ j a j q j {\displaystyle \mathbf {x} ~=~\sum _{j}a_{j}\mathbf {q} _{j}} . In the case of a single neuron trained by Oja's rule, we find the weight vector converges to q1, or the first principal component, as time or number of iterations approaches infinity. We can also define, given a set of input vectors Xi, that its correlation matrix Rij = XiXj has an associated eigenvector given by qj with eigenvalue λj. The variance of outputs of our Oja neuron σ2(n) = ⟨y2(n)⟩ then converges with time iterations to the principal eigenvalue, or lim n → ∞ σ 2 ( n ) = λ 1 {\displaystyle \lim _{n\rightarrow \infty }\sigma ^{2}(n)~=~\lambda _{1}} . These results are derived using Lyapunov function analysis, and they show that Oja's neuron necessarily converges on strictly the first principal component if certain conditions are met in our original learning rule. Most importantly, our learning rate η is allowed to vary with time, but only such that its sum is divergent but its power sum is convergent, that is ∑ n = 1 ∞ η ( n ) = ∞ , ∑ n = 1 ∞ η ( n ) p < ∞ , p > 1 {\displaystyle \sum _{n=1}^{\infty }\eta (n)=\infty ,~~~\sum _{n=1}^{\infty }\eta (n)^{p}<\infty ,~~~p>1} . Our output activation function y(x(n)) is also allowed to be nonlinear and nonstatic, but it must be continuously differentiable in both x and w and have derivatives bounded in time. == Applications == Oja's rule was originally described in Oja's 1982 paper, but the principle of self-organization to which it is applied is first attributed to Alan Turing in 1952. PCA has also had a long history of use before Oja's rule formalized its use in network computation in 1989. The model can thus be applied to any problem of self-organizing mapping, in particular those in which feature extraction is of primary interest. Therefore, Oja's rule has an important place in image and speech processing. It is also useful as it expands easily to higher dimensions of processing, thus being able to integrate multiple outputs quickly. A canonical example is its use in binocular vision. === Biology and Oja's subspace rule === There is clear evidence for both long-term potentiation and long-term depression in biological neural networks, along with a normalization effect in both input weights and neuron outputs. However, while there is no direct experimental evidence yet of Oja's rule active in a biological neural network, a biophysical derivation of a generalization of the rule is possible. Such a derivation requires retrograde signalling from the postsynaptic neuron, which is biologically plausible (see neural backpropagation), and takes the form of Δ w i j ∝ ⟨ x i y j ⟩ − ϵ ⟨ ( c p r e ∗ ∑ k w i k y k ) ⋅ ( c p o s t ∗ y j ) ⟩ , {\displaystyle \Delta w_{ij}~\propto ~\langle x_{i}y_{j}\rangle -\epsilon \left\langle \left(c_{\mathrm {pre} }\sum _{k}w_{ik}y_{k}\right)\cdot \left(c_{\mathrm {post} }y_{j}\right)\right\rangle ,} where as before wij is the synaptic weight between the ith input and jth output neurons, x is the input, y is the postsynaptic output, and we define ε to be a constant analogous the learning rate, and cpre and cpost are presynaptic and postsynaptic functions that model the weakening of signals over time. Note that the angle brackets denote the average and the ∗ operator is a convolution. By taking the pre- and post-synaptic functions into frequency space and combining integration terms with the convolution, we find that this gives an arbitrary-dimensional generalization of Oja's rule known as Oja's Subspace, namely Δ w = C x ⋅ w − w ⋅ C y . {\displaystyle \Delta w~=~Cx\cdot w-w\cdot Cy.}

