AI Coding Dark Factory

AI Coding Dark Factory — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

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  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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  Fitness function

  A fitness function is a particular type of objective or cost function that is used to summarize, as a single figure of merit, how close a given candidate solution is to achieving the set aims. It is an important component of evolutionary algorithms (EA), such as genetic programming, evolution strategies or genetic algorithms. An EA is a metaheuristic that reproduces the basic principles of biological evolution as a computer algorithm in order to solve challenging optimization or planning tasks, at least approximately. For this purpose, many candidate solutions are generated, which are evaluated using a fitness function in order to guide the evolutionary development towards the desired goal. Similar quality functions are also used in other metaheuristics, such as ant colony optimization or particle swarm optimization. In the field of EAs, each candidate solution, also called an individual, is commonly represented as a string of numbers (referred to as a chromosome). After each round of testing or simulation the idea is to delete the n worst individuals, and to breed n new ones from the best solutions. Each individual must therefore to be assigned a quality number indicating how close it has come to the overall specification, and this is generated by applying the fitness function to the test or simulation results obtained from that candidate solution. Two main classes of fitness functions exist: one where the fitness function does not change, as in optimizing a fixed function or testing with a fixed set of test cases; and one where the fitness function is mutable, as in niche differentiation or co-evolving the set of test cases. Another way of looking at fitness functions is in terms of a fitness landscape, which shows the fitness for each possible chromosome. In the following, it is assumed that the fitness is determined based on an evaluation that remains unchanged during an optimization run. A fitness function does not necessarily have to be able to calculate an absolute value, as it is sometimes sufficient to compare candidates in order to select the better one. A relative indication of fitness (candidate a is better than b) is sufficient in some cases, such as tournament selection or Pareto optimization. == Requirements of evaluation and fitness function == The quality of the evaluation and calculation of a fitness function is fundamental to the success of an EA optimisation. It implements Darwin's principle of "survival of the fittest". Without fitness-based selection mechanisms for mate selection and offspring acceptance, EA search would be blind and hardly distinguishable from the Monte Carlo method. When setting up a fitness function, one must always be aware that it is about more than just describing the desired target state. Rather, the evolutionary search on the way to the optimum should also be supported as much as possible (see also section on auxiliary objectives), if and insofar as this is not already done by the fitness function alone. If the fitness function is designed badly, the algorithm will either converge on an inappropriate solution, or will have difficulty converging at all. Definition of the fitness function is not straightforward in many cases and often is performed iteratively if the fittest solutions produced by an EA is not what is desired. Interactive genetic algorithms address this difficulty by outsourcing evaluation to external agents which are normally humans. == Computational efficiency == The fitness function should not only closely align with the designer's goal, but also be computationally efficient. Execution speed is crucial, as a typical evolutionary algorithm must be iterated many times in order to produce a usable result for a non-trivial problem. Fitness approximation may be appropriate, especially in the following cases: Fitness computation time of a single solution is extremely high Precise model for fitness computation is missing The fitness function is uncertain or noisy. Alternatively or also in addition to the fitness approximation, the fitness calculations can also be distributed to a parallel computer in order to reduce the execution times. Depending on the population model of the EA used, both the EA itself and the fitness calculations of all offspring of one generation can be executed in parallel. == Multi-objective optimization == Practical applications usually aim at optimizing multiple and at least partially conflicting objectives. Two fundamentally different approaches are often used for this purpose, Pareto optimization and optimization based on fitness calculated using the weighted sum. === Weighted sum and penalty functions === When optimizing with the weighted sum, the single values of the O {\displaystyle O} objectives are first normalized so that they can be compared. This can be done with the help of costs or by specifying target values and determining the current value as the degree of fulfillment. Costs or degrees of fulfillment can then be compared with each other and, if required, can also be mapped to a uniform fitness scale. Without loss of generality, fitness is assumed to represent a value to be maximized. Each objective o i {\displaystyle o_{i}} is assigned a weight w i {\displaystyle w_{i}} in the form of a percentage value so that the overall raw fitness f r a w {\displaystyle f_{raw}} can be calculated as a weighted sum: f r a w = ∑ i = 1 O o i ⋅ w i w i t h ∑ i = 1 O w i = 1 {\displaystyle f_{raw}=\sum _{i=1}^{O}{o_{i}\cdot w_{i}}\quad {\mathsf {with}}\quad \sum _{i=1}^{O}{w_{i}}=1} A violation of R {\displaystyle R} restrictions r j {\displaystyle r_{j}} can be included in the fitness determined in this way in the form of penalty functions. For this purpose, a function p f j ( r j ) {\displaystyle pf_{j}(r_{j})} can be defined for each restriction which returns a value between 0 {\displaystyle 0} and 1 {\displaystyle 1} depending on the degree of violation, with the result being 1 {\displaystyle 1} if there is no violation. The previously determined raw fitness is multiplied by the penalty function(s) and the result is then the final fitness f f i n a l {\displaystyle f_{final}} : f f i n a l = f r a w ⋅ ∏ j = 1 R p f j ( r j ) = ∑ i = 1 O ( o i ⋅ w i ) ⋅ ∏ j = 1 R p f j ( r j ) {\displaystyle f_{final}=f_{raw}\cdot \prod _{j=1}^{R}{pf_{j}(r_{j})}=\sum _{i=1}^{O}{(o_{i}\cdot w_{i})}\cdot \prod _{j=1}^{R}{pf_{j}(r_{j})}} This approach is simple and has the advantage of being able to combine any number of objectives and restrictions. The disadvantage is that different objectives can compensate each other and that the weights have to be defined before the optimization. This means that the compromise lines must be defined before optimization, which is why optimization with the weighted sum is also referred to as the a priori method. In addition, certain solutions may not be obtained, see the section on the comparison of both types of optimization. === Pareto optimization === A solution is called Pareto-optimal if the improvement of one objective is only possible with a deterioration of at least one other objective. The set of all Pareto-optimal solutions, also called Pareto set, represents the set of all optimal compromises between the objectives. The figure below on the right shows an example of the Pareto set of two objectives f 1 {\displaystyle f_{1}} and f 2 {\displaystyle f_{2}} to be maximized. The elements of the set form the Pareto front (green line). From this set, a human decision maker must subsequently select the desired compromise solution. Constraints are included in Pareto optimization in that solutions without constraint violations are per se better than those with violations. If two solutions to be compared each have constraint violations, the respective extent of the violations decides. It was recognized early on that EAs with their simultaneously considered solution set are well suited to finding solutions in one run that cover the Pareto front sufficiently well. They are therefore well suited as a-posteriori methods for multi-objective optimization, in which the final decision is made by a human decision maker after optimization and determination of the Pareto front. Besides the SPEA2, the NSGA-II and NSGA-III have established themselves as standard methods. The advantage of Pareto optimization is that, in contrast to the weighted sum, it provides all alternatives that are equivalent in terms of the objectives as an overall solution. The disadvantage is that a visualization of the alternatives becomes problematic or even impossible from four objectives on. Furthermore, the effort increases exponentially with the number of objectives. If there are more than three or four objectives, some have to be combined using the weighted sum or other aggregation methods. === Comparison of both types of assessment === With the help of the weighted sum, the total Pareto front can be obtained by a suitable choice of weights, provided that it is convex