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  • Collision detection

    Collision detection

    Collision detection is the computational problem of detecting an intersection of two or more objects in virtual space. More precisely, it deals with the questions of if, when, and where two or more objects intersect. Collision detection is a classic problem of computational geometry with applications in computer graphics, physical simulation, video games, robotics (including autonomous driving), and computational physics. Collision detection algorithms can be divided into operating on 2D or 3D spatial objects. == Overview == Collision detection is closely linked to calculating the distance between objects, as objects collide when the distance between them is less than or equal to zero. Negative distances indicate that one object has penetrated another. Performing collision detection requires more context than just the distance between the objects. Accurately identifying the points of contact on both objects' surfaces is also essential for computing a physically accurate collision response. The complexity of this task increases with the level of detail in the objects' representations: the more intricate the model, the greater the computational cost. Collision detection frequently involves dynamic objects, adding a temporal dimension to distance calculations. Instead of simply measuring distance between static objects, collision detection algorithms often aim to determine whether the objects' motion will bring them to a point in time when their distance is zero—an operation that adds significant computational overhead. In collision detection involving multiple objects, a naive approach would require detecting collisions for all pairwise combinations of objects. As the number of objects increases, the number of required comparisons grows rapidly: for n {\displaystyle n} objects, n ( n − 1 ) / 2 {n(n-1)}/{2} intersection tests are needed with a naive approach. This quadratic growth makes such an approach computationally expensive as n {\displaystyle n} increases. Due to the complexity mentioned above, collision detection is a computationally intensive process. Nevertheless, it is essential for interactive applications like video games, robotics, and real-time physics engines. To manage these computational demands, extensive efforts have gone into optimizing collision detection algorithms. A commonly used approach towards accelerating the required computations is to divide the process into two phases: the broad phase and the narrow phase. The broad phase aims to answer the question of whether objects might collide, using a conservative but efficient approach to rule out pairs that clearly do not intersect, thus avoiding unnecessary calculations. Objects that cannot be definitively separated in the broad phase are passed to the narrow phase. Here, more precise algorithms determine whether these objects actually intersect. If they do, the narrow phase often calculates the exact time and location of the intersection. == Broad phase == This phase aims at quickly finding objects or parts of objects for which it can be quickly determined that no further collision test is needed. A useful property of such approach is that it is output sensitive. In the context of collision detection this means that the time complexity of the collision detection is proportional to the number of objects that are close to each other. An early example of that is the I-COLLIDE where the number of required narrow phase collision tests was O ( n + m ) {\displaystyle O(n+m)} where n {\displaystyle n} is the number of objects and m {\displaystyle m} is the number of objects at close proximity. This is a significant improvement over the quadratic complexity of the naive approach. === Spatial partitioning === Several approaches can be grouped under the spatial partitioning umbrella, which includes octrees (for 3D), quadtrees (for 2D), binary space partitioning (or BSP trees) and other, similar approaches. If one splits space into a number of simple cells, and if two objects can be shown not to be in the same cell, then they need not be checked for intersection. Dynamic scenes and deformable objects require updating the partitioning which can add overhead. === Bounding volume hierarchy === Bounding Volume Hierarchy (BVH) is a tree structure over a set of bounding volumes. Collision is determined by doing a tree traversal starting from the root. If the bounding volume of the root doesn't intersect with the object of interest, the traversal can be stopped. If, however there is an intersection, the traversal proceeds and checks the branches for each there is an intersection. Branches for which there is no intersection with the bounding volume can be culled from further intersection test. Therefore, multiple objects can be determined to not intersect at once. BVH can be used with deformable objects such as cloth or soft-bodies but the volume hierarchy has to be adjusted as the shape deforms. For deformable objects we need to be concerned about self-collisions or self intersections. BVH can be used for that end as well. Collision between two objects is computed by computing intersection between the bounding volumes of the root of the tree as there are collision we dive into the sub-trees that intersect. Exact collisions between the actual objects, or its parts (often triangles of a triangle mesh) need to be computed only between intersecting leaves. The same approach works for pair wise collision and self-collisions. === Exploiting temporal coherence === During the broad-phase, when the objects in the world move or deform, the data-structures used to cull collisions have to be updated. In cases where the changes between two frames or time-steps are small and the objects can be approximated well with axis-aligned bounding boxes, the sweep and prune algorithm can be a suitable approach. Several key observation make the implementation efficient: Two bounding-boxes intersect if, and only if, there is overlap along all three axes; overlap can be determined, for each axis separately, by sorting the intervals for all the boxes; and lastly, between two frames updates are typically small (making sorting algorithms optimized for almost-sorted lists suitable for this application). The algorithm keeps track of currently intersecting boxes, and as objects move, re-sorting the intervals helps keep track of the status. === Pairwise pruning === Once a pair of physical bodies has been selected for further investigation, collisions need to be checked more carefully. However, in many applications, individual objects (if they are not too deformable) are described by a set of smaller primitives, mainly triangles. So there are two sets of triangles, S = S 1 , S 2 , … , S n {\displaystyle S={S_{1},S_{2},\dots ,S_{n}}} and T = T 1 , T 2 , … , T n {\displaystyle T={T_{1},T_{2},\dots ,T_{n}}} (for simplicity, each set has the same number of triangles.) The obvious thing to do is to check all triangles S j {\displaystyle S_{j}} against all triangles T k {\displaystyle T_{k}} for collisions, but this involves n 2 {\displaystyle n^{2}} comparisons, which is highly inefficient. If possible, it is desirable to use a pruning algorithm to reduce the number of pairs of triangles that need to be checked. The most widely used family of algorithms is known as the hierarchical bounding volumes method. As a preprocessing step, for each object (e.g., S {\displaystyle S} and T {\displaystyle T} ) calculates a hierarchy of bounding volumes. Then, at each time step, when collisions need to be checked between S {\displaystyle S} and T {\displaystyle T} , the hierarchical bounding volumes are used to reduce the number of pairs of triangles under consideration. For simplicity, provide an example using bounding spheres, although it has been noted that spheres are undesirable in many cases. If E {\displaystyle E} is a set of triangles, a bounding sphere is pre-calculated. B ( E ) {\displaystyle B(E)} . There are many ways of choosing B ( E ) {\displaystyle B(E)} , B ( E ) {\displaystyle B(E)} is a sphere that completely contains E {\displaystyle E} and is as small as possible. Ahead of time, B ( S ) {\displaystyle B(S)} and B ( T ) {\displaystyle B(T)} can be computed. Clearly, if these two spheres do not intersect (and that is very easy to test), then neither do S {\displaystyle S} and T {\displaystyle T} . This is not much better than an n-body pruning algorithm, however. If E = E 1 , E 2 , … , E m {\displaystyle E={E_{1},E_{2},\dots ,E_{m}}} is a set of triangles, then split it into two halves L ( E ) := E 1 , E 2 , … , E m / 2 {\displaystyle L(E):={E_{1},E_{2},\dots ,E_{m/2}}} and R ( E ) := E m / 2 + 1 , … , E m − 1 , E m {\displaystyle R(E):={E_{m/2+1},\dots ,E_{m-1},E_{m}}} . Apply this to S {\displaystyle S} and T {\displaystyle T} , and calculate (ahead of time) the bounding spheres B ( L ( S ) ) , B ( R ( S ) ) {\displaystyle B(L(S)),B(R(S))} and B ( L ( T ) ) , B ( R ( T ) ) {\displaystyle B(L(T)),B(R(T))} . T