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  Teacher forcing

  Teacher forcing is an algorithm for training the weights of recurrent neural networks (RNNs). It involves feeding observed sequence values (i.e. ground-truth samples) back into the RNN after each step, thus forcing the RNN to stay close to the ground-truth sequence. The term "teacher forcing" can be motivated by comparing the RNN to a human student taking a multi-part exam where the answer to each part (for example a mathematical calculation) depends on the answer to the preceding part. In this analogy, rather than grading every answer in the end, with the risk that the student fails every single part even though they only made a mistake in the first one, a teacher records the score for each individual part and then tells the student the correct answer, to be used in the next part. The use of an external teacher signal is in contrast to real-time recurrent learning (RTRL). Teacher signals are known from oscillator networks. The promise is, that teacher forcing helps to reduce the training time. The term "teacher forcing" was introduced in 1989 by Ronald J. Williams and David Zipser, who reported that the technique was already being "frequently used in dynamical supervised learning tasks" around that time. A NeurIPS 2016 paper introduced the related method of "professor forcing".

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  Neural gas

  Neural gas is an artificial neural network, inspired by the self-organizing map and introduced in 1991 by Thomas Martinetz and Klaus Schulten. The neural gas is a simple algorithm for finding optimal data representations based on feature vectors. The algorithm was coined "neural gas" because of the dynamics of the feature vectors during the adaptation process, which distribute themselves like a gas within the data space. It is applied where data compression or vector quantization is an issue, for example speech recognition, image processing or pattern recognition. As a robustly converging alternative to the k-means clustering it is also used for cluster analysis. == Algorithm == Suppose we want to model a probability distribution P ( x ) {\displaystyle P(x)} of data vectors x {\displaystyle x} using a finite number of feature vectors w i {\displaystyle w_{i}} , where i = 1 , ⋯ , N {\displaystyle i=1,\cdots ,N} . For each time step t {\displaystyle t} Sample data vector x {\displaystyle x} from P ( x ) {\displaystyle P(x)} Compute the distance between x {\displaystyle x} and each feature vector. Rank the distances. Let i 0 {\displaystyle i_{0}} be the index of the closest feature vector, i 1 {\displaystyle i_{1}} the index of the second closest feature vector, and so on. Update each feature vector by: w i k t + 1 = w i k t + ε ⋅ e − k / λ ⋅ ( x − w i k t ) , k = 0 , ⋯ , N − 1 {\displaystyle w_{i_{k}}^{t+1}=w_{i_{k}}^{t}+\varepsilon \cdot e^{-k/\lambda }\cdot (x-w_{i_{k}}^{t}),k=0,\cdots ,N-1} In the algorithm, ε {\displaystyle \varepsilon } can be understood as the learning rate, and λ {\displaystyle \lambda } as the neighborhood range. ε {\displaystyle \varepsilon } and λ {\displaystyle \lambda } are reduced with increasing t {\displaystyle t} so that the algorithm converges after many adaptation steps. The adaptation step of the neural gas can be interpreted as gradient descent on a cost function. By adapting not only the closest feature vector but all of them with a step size decreasing with increasing distance order, compared to (online) k-means clustering a much more robust convergence of the algorithm can be achieved. The neural gas model does not delete a node and also does not create new nodes. === Comparison with SOM === Compared to self-organized map, the neural gas model does not assume that some vectors are neighbors. If two vectors happen to be close together, they would tend to move together, and if two vectors happen to be apart, they would tend to not move together. In contrast, in an SOM, if two vectors are neighbors in the underlying graph, then they will always tend to move together, no matter whether the two vectors happen to be neighbors in the Euclidean space. The name "neural gas" is because one can imagine it to be what an SOM would be like if there is no underlying graph, and all points are free to move without the bonds that bind them together. == Variants == A number of variants of the neural gas algorithm exists in the literature so as to mitigate some of its shortcomings. More notable is perhaps Bernd Fritzke's growing neural gas, but also one should mention further elaborations such as the Growing When Required network and also the incremental growing neural gas. A performance-oriented approach that avoids the risk of overfitting is the Plastic Neural gas model. === Growing neural gas === Fritzke describes the growing neural gas (GNG) as an incremental network model that learns topological relations by using a "Hebb-like learning rule", only, unlike the neural gas, it has no parameters that change over time and it is capable of continuous learning, i.e. learning on data streams. GNG has been widely used in several domains, demonstrating its capabilities for clustering data incrementally. The GNG is initialized with two randomly positioned nodes which are initially connected with a zero age edge and whose errors are set to 0. Since in the GNG input data is presented sequentially one by one, the following steps are followed at each iteration: It is calculating the errors (distances) between the two closest nodes to the current input data. The error of the winner node (only the closest one) is respectively accumulated. The winner node and its topological neighbors (connected by an edge) are moving towards the current input by different fractions of their respective errors. The age of all edges connected to the winner node are incremented. If