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  • Policy gradient method

    Policy gradient method

    Policy gradient methods are a class of reinforcement learning algorithms and a sub-class of policy optimization methods. Unlike value-based methods which learn a value function to derive a policy, policy optimization methods directly learn a policy function π {\displaystyle \pi } that selects actions without consulting a value function. For policy gradient to apply, the policy function π θ {\displaystyle \pi _{\theta }} is parameterized by a differentiable parameter θ {\displaystyle \theta } . == Overview == In policy-based RL, the actor is a parameterized policy function π θ {\displaystyle \pi _{\theta }} , where θ {\displaystyle \theta } are the parameters of the actor. The actor takes as argument the state of the environment s {\displaystyle s} and produces a probability distribution π θ ( ⋅ ∣ s ) {\displaystyle \pi _{\theta }(\cdot \mid s)} . If the action space is discrete, then ∑ a π θ ( a ∣ s ) = 1 {\displaystyle \sum _{a}\pi _{\theta }(a\mid s)=1} . If the action space is continuous, then ∫ a π θ ( a ∣ s ) d a = 1 {\displaystyle \int _{a}\pi _{\theta }(a\mid s)\mathrm {d} a=1} . The goal of policy optimization is to find some θ {\displaystyle \theta } that maximizes the expected episodic reward J ( θ ) {\displaystyle J(\theta )} : J ( θ ) = E π θ [ ∑ t = 0 T γ t R t | S 0 = s 0 ] {\displaystyle J(\theta )=\mathbb {E} _{\pi _{\theta }}\left[\sum _{t=0}^{T}\gamma ^{t}R_{t}{\Big |}S_{0}=s_{0}\right]} where γ {\displaystyle \gamma } is the discount factor, R t {\displaystyle R_{t}} is the reward at step t {\displaystyle t} , s 0 {\displaystyle s_{0}} is the starting state, and T {\displaystyle T} is the time-horizon (which can be infinite). The policy gradient is defined as ∇ θ J ( θ ) {\displaystyle \nabla _{\theta }J(\theta )} . Different policy gradient methods stochastically estimate the policy gradient in different ways. The goal of any policy gradient method is to iteratively maximize J ( θ ) {\displaystyle J(\theta )} by gradient ascent. Since the key part of any policy gradient method is the stochastic estimation of the policy gradient, they are also studied under the title of "Monte Carlo gradient estimation". == REINFORCE == === Policy gradient === The REINFORCE algorithm, introduced by Ronald J. Williams in 1992, was the first policy gradient method. It is based on the identity for the policy gradient ∇ θ J ( θ ) = E π θ [ ∑ t = 0 T ∇ θ ln ⁡ π θ ( A t ∣ S t ) ∑ t = 0 T ( γ t R t ) | S 0 = s 0 ] {\displaystyle \nabla _{\theta }J(\theta )=\mathbb {E} _{\pi _{\theta }}\left[\sum _{t=0}^{T}\nabla _{\theta }\ln \pi _{\theta }(A_{t}\mid S_{t})\;\sum _{t=0}^{T}(\gamma ^{t}R_{t}){\Big |}S_{0}=s_{0}\right]} which can be improved via the "causality trick" ∇ θ J ( θ ) = E π θ [ ∑ t = 0 T ∇ θ ln ⁡ π θ ( A t ∣ S t ) ∑ τ = t T ( γ τ R τ ) | S 0 = s 0 ] {\displaystyle \nabla _{\theta }J(\theta )=\mathbb {E} _{\pi _{\theta }}\left[\sum _{t=0}^{T}\nabla _{\theta }\ln \pi _{\theta }(A_{t}\mid S_{t})\sum _{\tau =t}^{T}(\gamma ^{\tau }R_{\tau }){\Big |}S_{0}=s_{0}\right]} Thus, we have an unbiased estimator of the policy gradient: ∇ θ J ( θ ) ≈ 1 N ∑ n = 1 N [ ∑ t = 0 T ∇ θ ln ⁡ π θ ( A t , n ∣ S t , n ) ∑ τ = t T ( γ τ − t R τ , n ) ] {\displaystyle \nabla _{\theta }J(\theta )\approx {\frac {1}{N}}\sum _{n=1}^{N}\left[\sum _{t=0}^{T}\nabla _{\theta }\ln \pi _{\theta }(A_{t,n}\mid S_{t,n})\sum _{\tau =t}^{T}(\gamma ^{\tau -t}R_{\tau ,n})\right]} where the index n {\displaystyle n} ranges over N {\displaystyle N} rollout trajectories using the policy π θ {\displaystyle \pi _{\theta }} . The score function ∇ θ ln ⁡ π θ ( A t ∣ S t ) {\displaystyle \nabla _{\theta }\ln \pi _{\theta }(A_{t}\mid S_{t})} can be interpreted as the direction in the parameter space that increases the probability of taking action A t {\displaystyle A_{t}} in state S t {\displaystyle S_{t}} . The policy gradient, then, is a weighted average of all possible directions to increase the probability of taking any action in any state, but weighted by reward signals, so that if taking a certain action in a certain state is associated with high reward, then that direction would be highly reinforced, and vice versa. === Algorithm === The REINFORCE algorithm is a loop: Rollout N {\displaystyle N} trajectories in the environment, using π θ t {\displaystyle \pi _{\theta _{t}}} as the policy function. Compute the policy gradient estimation: g i ← 1 N ∑ n = 1 N [ ∑ t = 0 T ∇ θ t ln ⁡ π θ ( A t , n ∣ S t , n ) ∑ τ = t T ( γ τ R τ , n ) ] {\displaystyle g_{i}\leftarrow {\frac {1}{N}}\sum _{n=1}^{N}\left[\sum _{t=0}^{T}\nabla _{\theta _{t}}\ln \pi _{\theta }(A_{t,n}\mid S_{t,n})\sum _{\tau =t}^{T}(\gamma ^{\tau }R_{\tau ,n})\right]} Update the policy by gradient ascent: θ i + 1 ← θ i + α i g i {\displaystyle \theta _{i+1}\leftarrow \theta _{i}+\alpha _{i}g_{i}} Here, α i {\displaystyle \alpha _{i}} is the learning rate at update step i {\displaystyle i} . == Variance reduction == REINFORCE is an on-policy algorithm, meaning that the trajectories used for the update must be sampled from the current policy π θ {\displaystyle \pi _{\theta }} . This can lead to high variance in the updates, as the returns R ( τ ) {\displaystyle R(\tau )} can vary significantly between trajectories. Many variants of REINFORCE have been introduced, under the title of variance reduction. === REINFORCE with baseline === A common way for reducing variance is the REINFORCE with baseline algorithm, based on the following identity: ∇ θ J ( θ ) = E π θ [ ∑ t = 0 T ∇ θ ln ⁡ π θ ( A t | S t ) ( ∑ τ = t T ( γ τ R τ ) − b ( S t ) ) | S 0 = s 0 ] {\displaystyle \nabla _{\theta }J(\theta )=\mathbb {E} _{\pi _{\theta }}\left[\sum _{t=0}^{T}\nabla _{\theta }\ln \pi _{\theta }(A_{t}|S_{t})\left(\sum _{\tau =t}^{T}(\gamma ^{\tau }R_{\tau })-b(S_{t})\right){\Big |}S_{0}=s_{0}\right]} for any function b : States → R {\displaystyle b:{\text{States}}\to \mathbb {R} } . This can be proven by applying the previous lemma. The algorithm uses the modified gradient estimator g i ← 1 N ∑ n = 1 N [ ∑ t = 0 T ∇ θ t ln ⁡ π θ ( A t , n | S t , n ) ( ∑ τ = t T ( γ τ R τ , n ) − b i ( S t , n ) ) ] {\displaystyle g_{i}\leftarrow {\frac {1}{N}}\sum _{n=1}^{N}\left[\sum _{t=0}^{T}\nabla _{\theta _{t}}\ln \pi _{\theta }(A_{t,n}|S_{t,n})\left(\sum _{\tau =t}^{T}(\gamma ^{\tau }R_{\tau ,n})-b_{i}(S_{t,n})\right)\right]} and the original REINFORCE algorithm is the special case where b i ≡ 0 {\displaystyle b_{i}\equiv 0} . === Actor-critic methods === If b i {\textstyle b_{i}} is chosen well, such that b i ( S t ) ≈ ∑ τ = t T ( γ τ R τ ) = γ t V π θ i ( S t ) {\textstyle b_{i}(S_{t})\approx \sum _{\tau =t}^{T}(\gamma ^{\tau }R_{\tau })=\gamma ^{t}V^{\pi _{\theta _{i}}}(S_{t})} , this could significantly decrease variance in the gradient estimation. That is, the baseline should be as close to the value function V π θ i ( S t ) {\displaystyle V^{\pi _{\theta _{i}}}(S_{t})} as possible, approaching the ideal of: ∇ θ J ( θ ) = E π θ [ ∑ t = 0 T ∇ θ ln ⁡ π θ ( A t | S t ) ( ∑ τ = t T ( γ τ R τ ) − γ t V π θ ( S t ) ) | S 0 = s 0 ] {\displaystyle \nabla _{\theta }J(\theta )=\mathbb {E} _{\pi _{\theta }}\left[\sum _{t=0}^{T}\nabla _{\theta }\ln \pi _{\theta }(A_{t}|S_{t})\left(\sum _{\tau =t}^{T}(\gamma ^{\tau }R_{\tau })-\gamma ^{t}V^{\pi _{\theta }}(S_{t})\right){\Big |}S_{0}=s_{0}\right]} Note that, as the policy π θ t {\displaystyle \pi _{\theta _{t}}} updates, the value function V π θ i ( S t ) {\displaystyle V^{\pi _{\theta _{i}}}(S_{t})} updates as well, so the baseline should also be updated. One common approach is to train a separate function that estimates the value function, and use that as the baseline. This is one of the actor-critic methods, where the policy function is the actor and the value function is the critic. The Q-function Q π {\displaystyle Q^{\pi }} can also be used as the critic, since ∇ θ J ( θ ) = E π θ [ ∑ 0 ≤ t ≤ T γ t ∇ θ ln ⁡ π θ ( A t | S t ) ⋅ Q π θ ( S t , A t ) | S 0 = s 0 ] {\displaystyle \nabla _{\theta }J(\theta )=E_{\pi _{\theta }}\left[\sum _{0\leq t\leq T}\gamma ^{t}\nabla _{\theta }\ln \pi _{\theta }(A_{t}|S_{t})\cdot Q^{\pi _{\theta }}(S_{t},A_{t}){\Big |}S_{0}=s_{0}\right]} by a similar argument using the tower law. Subtracting the value function as a baseline, we find that the advantage function A π ( S , A ) = Q π ( S , A ) − V π ( S ) {\displaystyle A^{\pi }(S,A)=Q^{\pi }(S,A)-V^{\pi }(S)} can be used as the critic as well: ∇ θ J ( θ ) = E π θ [ ∑ 0 ≤ t ≤ T γ t ∇ θ ln ⁡ π θ ( A t | S t ) ⋅ A π θ ( S t , A t ) | S 0 = s 0 ] {\displaystyle \nabla _{\theta }J(\theta )=E_{\pi _{\theta }}\left[\sum _{0\leq t\leq T}\gamma ^{t}\nabla _{\theta }\ln \pi _{\theta }(A_{t}|S_{t})\cdot A^{\pi _{\theta }}(S_{t},A_{t}){\Big |}S_{0}=s_{0}\right]} In summary, there are many unbiased estimators for ∇ θ J θ {\textstyle \nabla _{\theta }J_{\theta }} , all in the form of: ∇ θ J ( θ ) = E π θ [ ∑ 0 ≤ t ≤ T ∇ θ ln ⁡ π θ ( A t | S t ) ⋅ Ψ t | S 0 = s 0 ] {\displaystyle \nabla _{\theta }J(\theta )=E_{\pi _{\theta }}\left[\su

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  • Julia (programming language)

    Julia (programming language)

    Julia is a dynamic general-purpose programming language. As a high-level language, distinctive aspects of Julia's design include a type system with parametric polymorphism, the use of multiple dispatch as a core programming paradigm, just-in-time compilation and a parallel garbage collection implementation. Notably, Julia does not support classes with encapsulated methods but instead relies on the types of all of a function's arguments to determine which method will be called. By default, Julia is run similarly to scripting languages, using its runtime, and allows for interactions, but Julia programs can also be compiled to small binary standalone executables (or to small libraries for e.g. Python), with e.g. the JuliaC.jl compiler. Julia programs can reuse libraries from other languages, and vice versa. Julia has interoperability with C, C++, Fortran, Rust, Python, and R. Additionally, some Julia packages have bindings to be used from Python and R as libraries. Julia is supported by programmer tools like IDEs (see below) and by notebooks like Pluto.jl, Jupyter, and since 2025, Google Colab officially supports Julia natively. Julia is sometimes used in embedded systems (e.g. has been used in a satellite in space on a Raspberry Pi Compute Module 4; 64-bit Pis work best with Julia, and Julia is supported in Raspbian). == History == Work on Julia began in 2009, when Jeff Bezanson, Stefan Karpinski, Viral B. Shah, and Alan Edelman set out to create a free language that was both high-level and fast. On 14 February 2012, the team launched a website with a blog post explaining the language's mission. In an interview with InfoWorld in April 2012, Karpinski said about the name of the language, Julia: "There's no good reason, really. It just seemed like a pretty name." Bezanson said he chose the name on the recommendation of a friend, then years later wrote: Maybe julia stands for "Jeff's uncommon lisp is automated"? Julia's syntax is stable, since version 1.0 in 2018, and Julia has a backward compatibility guarantee for 1.x and also a stability promise for the documented (stable) API, while in the years before in the early development prior to 0.7 the syntax (and semantics) was changed in new versions. All of the (registered package) ecosystem uses the new and improved syntax, and in most cases relies on new APIs that have been added regularly, and in some cases minor additional syntax added in a forward compatible way e.g. in Julia 1.7. In the 10 years since the 2012 launch of pre-1.0 Julia, the community has grown. The Julia package ecosystem has over 11.8 million lines of code (including docs and tests). The JuliaCon academic conference for Julia users and developers has been held annually since 2014 with JuliaCon2020 welcoming over 28,900 unique viewers, and then JuliaCon2021 breaking all previous records (with more than 300 JuliaCon2021 presentations available for free on YouTube, up from 162 the year before), and 43,000 unique viewers during the conference. Three of the Julia co-creators are the recipients of the 2019 James H. Wilkinson Prize for Numerical Software (awarded every four years) "for the creation of Julia, an innovative environment for the creation of high-performance tools that enable the analysis and solution of computational science problems." Also, Alan Edelman, professor of applied mathematics at MIT, has been selected to receive the 2019 IEEE Computer Society Sidney Fernbach Award "for outstanding breakthroughs in high-performance computing, linear algebra, and computational science and for contributions to the Julia programming language." Version 0.3 was released in August 2014. Both Julia 0.7 and version 1.0 were released on 8 August 2018. Julia 1.4 added syntax for generic array indexing to handle e.g. 0-based arrays. The memory model was also changed. Julia 1.5 released in August 2020 added record and replay debugging support, for Mozilla's rr tool. The release changed the behavior in the REPL (to soft scope) to the one used in Jupyter, but keeps full compatible with non-REPL code (that retains hard scope). Julia 1.6 was the largest release since 1.0, and it was the long-term support (LTS) version for the longest time. Since Julia 1.7 development is back to time-based releases, and it was released in November 2021 with e.g. a new default random-number generator and Julia 1.7.3 fixed at least one security issue. Julia 1.8 added options for hiding source code when compiling Julia source code to executables. Julia 1.9 has added the ability to precompile packages to native machine code, done automatically; to improve precompilation of packages a new package PrecompileTools.jl was introduced, for use by package developers. Julia 1.10 was released on 25 December 2023 with new features such as parallel garbage collection. Julia 1.11 was released on 7 October 2024, and with it 1.10.5 became the next long-term support (LTS) version (i.e. those became the only two supported versions), since replaced by 1.10.10 released on 27 June, and 1.6 is no longer an LTS version. Julia 1.11 adds e.g. the new public keyword to signal safe public API (Julia users are advised to use such API, not internals, of Julia or packages, and package authors advised to use the keyword, generally indirectly, e.g. prefixed with the @compat macro, from Compat.jl, to also support older Julia versions, at least the LTS version). Julia 1.12 was released on 7 October 2025 (and 1.12.5 on 9 February 2026), and with it a JuliaC.jl package including the juliac compiler that works with it, for making rather small binary executables (much smaller than was possible before; through the use of new so-called trimming feature). Julia 1.10 LTS is an officially still-supported branch, but the 1.11 branch has also been maintained after 1.12 release, with 1.11.8 released and then 1.11.9 released on 8 February 2026. === JuliaCon === Since 2014, the Julia Community has hosted an annual Julia Conference focused on developers and users. The first JuliaCon took place in Chicago and kickstarted the annual occurrence of the conference. Since 2014, the conference has taken place across a number of locations including MIT and the University of Maryland, Baltimore. The event audience has grown from a few dozen people to over 28,900 unique attendees during JuliaCon 2020, which took place virtually. JuliaCon 2021 also took place virtually with keynote addresses from professors William Kahan, the primary architect of the IEEE 754 floating-point standard (which virtually all CPUs and languages, including Julia, use), Jan Vitek, Xiaoye Sherry Li, and Soumith Chintala, a co-creator of PyTorch. JuliaCon grew to 43,000 unique attendees and more than 300 presentations (still freely accessible, plus for older years). JuliaCon 2022 will also be virtual held between July 27 and July 29, 2022, for the first time in several languages, not just in English. === Sponsors === The Julia language became a NumFOCUS fiscally sponsored project in 2014 in an effort to ensure the project's long-term sustainability. Jeremy Kepner at MIT Lincoln Laboratory was the founding sponsor of the Julia project in its early days. In addition, funds from the Gordon and Betty Moore Foundation, the Alfred P. Sloan Foundation, Intel, and agencies such as NSF, DARPA, NIH, NASA, and FAA have been essential to the development of Julia. Mozilla, the maker of Firefox web browser, with its research grants for H1 2019, sponsored "a member of the official Julia team" for the project "Bringing Julia to the Browser", meaning to Firefox and other web browsers. The Julia language is also supported by individual donors on GitHub. === The Julia company === JuliaHub, Inc. was founded in 2015 as Julia Computing, Inc. by Viral B. Shah, Deepak Vinchhi, Alan Edelman, Jeff Bezanson, Stefan Karpinski and Keno Fischer. In June 2017, Julia Computing raised US$4.6 million in seed funding from General Catalyst and Founder Collective, the same month was "granted $910,000 by the Alfred P. Sloan Foundation to support open-source Julia development, including $160,000 to promote diversity in the Julia community", and in December 2019 the company got $1.1 million funding from the US government to "develop a neural component machine learning tool to reduce the total energy consumption of heating, ventilation, and air conditioning (HVAC) systems in buildings". In July 2021, Julia Computing announced they raised a $24 million Series A round led by Dorilton Ventures, which also owns Formula One team Williams Racing, that partnered with Julia Computing. Williams' Commercial Director said: "Investing in companies building best-in-class cloud technology is a strategic focus for Dorilton and Julia's versatile platform, with revolutionary capabilities in simulation and modelling, is hugely relevant to our business. We look forward to embedding Julia Computing in the world's most technologically advanced sport". In June 2023, JuliaHub received (again, now

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  • Weighted majority algorithm (machine learning)

    Weighted majority algorithm (machine learning)

    In machine learning, weighted majority algorithm (WMA) is a meta learning algorithm used to construct a compound algorithm from a pool of prediction algorithms, which could be any type of learning algorithms, classifiers, or even real human experts. The algorithm assumes that we have no prior knowledge about the accuracy of the algorithms in the pool, but there are sufficient reasons to believe that one or more will perform well. Assume that the problem is a binary decision problem. To construct the compound algorithm, a positive weight is given to each of the algorithms in the pool. The compound algorithm then collects weighted votes from all the algorithms in the pool, and gives the prediction that has a higher vote. If the compound algorithm makes a mistake, the algorithms in the pool that contributed to the wrong predicting will be discounted by a certain ratio β where 0<β<1. It can be shown that the upper bounds on the number of mistakes made in a given sequence of predictions from a pool of algorithms A {\displaystyle \mathbf {A} } is O ( l o g | A | + m ) {\displaystyle \mathbf {O(log|A|+m)} } if one algorithm in x i {\displaystyle \mathbf {x} _{i}} makes at most m {\displaystyle \mathbf {m} } mistakes. There are many variations of the weighted majority algorithm to handle different situations, like shifting targets, infinite pools, or randomized predictions. The core mechanism remains similar, with the final performances of the compound algorithm bounded by a function of the performance of the specialist (best performing algorithm) in the pool.

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  • Artificial intelligence in hiring