the winner node and the second-winner are connected by an edge, such an edge is set to 0. Else, an edge is created between them. If there are edges with an age larger than a threshold, they are removed. Nodes without connections are eliminated. If the current iteration is an integer multiple of a predefined frequency-creation threshold, a new node is inserted between the node with the largest error (among all) and its topological neighbor presenting the highest error. The link between the former and the latter nodes is eliminated (their errors are decreased by a given factor) and the new node is connected to both of them. The error of the new node is initialized as the updated error of the node which had the largest error (among all). The accumulated error of all nodes is decreased by a given factor. If the stopping criterion is not met, the algorithm takes a following input. The criterion might be a given number of epochs, i.e., a pre-set number of times where all data is presented, or the reach of a maximum number of nodes. === Incremental growing neural gas === Another neural gas variant inspired by the GNG algorithm is the incremental growing neural gas (IGNG). The authors propose the main advantage of this algorithm to be "learning new data (plasticity) without degrading the previously trained network and forgetting the old input data (stability)." === Growing when required === Having a network with a growing set of nodes, like the one implemented by the GNG algorithm was seen as a great advantage, however some limitation on the learning was seen by the introduction of the parameter λ, in which the network would only be able to grow when iterations were a multiple of this parameter. The proposal to mitigate this problem was a new algorithm, the Growing When Required network (GWR), which would have the network grow more quickly, by adding nodes as quickly as possible whenever the network identified that the existing nodes would not describe the input well enough. === Plastic neural gas === The ability to only grow a network may quickly introduce overfitting; on the other hand, removing nodes on the basis of age only, as in the GNG model, does not ensure that the removed nodes are actually useless, because removal depends on a model parameter that should be carefully tuned to the "memory length" of the stream of input data. The "Plastic Neural Gas" model solves this problem by making decisions to add or remove nodes using an unsupervised version of cross-validation, which controls an equivalent notion of "generalization ability" for the unsupervised setting. While growing-only methods only cater for the incremental learning scenario, the ability to grow and shrink is suited to the more general streaming data problem. == Implementations == To find the ranking i 0 , i 1 , … , i N − 1 {\displaystyle i_{0},i_{1},\ldots ,i_{N-1}} of the feature vectors, the neural gas algorithm involves sorting, which is a procedure that does not lend itself easily to parallelization or implementation in analog hardware. However, implementations in both parallel software and analog hardware were actually designed.

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  Facial age estimation

  Facial age estimation is the use of artificial intelligence to estimate the age of a person based on their facial features. Computer vision techniques are used to analyse the facial features in the images of millions of people whose age is known and then deep learning is used to create an algorithm that tries to predict the age of an unknown person. The key use of the technology is to prevent access to age-restricted goods and services. Examples include restricting children from accessing internet pornography, checking that they meet a mandatory minimum age when registering for an account on social media, or preventing adults from accessing websites, online chat or games designed only for use by children. The technology is distinct from facial recognition systems as the software does not attempt to uniquely identify the individual. Researchers have applied neural networks for age estimation since at least 2010. == Evaluation == An ongoing study by the National Institute of Standards and Technology (NIST) entitled 'Face Analysis Technology Evaluation' seeks to establish the technical performance of prototype age estimation algorithms submitted by academic teams and software vendors including Brno University of Technology, Czech Technical University in Prague, Dermalog, IDEMIA, Incode Technologies Inc, Jumio, Nominder, Rank One Computing, Unissey and Yoti. == Public sector use == The UK government has explored using facial age estimation at the UK border as an alternative to bone X-rays and MRI scans when determining child status of asylum seekers. == Commercial use == Commercial users of facial age estimation include Instagram and OnlyFans. In January 2025, John Lewis & Partners announced that had started using the technology to check the age of people shopping for knives on its website, to comply with UK legislation to limit knife crime. In the UK, several supermarket chains have taken part in Home Office trials of the technology to automate the checking of a customer's age when buying age-restricted goods such as alcohol. UK legislation introduced in January 2025 mandates robust forms of age verification hosting adult content viewable in the UK by July 2025. Allowable methods include facial age estimation. == Criticism == Adam Schwartz, a lawyer for the Electronic Frontier Foundation, criticized the use of facial age estimation software, noting its inaccuracy especially in cases of minorities and women, as was found in NIST's 2024 report. Twenty organisations jointly under European Digital Rights called the practice a "systematic and invasive processing of young people's data" that risks discriminatory profiling.