    Artificial intelligence in hiring

    Artificial intelligence can be used to automate aspects of the job recruitment process. Advances in artificial intelligence, such as the advent of machine learning and the growth of big data, enable AI to be utilized to recruit, screen, and predict the success of applicants. Proponents of artificial intelligence in hiring claim it reduces bias, assists with finding qualified candidates, and frees up human resource workers' time for other tasks, while opponents worry that AI perpetuates inequalities in the workplace and will eliminate jobs. Despite the potential benefits, the ethical implications of AI in hiring remain a subject of debate, with concerns about algorithmic transparency, accountability, and the need for ongoing oversight to ensure fair and unbiased decision-making throughout the recruitment process. == Background == It is common for companies to use AI to automate aspects of their hiring process, especially the hospitality, finance, and tech industries. == Uses == === Screeners === Screeners are tests that allow companies to sift through a large applicant pool and extract applicants that have desirable features. What factors are used to screen applicants is a concern to ethicists and civil rights activists. A screener that favors people who have similar characteristics to those already employed at a company may perpetuate inequalities. For example, if a company that is predominantly white and male uses its employees' data to train its screener it may accidentally create a screening process that favors white, male applicants. The automation of screeners also has the potential to reduce biases. Biases against applicants with African American sounding names have been shown in multiple studies. An AI screener has the potential to limit human bias and error in the hiring process, allowing more minority applicants to be successful. === Recruitment === Recruitment involves the identification of potential applicants and the marketing of positions. AI is commonly utilized in the recruitment process because it can help boost the number of qualified applicants for positions. Companies are able to use AI to target their marketing to applicants who are likely to be good fits for a position. This often involves the use of social media sites advertising tools, which rely on AI. Facebook allows advertisers to target ads based on demographics, location, interests, behavior, and connections. Facebook also allows companies to target a "look-a-like" audience, that is the company supplies Facebook with a data set, typically the company's current employees, and Facebook will target the ad to profiles that are similar to the profiles in the data set. Additionally, job sites like Indeed, Glassdoor, and ZipRecruiter target job listings to applicants that have certain characteristics employers are looking for. Targeted advertising has many advantages for companies trying to recruit such being a more efficient use of resources, reaching a desired audience, and boosting qualified applicants. This has helped make it a mainstay in modern hiring. Who receives a targeted ad can be controversial. In hiring, the implications of targeted ads have to do with who is able to find out about and then apply to a position. Most targeted ad algorithms are proprietary information. Some platforms, like Facebook and Google, allow users to see why they were shown a specific ad, but users who do not receive the ad likely never know of its existence and also have no way of knowing why they were not shown the ad. === Interviews === Chatbots were one of the first applications of AI and are commonly used in the hiring process. Interviewees interact with chatbots to answer interview questions, and an analysis of their responses can be generated by AI. HireVue has created technology that analyzes interviewees' responses and gestures during recorded video interviews. Over 12 million interviewees have been screened by the more than 700 companies that utilize the service. == Controversies == Artificial intelligence in hiring confers many benefits, but it also has some challenges that have concerned experts. AI is only as good as the data it is using. Biases can inadvertently be baked into the data used in AI. Often companies will use data from their employees to decide what people to recruit or hire. This can perpetuate bias and lead to more homogenous workforces. Facebook Ads was an example of a platform that created such controversy for allowing business owners to specify what type of employee they are looking for. For example, job advertisements for nursing and teach could be set such that only women of a specific age group would see the advertisements. Facebook Ads has since then removed this function from its platform, citing the potential problems with the function in perpetuating biases and stereotypes against minorities. The growing use of Artificial Intelligence-enabled hiring systems has become an important component of modern talent hiring, particularly through social networks such as LinkedIn and Facebook. However, data overflow embedded in the hiring systems, based on Natural Language Processing (NLP) methods, may result in unconscious gender bias. Utilizing data driven methods may mitigate some bias generated from these systems It can also be hard to quantify what makes a good employee. This poses a challenge for training AI to predict which employees will be best. Commonly used metrics like performance reviews can be subjective and have been shown to favor white employees over black employees and men over women. Another challenge is the limited amount of available data. Employers only collect certain details about candidates during the initial stages of the hiring process. This requires AI to make determinations about candidates with very limited information to go off of. Additionally, many employers do not hire employees frequently and so have limited firm specific data to go off. To combat this, many firms will use algorithms and data from other firms in their industry. AI's reliance on applicant and current employees personal data raises privacy issues. These issues effect both the applicants and current employees, but also may have implications for third parties who are linked through social media to applicants or current employees. For example, a sweep of someone's social media will also show their friends and people they have tagged in photos or posts. == AI and the future of hiring == Artificial intelligence along with other technological advances such as improvements in robotics have placed 47% of jobs at risk of being eliminated in the near future. In 2016 the founder of the World Economic Forum, Klaus Schwab, called AI and related technology the "Fourth Industrial Revolution". According to some scholars, however, the transformative impact of AI on labor has been overstated. The "no-real-change" theory holds that an IT revolution has already occurred, but that the benefits of implementing new technologies does not outweigh the costs associated with adopting them. This theory claims that the result of the IT revolution is thus much less impactful than had originally been forecasted. Other scholars refute this theory claiming that AI has already led to significant job loss for unskilled labor and that it will eliminate middle skill and high skill jobs in the future. This position is based around the idea that AI is not yet a technology of general use and that any potential 4th industrial revolution has not fully occurred. A third theory holds that the effect of AI and other technological advances is too complicated to yet be understood. This theory is centered around the idea that while AI will likely eliminate jobs in the short term it will also likely increase the demand for other jobs. The question then becomes will the new jobs be accessible to people and will they emerge near when jobs are eliminated. == AI use in hiring for candidates == Job seekers now commonly encounter AI-driven tools at multiple stages, including automated resume parsing, video interview analysis, chatbots for frequently asked questions, and real‑time application updates. Some candidates also employ AI career agents, designed to optimize job searches, tailor applications, and interface with hiring teams. A 2025 Australian study found that AI-driven video interviews exhibited transcription error rates of up to 22% for non‑native speakers and those with speech-related disabilities, raising concerns of discrimination. A 2017 study in the Journal of Sociology found persistent gender and racial disparities in AI screening tools, even when fairness interventions are applied. Industry observers describe a growing “AI arms race” in recruitment, where both employers and candidates increasingly rely on automated agents. Employers use recruiting systems to source and filter applicants, while candidates deploy AI agents to prepare and submit applications. == Regulations == The Artifici

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  • Jackknife variance estimates for random forest