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  Inductive logic programming

  Inductive logic programming (ILP) is a subfield of symbolic artificial intelligence which uses logic programming as a uniform representation for examples, background knowledge and hypotheses. The term "inductive" here refers to philosophical (i.e. suggesting a theory to explain observed facts) rather than mathematical (i.e. proving a property for all members of a well-ordered set) induction. Given an encoding of the known background knowledge and a set of examples represented as a logical database of facts, an ILP system will derive a hypothesised logic program which entails all the positive and none of the negative examples. Schema: positive examples + negative examples + background knowledge ⇒ hypothesis. Bioinformatics and drug design have been highlighted as a principal application area of inductive logic programming techniques. == History == Building on earlier work on Inductive inference, Gordon Plotkin was the first to formalise induction in a clausal setting around 1970, adopting an approach of generalising from examples. In 1981, Ehud Shapiro introduced several ideas that would shape the field in his new approach of model inference, an algorithm employing refinement and backtracing to search for a complete axiomatisation of given examples. His first implementation was the Model Inference System in 1981: a Prolog program that inductively inferred Horn clause logic programs from positive and negative examples. The term Inductive Logic Programming was first introduced in a paper by Stephen Muggleton in 1990, defined as the intersection of machine learning and logic programming. Muggleton and Wray Buntine introduced predicate invention and inverse resolution in 1988. Several inductive logic programming systems that proved influential appeared in the early 1990s. FOIL, introduced by Ross Quinlan in 1990 was based on upgrading propositional learning algorithms AQ and ID3. Golem, introduced by Muggleton and Feng in 1990, went back to a restricted form of Plotkin's least generalisation algorithm. The Progol system, introduced by Muggleton in 1995, first implemented inverse entailment, and inspired many later systems. Aleph, a descendant of Progol introduced by Ashwin Srinivasan in 2001, is still one of the most widely used systems as of 2022. At around the same time, the first practical applications emerged, particularly in bioinformatics, where by 2000 inductive logic programming had been successfully applied to drug design, carcinogenicity and mutagenicity prediction, and elucidation of the structure and function of proteins. Unlike the focus on automatic programming inherent in the early work, these fields used inductive logic programming techniques from a viewpoint of relational data mining. The success of those initial applications and the lack of progress in recovering larger traditional logic programs shaped the focus of the field. Recently, classical tasks from automated programming have moved back into focus, as the introduction of meta-interpretative learning makes predicate invention and learning recursive programs more feasible. This technique was pioneered with the Metagol system introduced by Muggleton, Dianhuan Lin, Niels Pahlavi and Alireza Tamaddoni-Nezhad in 2014. This allows ILP systems to work with fewer examples, and brought successes in learning string transformation programs, answer set grammars and general algorithms. == Setting == Inductive logic programming has adopted several different learning settings, the most common of which are learning from entailment and learning from interpretations. In both cases, the input is provided in the form of background knowledge B, a logical theory (commonly in the form of clauses used in logic programming), as well as positive and negative examples, denoted E + {\textstyle E^{+}} and E − {\textstyle E^{-}} respectively. The output is given as a hypothesis H, itself a logical theory that typically consists of one or more clauses. The two settings differ in the format of examples presented. === Learning from entailment === As of 2022, learning from entailment is by far the most popular setting for inductive logic programming. In this setting, the positive and negative examples are given as finite sets E + {\textstyle E^{+}} and E − {\textstyle E^{-}} of positive and negated ground literals, respectively. A correct hypothesis H is a set of clauses satisfying the following requirements, where the turnstile symbol ⊨ {\displaystyle \models } stands for logical entailment: Completeness: B ∪ H ⊨ E + Consistency: B ∪ H ∪ E − ⊭ false {\displaystyle {\begin{array}{llll}{\text{Completeness:}}&B\cup H&\models &E^{+}\\{\text{Consistency: }}&B\cup H\cup E^{-}&\not \models &{\textit {false}}\end{array}}} Completeness requires any generated hypothesis H to explain all positive examples E + {\textstyle E^{+}} , and consistency forbids generation of any hypothesis H that is inconsistent with the negative examples E − {\textstyle E^{-}} , both given the background knowledge B. In Muggleton's setting of concept learning, "completeness" is referred to as "sufficiency", and "consistency" as "strong consistency". Two further conditions are added: "Necessity", which postulates that B does not entail E + {\textstyle E^{+}} , does not impose a restriction on H, but forbids any generation of a hypothesis as long as the positive facts are explainable without it. "Weak consistency", which states that no contradiction can be derived from B ∧ H {\textstyle B\land H} , forbids generation of any hypothesis H that contradicts the background knowledge B. Weak consistency is implied by strong consistency; if no negative examples are given, both requirements coincide. Weak consistency is particularly important in the case of noisy data, where completeness and strong consistency cannot be guaranteed. === Learning from interpretations === In learning from interpretations, the positive and negative examples are given as a set of complete or partial Herbrand structures, each of which are themselves a finite set of ground literals. Such a structure e is said to be a model of the set of clauses B ∪ H {\textstyle B\cup H} if for any substitution θ {\textstyle \theta } and any clause h e a d ← b o d y {\textstyle \mathrm {head} \leftarrow \mathrm {body} } in B ∪ H {\textstyle B\cup H} such that b o d y θ ⊆ e {\textstyle \mathrm {body} \theta \subseteq e} , h e a d θ ⊆ e {\displaystyle \mathrm {head} \theta \subseteq e} also holds. The goal is then to output a hypothesis that is complete, meaning every positive example is a model of B ∪ H {\textstyle B\cup H} , and consistent, meaning that no negative example is a model of B ∪ H {\textstyle B\cup H} . == Approaches to ILP == An inductive logic programming system is a program that takes as an input logic theories B , E + , E − {\displaystyle B,E^{+},E^{-}} and outputs a correct hypothesis H with respect to theories B , E + , E − {\displaystyle B,E^{+},E^{-}} . A system is complete if and only if for any input logic theories B , E + , E − {\displaystyle B,E^{+},E^{-}} any correct hypothesis H with respect to these input theories can be found with its hypothesis search procedure. Inductive logic programming systems can be roughly divided into two classes, search-based and meta-interpretative systems. Search-based systems exploit that the space of possible clauses forms a complete lattice under the subsumption relation, where one clause C 1 {\textstyle C_{1}} subsumes another clause C 2 {\textstyle C_{2}} if there is a substitution θ {\textstyle \theta } such that C 1 θ {\textstyle C_{1}\theta } , the result of applying θ {\textstyle \theta } to C 1 {\textstyle C_{1}} , is a subset of C 2 {\textstyle C_{2}} . This lattice can be traversed either bottom-up or top-down. === Bottom-up search === Bottom-up methods to search the subsumption lattice have been investigated since Plotkin's first work on formalising induction in clausal logic in 1970. Techniques used include least general generalisation, based on anti-unification, and inverse resolution, based on inverting the resolution inference rule. ==== Least general generalisation ==== A least general generalisation algorithm takes as input two clauses C 1 {\textstyle C_{1}} and C 2 {\textstyle C_{2}} and outputs the least general generalisation of C 1 {\textstyle C_{1}} and C 2 {\textstyle C_{2}} , that is, a clause C {\textstyle C} that subsumes C 1 {\textstyle C_{1}} and C 2 {\textstyle C_{2}} , and that is subsumed by every other clause that subsumes C 1 {\textstyle C_{1}} and C 2 {\textstyle C_{2}} . The least general generalisation can be computed by first computing all selections from C 1 {\textstyle C_{1}} and C 2 {\textstyle C_{2}} , which are pairs of literals ( L , M ) ∈ ( C 1 × C 2 ) {\displaystyle (L,M)\in (C_{1}\times C_{2})} sharing the same predicate symbol and negated/unnegated status. Then, the least general generalisation is obtained as the disjunction of the least general generalisations of the indi

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  Ordination (statistics)

  Ordination or gradient analysis, in multivariate analysis, is a method complementary to data clustering, and used mainly in exploratory data analysis (rather than in hypothesis testing). In contrast to cluster analysis, ordination orders quantities in a (usually lower-dimensional) latent space. In the ordination space, quantities that are near each other share attributes (i.e., are similar to some degree), and dissimilar objects are farther from each other. Such relationships between the objects, on each of several axes or latent variables, are then characterized numerically and/or graphically in a biplot. The first ordination method, principal components analysis, was suggested by Karl Pearson in 1901. == Methods == Ordination methods can broadly be categorized in eigenvector-, algorithm-, or model-based methods. Many classical ordination techniques, including principal components analysis, correspondence analysis (CA) and its derivatives (detrended correspondence analysis, canonical correspondence analysis, and redundancy analysis, belong to the first group). The second group includes some distance-based methods such as non-metric multidimensional scaling, and machine learning methods such as T-distributed stochastic neighbor embedding and nonlinear dimensionality reduction. The third group includes model-based ordination methods, which can be considered as multivariate extensions of Generalized Linear Models. Model-based ordination methods are more flexible in their application than classical ordination methods, so that it is for example possible to include random-effects. Unlike in the aforementioned two groups, there is no (implicit or explicit) distance measure in the ordination. Instead, a distribution needs to be specified for the responses as is typical for statistical models. These and other assumptions, such as the assumed mean-variance relationship, can be validated with the use of residual diagnostics, unlike in other ordination methods. == Applications == Ordination can be used on the analysis of any set of multivariate objects. It is frequently used in several environmental or ecological sciences, particularly plant community ecology. It is also used in genetics and systems biology for microarray data analysis and in psychometrics.

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  Pruning (artificial neural network)

  In deep learning, pruning is the practice of removing parameters from an existing artificial neural network. The goal of this process is to reduce the size (parameter count) of the neural network (and therefore the computational resources required to run it) whilst maintaining accuracy. This can be compared to the biological process of synaptic pruning which takes place in mammalian brains during development. == Node (neuron) pruning == A basic algorithm for pruning is as follows: Evaluate the importance of each neuron. Rank the neurons according to their importance (assuming there is a clearly defined measure for "importance"). Remove the least important neuron. Check a termination condition (to be determined by the user) to see whether to continue pruning. == Edge (weight) pruning == Most work on neural network pruning does not remove full neurons or layers (structured pruning). Instead, it focuses on removing the most insignificant weights (unstructured pruning), namely, setting their values to zero. This can either be done globally by comparing weights from all layers in the network or locally by comparing weights in each layer separately. Different metrics can be used to measure the importance of each weight. Weight magnitude as well as combinations of weight and gradient information are commonly used metrics. Early work suggested also to change the values of non-pruned weights. == When to prune the neural network? == Pruning can be applied at three different stages: before training, during training, or after training. When pruning is performed during or after training, additional fine-tuning epochs are typically required. Each approach involves different trade-offs between accuracy and computational cost.