    Jackknife variance estimates for random forest

    In statistics, jackknife variance estimates for random forest are a way to estimate the variance in random forest models, in order to eliminate the bootstrap effects. == Jackknife variance estimates == The sampling variance of bagged learners is: V ( x ) = V a r [ θ ^ ∞ ( x ) ] {\displaystyle V(x)=Var[{\hat {\theta }}^{\infty }(x)]} Jackknife estimates can be considered to eliminate the bootstrap effects. The jackknife variance estimator is defined as: V ^ j = n − 1 n ∑ i = 1 n ( θ ^ ( − i ) − θ ¯ ) 2 {\displaystyle {\hat {V}}_{j}={\frac {n-1}{n}}\sum _{i=1}^{n}({\hat {\theta }}_{(-i)}-{\overline {\theta }})^{2}} In some classification problems, when random forest is used to fit models, jackknife estimated variance is defined as: V ^ j = n − 1 n ∑ i = 1 n ( t ¯ ( − i ) ⋆ ( x ) − t ¯ ⋆ ( x ) ) 2 {\displaystyle {\hat {V}}_{j}={\frac {n-1}{n}}\sum _{i=1}^{n}({\overline {t}}_{(-i)}^{\star }(x)-{\overline {t}}^{\star }(x))^{2}} Here, t ⋆ {\displaystyle t^{\star }} denotes a decision tree after training, t ( − i ) ⋆ {\displaystyle t_{(-i)}^{\star }} denotes the result based on samples without i t h {\displaystyle ith} observation. == Examples == E-mail spam problem is a common classification problem, in this problem, 57 features are used to classify spam e-mail and non-spam e-mail. Applying IJ-U variance formula to evaluate the accuracy of models with m=15,19 and 57. The results shows in paper( Confidence Intervals for Random Forests: The jackknife and the Infinitesimal Jackknife ) that m = 57 random forest appears to be quite unstable, while predictions made by m=5 random forest appear to be quite stable, this results is corresponding to the evaluation made by error percentage, in which the accuracy of model with m=5 is high and m=57 is low. Here, accuracy is measured by error rate, which is defined as: E r r o r R a t e = 1 N ∑ i = 1 N ∑ j = 1 M y i j , {\displaystyle ErrorRate={\frac {1}{N}}\sum _{i=1}^{N}\sum _{j=1}^{M}y_{ij},} Here N is also the number of samples, M is the number of classes, y i j {\displaystyle y_{ij}} is the indicator function which equals 1 when i t h {\displaystyle ith} observation is in class j, equals 0 when in other classes. No probability is considered here. There is another method which is similar to error rate to measure accuracy: l o g l o s s = 1 N ∑ i = 1 N ∑ j = 1 M y i j l o g ( p i j ) {\displaystyle logloss={\frac {1}{N}}\sum _{i=1}^{N}\sum _{j=1}^{M}y_{ij}log(p_{ij})} Here N is the number of samples, M is the number of classes, y i j {\displaystyle y_{ij}} is the indicator function which equals 1 when i t h {\displaystyle ith} observation is in class j, equals 0 when in other classes. p i j {\displaystyle p_{ij}} is the predicted probability of i t h {\displaystyle ith} observation in class j {\displaystyle j} .This method is used in Kaggle These two methods are very similar. == Modification for bias == When using Monte Carlo MSEs for estimating V I J ∞ {\displaystyle V_{IJ}^{\infty }} and V J ∞ {\displaystyle V_{J}^{\infty }} , a problem about the Monte Carlo bias should be considered, especially when n is large, the bias is getting large: E [ V ^ I J B ] − V ^ I J ∞ ≈ n ∑ b = 1 B ( t b ⋆ − t ¯ ⋆ ) 2 B {\displaystyle E[{\hat {V}}_{IJ}^{B}]-{\hat {V}}_{IJ}^{\infty }\approx {\frac {n\sum _{b=1}^{B}(t_{b}^{\star }-{\bar {t}}^{\star })^{2}}{B}}} To eliminate this influence, bias-corrected modifications are suggested: V ^ I J − U B = V ^ I J B − n ∑ b = 1 B ( t b ⋆ − t ¯ ⋆ ) 2 B {\displaystyle {\hat {V}}_{IJ-U}^{B}={\hat {V}}_{IJ}^{B}-{\frac {n\sum _{b=1}^{B}(t_{b}^{\star }-{\bar {t}}^{\star })^{2}}{B}}} V ^ J − U B = V ^ J B − ( e − 1 ) n ∑ b = 1 B ( t b ⋆ − t ¯ ⋆ ) 2 B {\displaystyle {\hat {V}}_{J-U}^{B}={\hat {V}}_{J}^{B}-(e-1){\frac {n\sum _{b=1}^{B}(t_{b}^{\star }-{\bar {t}}^{\star })^{2}}{B}}}