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  Exploration–exploitation dilemma

  The exploration–exploitation dilemma, also known as the explore–exploit tradeoff, is a fundamental concept in decision-making that arises in many domains. It is depicted as the balancing act between two opposing strategies. Exploitation involves choosing the best option based on current knowledge of the system (which may be incomplete or misleading), while exploration involves trying out new options that may lead to better outcomes in the future at the expense of an exploitation opportunity. Finding the optimal balance between these two strategies is a crucial challenge in many decision-making problems whose goal is to maximize long-term benefits. == Application in machine learning == In the context of machine learning, the exploration–exploitation tradeoff is fundamental in reinforcement learning (RL), a type of machine learning that involves training agents to make decisions based on feedback from the environment. Crucially, this feedback may be incomplete or delayed. The agent must decide whether to exploit the current best-known policy or explore new policies to improve its performance. === Multi-armed bandit methods === The multi-armed bandit (MAB) problem was a classic example of the tradeoff, and many methods were developed for it, such as epsilon-greedy, Thompson sampling, and the upper confidence bound (UCB). See the page on MAB for details. In more complex RL situations than the MAB problem, the agent can treat each choice as a MAB, where the payoff is the expected future reward. For example, if the agent performs an epsilon-greedy method, then the agent will often "pull the best lever" by picking the action that had the best predicted expected reward (exploit). However, it would pick a random action with probability epsilon (explore). Monte Carlo tree search, for example, uses a variant of the UCB method. === Exploration problems === There are some problems that make exploration difficult. Sparse reward. If rewards occur only once a long while, then the agent might not persist in exploring. Furthermore, if the space of actions is large, then the sparse reward would mean the agent would not be guided by the reward to find a good direction for deeper exploration. A standard example is Montezuma's Revenge. Deceptive reward. If some early actions give immediate small reward, but other actions give later large reward, then the agent might be lured away from exploring the other actions. Noisy TV problem. If certain observations are irreducibly noisy (such as a television showing random images), then the agent might be trapped exploring those observations (watching the television). === Exploration reward === This section based on. The exploration reward (also called exploration bonus) methods convert the exploration-exploitation dilemma into a balance of exploitations. That is, instead of trying to get the agent to balance exploration and exploitation, exploration is simply treated as another form of exploitation, and the agent simply attempts to maximize the sum of rewards from exploration and exploitation. The exploration reward can be treated as a form of intrinsic reward. We write these as r t i , r t e {\displaystyle r_{t}^{i},r_{t}^{e}} , meaning the intrinsic and extrinsic rewards at time step t {\displaystyle t} . However, exploration reward is different from exploitation in two regards: The reward of exploitation is not freely chosen, but given by the environment, but the reward of exploration may be picked freely. Indeed, there are many different ways to design r t i {\displaystyle r_{t}^{i}} described below. The reward of exploitation is usually stationary (i.e. the same action in the same state gives the same reward), but the reward of exploration is non-stationary (i.e. the same action in the same state should give less and less reward). Count-based exploration uses N n ( s ) {\displaystyle N_{n}(s)} , the number of visits to a state s {\displaystyle s} during the time-steps 1 : n {\displaystyle 1:n} , to calculate the exploration reward. This is only possible in small and discrete state space. Density-based exploration extends count-based exploration by using a density model ρ n ( s ) {\displaystyle \rho _{n}(s)} . The idea is that, if a state has been visited, then nearby states are also partly-visited. In maximum entropy exploration, the entropy of the agent's policy π {\displaystyle \pi } is included as a term in the intrinsic reward. That is, r t i = − ∑ a π ( a | s t ) ln ⁡ π ( a | s t ) + ⋯ {\displaystyle r_{t}^{i}=-\sum _{a}\pi (a|s_{t})\ln \pi (a|s_{t})+\cdots } . === Prediction-based === This section based on. The forward dynamics model is a function for predicting the next state based on the current state and the current action: f : ( s t , a t ) ↦ s t + 1 {\displaystyle f:(s_{t},a_{t})\mapsto s_{t+1}} . The forward dynamics model is trained as the agent plays. The model becomes better at predicting state transition for state-action pairs that had been done many times. A forward dynamics model can define an exploration reward by r t i = ‖ f ( s t , a t ) − s t + 1 ‖ 2 2 {\displaystyle r_{t}^{i}=\|f(s_{t},a_{t})-s_{t+1}\|_{2}^{2}} . That is, the reward is the squared-error of the prediction compared to reality. This rewards the agent to perform state-action pairs that had not been done many times. This is however susceptible to the noisy TV problem. Dynamics model can be run in latent space. That is, r t i = ‖ f ( s t , a t ) − ϕ ( s t + 1 ) ‖ 2 2 {\displaystyle r_{t}^{i}=\|f(s_{t},a_{t})-\phi (s_{t+1})\|_{2}^{2}} for some featurizer ϕ {\displaystyle \phi } . The featurizer can be the identity function (i.e. ϕ ( x ) = x {\displaystyle \phi (x)=x} ), randomly generated, the encoder-half of a variational autoencoder, etc. A good featurizer improves forward dynamics exploration. The Intrinsic Curiosity Module (ICM) method trains simultaneously a forward dynamics model and a featurizer. The featurizer is trained by an inverse dynamics model, which is a function for predicting the current action based on the features of the current and the next state: g : ( ϕ ( s t ) , ϕ ( s t + 1 ) ) ↦ a t {\displaystyle g:(\phi (s_{t}),\phi (s_{t+1}))\mapsto a_{t}} . By optimizing the inverse dynamics, both the inverse dynamics model and the featurizer are improved. Then, the improved featurizer improves the forward dynamics model, which improves the exploration of the agent. Random Network Distillation (RND) method attempts to solve this problem by teacher–student distillation. Instead of a forward dynamics model, it has two models f , f ′ {\displaystyle f,f'} . The f ′ {\displaystyle f'} teacher model is fixed, and the f {\displaystyle f} student model is trained to minimize ‖ f ( s ) − f ′ ( s ) ‖ 2 2 {\displaystyle \|f(s)-f'(s)\|_{2}^{2}} on states s {\displaystyle s} . As a state is visited more and more, the student network becomes better at predicting the teacher. Meanwhile, the prediction error is also an exploration reward for the agent, and so the agent learns to perform actions that result in higher prediction error. Thus, we have a student network attempting to minimize the prediction error, while the agent attempting to maximize it, resulting in exploration. The states are normalized by subtracting a running average and dividing a running variance, which is necessary since the teacher model is frozen. The rewards are normalized by dividing with a running variance. Exploration by disagreement trains an ensemble of forward dynamics models, each on a random subset of all ( s t , a t , s t + 1 ) {\displaystyle (s_{t},a_{t},s_{t+1})} tuples. The exploration reward is the variance of the models' predictions. === Noise === For neural network–based agents, the NoisyNet method changes some of its neural network modules by noisy versions. That is, some network parameters are random variables from a probability distribution. The parameters of the distribution are themselves learnable. For example, in a linear layer y = W x + b {\displaystyle y=Wx+b} , both W , b {\displaystyle W,b} are sampled from Gaussian distributions N ( μ W , Σ W ) , N ( μ b , Σ b ) {\displaystyle {\mathcal {N}}(\mu _{W},\Sigma _{W}),{\mathcal {N}}(\mu _{b},\Sigma _{b})} at every step, and the parameters μ W , Σ W , μ b , Σ b {\displaystyle \mu _{W},\Sigma _{W},\mu _{b},\Sigma _{b}} are learned via the reparameterization trick.

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  Vapnik–Chervonenkis theory

  Vapnik–Chervonenkis theory (also known as VC theory) was developed during 1960–1990 by Vladimir Vapnik and Alexey Chervonenkis. The theory is a form of computational learning theory, which attempts to explain the learning process from a statistical point of view. == Introduction == VC theory covers at least four parts (as explained in The Nature of Statistical Learning Theory): Theory of consistency of learning processes What are (necessary and sufficient) conditions for consistency of a learning process based on the empirical risk minimization principle? Nonasymptotic theory of the rate of convergence of learning processes How fast is the rate of convergence of the learning process? Theory of controlling the generalization ability of learning processes How can one control the rate of convergence (the generalization ability) of the learning process? Theory of constructing learning machines How can one construct algorithms that can control the generalization ability? VC Theory is a major subbranch of statistical learning theory. One of its main applications in statistical learning theory is to provide generalization conditions for learning algorithms. From this point of view, VC theory is related to stability, which is an alternative approach for characterizing generalization. In addition, VC theory and VC dimension are instrumental in the theory of empirical processes, in the case of processes indexed by VC classes. Arguably these are the most important applications of the VC theory, and are employed in proving generalization. Several techniques will be introduced that are widely used in the empirical process and VC theory. The discussion is mainly based on the book Weak Convergence and Empirical Processes: With Applications to Statistics. == Overview of VC theory in empirical processes == === Background on empirical processes === Let ( X , A ) {\displaystyle ({\mathcal {X}},{\mathcal {A}})} be a measurable space. For any measure Q {\displaystyle Q} on ( X , A ) {\displaystyle ({\mathcal {X}},{\mathcal {A}})} , and any measurable functions f : X → R {\displaystyle f:{\mathcal {X}}\to \mathbf {R} } , define Q f = ∫ f d Q {\displaystyle Qf=\int fdQ} Measurability issues will be ignored here, for more technical detail see. Let F {\displaystyle {\mathcal {F}}} be a class of measurable functions f : X → R {\displaystyle f:{\mathcal {X}}\to \mathbf {R} } and define: ‖ Q ‖ F = sup { | Q f | : f ∈ F } . {\displaystyle \|Q\|_{\mathcal {F}}=\sup\{\vert Qf\vert \ :\ f\in {\mathcal {F}}\}.} Let X 1 , … , X n {\displaystyle X_{1},\ldots ,X_{n}} be independent, identically distributed random elements of ( X , A ) {\displaystyle ({\mathcal {X}},{\mathcal {A}})} . Then define the empirical measure P n = n − 1 ∑ i = 1 n δ X i , {\displaystyle \mathbb {P} _{n}=n^{-1}\sum _{i=1}^{n}\delta _{X_{i}},} where δ here stands for the Dirac measure. The empirical measure induces a map F → R {\displaystyle {\mathcal {F}}\to \mathbf {R} } given by: f ↦ P n f = 1 n ( f ( X 1 ) + . . . + f ( X n ) ) {\displaystyle f\mapsto \mathbb {P} _{n}f={\frac {1}{n}}(f(X_{1})+...+f(X_{n}))} Now suppose P is the underlying true distribution of the data, which is unknown. Empirical Processes theory aims at identifying classes F {\displaystyle {\mathcal {F}}} for which statements such as the following hold: uniform law of large numbers: ‖ P n − P ‖ F → n 0 , {\displaystyle \|\mathbb {P} _{n}-P\|_{\mathcal {F}}{\underset {n}{\to }}0,} That is, as n → ∞ {\displaystyle n\to \infty } , | 1 n ( f ( X 1 ) + . . . + f ( X n ) ) − ∫ f d P | → 0 {\displaystyle \left|{\frac {1}{n}}(f(X_{1})+...+f(X_{n}))-\int fdP\right|\to 0} uniformly for all f ∈ F {\displaystyle f\in {\mathcal {F}}} . uniform central limit theorem: G n = n ( P n − P ) ⇝ G , in ℓ ∞ ( F ) {\displaystyle \mathbb {G} _{n}={\sqrt {n}}(\mathbb {P} _{n}-P)\rightsquigarrow \mathbb {G} ,\quad {\text{in }}\ell ^{\infty }({\mathcal {F}})} In the former case F {\displaystyle {\mathcal {F}}} is called Glivenko–Cantelli class, and in the latter case (under the assumption ∀ x , sup f ∈ F | f ( x ) − P f | < ∞ {\displaystyle \forall x,\sup \nolimits _{f\in {\mathcal {F}}}\vert f(x)-Pf\vert <\infty } ) the class F {\displaystyle {\mathcal {F}}} is called Donsker or P-Donsker. A Donsker class is Glivenko–Cantelli in probability by an application of Slutsky's