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  • Cross-entropy

    Cross-entropy

    In information theory, the cross-entropy between two probability distributions p {\displaystyle p} and q {\displaystyle q} , over the same underlying set of events, measures the average number of bits needed to identify an event drawn from the set when the coding scheme used for the set is optimized for an estimated probability distribution q {\displaystyle q} , rather than the true distribution p {\displaystyle p} . == Definition == The cross-entropy of the distribution q {\displaystyle q} relative to a distribution p {\displaystyle p} over a given set is defined as follows: H ( p , q ) = − E p ⁡ [ log ⁡ q ] , {\displaystyle H(p,q)=-\operatorname {E} _{p}[\log q],} where E p ⁡ [ ⋅ ] {\displaystyle \operatorname {E} _{p}[\cdot ]} is the expected value operator with respect to the distribution p {\displaystyle p} . The definition may be formulated using the Kullback–Leibler divergence D K L ( p ∥ q ) {\displaystyle D_{\mathrm {KL} }(p\parallel q)} , divergence of p {\displaystyle p} from q {\displaystyle q} (also known as the relative entropy of p {\displaystyle p} with respect to q {\displaystyle q} ). H ( p , q ) = H ( p ) + D K L ( p ∥ q ) , {\displaystyle H(p,q)=H(p)+D_{\mathrm {KL} }(p\parallel q),} where H ( p ) {\displaystyle H(p)} is the entropy of p {\displaystyle p} . For discrete probability distributions p {\displaystyle p} and q {\displaystyle q} with the same support X {\displaystyle {\mathcal {X}}} , this means The situation for continuous distributions is analogous. We have to assume that p {\displaystyle p} and q {\displaystyle q} are absolutely continuous with respect to some reference measure r {\displaystyle r} (usually r {\displaystyle r} is a Lebesgue measure on a Borel σ-algebra). Let P {\displaystyle P} and Q {\displaystyle Q} be probability density functions of p {\displaystyle p} and q {\displaystyle q} with respect to r {\displaystyle r} . Then − ∫ X P ( x ) log ⁡ Q ( x ) d x = E p ⁡ [ − log ⁡ Q ] , {\displaystyle -\int _{\mathcal {X}}P(x)\,\log Q(x)\,\mathrm {d} x=\operatorname {E} _{p}[-\log Q],} and therefore NB: The notation H ( p , q ) {\displaystyle H(p,q)} is also used for a different concept, the joint entropy of p {\displaystyle p} and q {\displaystyle q} . == Motivation == In information theory, the Kraft–McMillan theorem establishes that any directly decodable coding scheme for coding a message to identify one value x i {\displaystyle x_{i}} out of a set of possibilities { x 1 , … , x n } {\displaystyle \{x_{1},\ldots ,x_{n}\}} can be seen as representing an implicit probability distribution q ( x i ) = ( 1 2 ) ℓ i {\displaystyle q(x_{i})=\left({\frac {1}{2}}\right)^{\ell _{i}}} over { x 1 , … , x n } {\displaystyle \{x_{1},\ldots ,x_{n}\}} , where ℓ i {\displaystyle \ell _{i}} is the length of the code for x i {\displaystyle x_{i}} in bits. Therefore, cross-entropy can be interpreted as the expected message-length per datum when a wrong distribution q {\displaystyle q} is assumed while the data actually follows a distribution p {\displaystyle p} . That is why the expectation is taken over the true probability distribution p {\displaystyle p} and not q . {\displaystyle q.} Indeed the expected message-length under the true distribution p {\displaystyle p} is E p ⁡ [ ℓ ] = − E p ⁡ [ ln ⁡ q ( x ) ln ⁡ ( 2 ) ] = − E p ⁡ [ log 2 ⁡ q ( x ) ] = − ∑ x i p ( x i ) log 2 ⁡ q ( x i ) = − ∑ x p ( x ) log 2 ⁡ q ( x ) = H ( p , q ) . {\displaystyle {\begin{aligned}\operatorname {E} _{p}[\ell ]&=-\operatorname {E} _{p}\left[{\frac {\ln {q(x)}}{\ln(2)}}\right]\\[1ex]&=-\operatorname {E} _{p}\left[\log _{2}{q(x)}\right]\\[1ex]&=-\sum _{x_{i}}p(x_{i})\,\log _{2}q(x_{i})\\[1ex]&=-\sum _{x}p(x)\,\log _{2}q(x)=H(p,q).\end{aligned}}} == Estimation == There are many situations where cross-entropy needs to be measured but the distribution of p {\displaystyle p} is unknown. An example is language modeling, where a model is created based on a training set T {\displaystyle T} , and then its cross-entropy is measured on a test set to assess how accurate the model is in predicting the test data. In this example, p {\displaystyle p} is the true distribution of words in any corpus, and q {\displaystyle q} is the distribution of words as predicted by the model. Since the true distribution is unknown, cross-entropy cannot be directly calculated. In these cases, an estimate of cross-entropy is calculated using the following formula: H ( T , q ) = − ∑ i = 1 N 1 N log 2 ⁡ q ( x i ) {\displaystyle H(T,q)=-\sum _{i=1}^{N}{\frac {1}{N}}\log _{2}q(x_{i})} where N {\displaystyle N} is the size of the test set, and q ( x ) {\displaystyle q(x)} is the probability of event x {\displaystyle x} estimated from the training set. In other words, q ( x i ) {\displaystyle q(x_{i})} is the probability estimate of the model that the i-th word of the text is x i {\displaystyle x_{i}} . The sum is averaged over the N {\displaystyle N} words of the test. This is a Monte Carlo estimate of the true cross-entropy, where the test set is treated as samples from p ( x ) {\displaystyle p(x)} . == Relation to maximum likelihood == The cross entropy arises in classification problems when introducing a logarithm in the guise of the log-likelihood function. This section concerns the estimation of the probabilities of different discrete outcomes. To this end, denote a parametrized family of distributions by q θ {\displaystyle q_{\theta }} , with θ {\displaystyle \theta } subject to the optimization effort. Consider a given finite sequence of N {\displaystyle N} values x i {\displaystyle x_{i}} from a training set, obtained from conditionally independent sampling. The likelihood assigned to any considered parameter θ {\displaystyle \theta } of the model is then given by the product over all probabilities q θ ( X = x i ) {\displaystyle q_{\theta }(X=x_{i})} . Repeated occurrences are possible, leading to equal factors in the product. If the count of occurrences of the value equal to x {\displaystyle x} is denoted by # x {\displaystyle \#x} , then the frequency of that value equals # x / N {\displaystyle \#x/N} . If p ( X = x ) {\displaystyle p(X=x)} is the underlying probability distribution, for large N {\displaystyle N} we expect p ( X = x ) ≈ # x / N {\displaystyle p(X=x)\approx \#x/N} , by the law of large numbers. Writing our likelihood function as the product of observations from the distribution q θ {\displaystyle q_{\theta }} : L ( θ ; x ) = ∏ i q θ ( X = x i ) = ∏ x q θ ( X = x ) # x ≈ ∏ x q θ ( X = x ) N ⋅ p ( X = x ) = exp ⁡ log ⁡ [ ∏ x q θ ( X = x ) N ⋅ p ( X = x ) ] = exp ⁡ ( ∑ x N ⋅ p ( X = x ) log ⁡ q θ ( X = x ) ) , {\displaystyle {\begin{aligned}{\mathcal {L}}(\theta ;{\mathbf {x} })&=\prod _{i}q_{\theta }(X=x_{i})=\prod _{x}q_{\theta }(X=x)^{\#x}\\&\approx \prod _{x}q_{\theta }(X=x)^{N\cdot p(X=x)}=\exp \log \left[\prod _{x}q_{\theta }(X=x)^{N\cdot p(X=x)}\right]\\&=\exp \left(\sum _{x}N\cdot p(X=x)\log q_{\theta }(X=x)^{}\right),\end{aligned}}} where we have used the calculation rules for the logarithm in the final line. Notice how the exponent contains a − H ( p , q θ ) {\displaystyle -H(p,q_{\theta })} term. Taking the logarithm of both sides gives: log ⁡ L ( θ ; x ) = − N ⋅ H ( p , q θ ) . {\displaystyle \log {\mathcal {L}}(\theta ;{\mathbf {x} })=-N\cdot H(p,q_{\theta }).} Since the logarithm is a monotonically increasing function, the maximizing value of θ {\displaystyle \theta } is unaffected by this final step. Similarly, the maximizing value of θ {\displaystyle \theta } is unaffected by the factor of N {\displaystyle N} . So we observe that the likelihood maximization amounts to minimization of the cross-entropy. == Cross-entropy minimization == Cross-entropy minimization is frequently used in optimization and rare-event probability estimation. When comparing a distribution q {\displaystyle q} against a fixed reference distribution p {\displaystyle p} , cross-entropy and KL divergence are identical up to an additive constant (since p {\displaystyle p} is fixed): According to the Gibbs' inequality, both take on their minimal values when p = q {\displaystyle p=q} , which is 0 {\displaystyle 0} for KL divergence, and H ( p ) {\displaystyle \mathrm {H} (p)} for cross-entropy. In the engineering literature, the principle of minimizing KL divergence (Kullback's "Principle of Minimum Discrimination Information") is often called the Principle of Minimum Cross-Entropy (MCE), or Minxent. However, as discussed in the article Kullback–Leibler divergence, sometimes the distribution q {\displaystyle q} is the fixed prior reference distribution, and the distribution p {\displaystyle p} is optimized to be as close to q {\displaystyle q} as possible, subject to some constraint. In this case the two minimizations are not equivalent. This has led to some ambiguity in the literature, with some authors attempting to resolve the inconsistency by restating cross-entropy to be D K L ( p ∥ q ) {\displaystyle D_{\mathrm {KL} }(p\parallel q)} , rather than H (

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