theorem. These statements are true for a single f {\displaystyle f} , by standard LLN, CLT arguments under regularity conditions, and the difficulty in the Empirical Processes comes in because joint statements are being made for all f ∈ F {\displaystyle f\in {\mathcal {F}}} . Intuitively then, the set F {\displaystyle {\mathcal {F}}} cannot be too large, and as it turns out that the geometry of F {\displaystyle {\mathcal {F}}} plays a very important role. One way of measuring how big the function set F {\displaystyle {\mathcal {F}}} is to use the so-called covering numbers. The covering number N ( ε , F , ‖ ⋅ ‖ ) {\displaystyle N(\varepsilon ,{\mathcal {F}},\|\cdot \|)} is the minimal number of balls { g : ‖ g − f ‖ < ε } {\displaystyle \{g:\|g-f\|<\varepsilon \}} needed to cover the set F {\displaystyle {\mathcal {F}}} (here it is obviously assumed that there is an underlying norm on F {\displaystyle {\mathcal {F}}} ). The entropy is the logarithm of the covering number. Two sufficient conditions are provided below, under which it can be proved that the set F {\displaystyle {\mathcal {F}}} is Glivenko–Cantelli or Donsker. A class F {\displaystyle {\mathcal {F}}} is P-Glivenko–Cantelli if it is P-measurable with envelope F such that P ∗ F < ∞ {\displaystyle P^{\ast }F<\infty } and satisfies: ∀ ε > 0 sup Q N ( ε ‖ F ‖ Q , F , L 1 ( Q ) ) < ∞ . {\displaystyle \forall \varepsilon >0\quad \sup \nolimits _{Q}N(\varepsilon \|F\|_{Q},{\mathcal {F}},L_{1}(Q))<\infty .} The next condition is a version of Dudley's theorem. If F {\displaystyle {\mathcal {F}}} is a class of functions such that ∫ 0 ∞ sup Q log ⁡ N ( ε ‖ F ‖ Q , 2 , F , L 2 ( Q ) ) d ε < ∞ {\displaystyle \int _{0}^{\infty }\sup \nolimits _{Q}{\sqrt {\log N\left(\varepsilon \|F\|_{Q,2},{\mathcal {F}},L_{2}(Q)\right)}}d\varepsilon <\infty } then F {\displaystyle {\mathcal {F}}} is P-Donsker for every probability measure P such that P ∗ F 2 < ∞ {\displaystyle P^{\ast }F^{2}<\infty } . In the last integral, the notation means ‖ f ‖ Q , 2 = ( ∫ | f | 2 d Q ) 1 2 {\displaystyle \|f\|_{Q,2}=\left(\int |f|^{2}dQ\right)^{\frac {1}{2}}} . === Symmetrization === The majority of the arguments about how to bound the empirical process rely on symmetrization, maximal and concentration inequalities, and chaining. Symmetrization is usually the first step of the proofs, and since it is used in many machine learning proofs on bounding empirical loss functions (including the proof of the VC inequality which is discussed in the next section). It is presented here: Consider the empirical process: f ↦ ( P n − P ) f = 1 n ∑ i = 1 n ( f ( X i ) − P f ) {\displaystyle f\mapsto (\mathbb {P} _{n}-P)f={\dfrac {1}{n}}\sum _{i=1}^{n}(f(X_{i})-Pf)} Turns out that there is a connection between the empirical and the following symmetrized process: f ↦ P n 0 f = 1 n ∑ i = 1 n ε i f ( X i ) {\displaystyle f\mapsto \mathbb {P} _{n}^{0}f={\dfrac {1}{n}}\sum _{i=1}^{n}\varepsilon _{i}f(X_{i})} The symmetrized process is a Rademacher process, conditionally on the data X i {\displaystyle X_{i}} . Therefore, it is a sub-Gaussian process by Hoeffding's inequality. Lemma (Symmetrization). For every nondecreasing, convex Φ: R → R and class of measurable functions F {\displaystyle {\mathcal {F}}} , E Φ ( ‖ P n − P ‖ F ) ≤ E Φ ( 2 ‖ P n 0 ‖ F ) {\displaystyle \mathbb {E} \Phi (\|\mathbb {P} _{n}-P\|_{\mathcal {F}})\leq \mathbb {E} \Phi \left(2\left\|\mathbb {P} _{n}^{0}\right\|_{\mathcal {F}}\right)} The proof of the Symmetrization lemma relies on introducing independent copies of the original variables X i {\displaystyle X_{i}} (sometimes referred to as a ghost sample) and replacing the inner expectation of the LHS by these copies. After an application of Jensen's inequality different signs could be introduced (hence the name symmetrization) without changing the expectation. The proof can be found below because of its instructive nature. The same proof method can be used to prove the Glivenko–Cantelli theorem. A typical way of proving empirical CLTs, first uses symmetrization to pass the empirical process to P n 0 {\displaystyle \mathbb {P} _{n}^{0}} and then argue conditionally on the data, using the fact that Rademacher processes are simple processes with nice properties. === VC Connection === It turns out that there is a fascinating connection between certain combinatorial properties of the set F {\displaystyle {\mathcal {F}}} and the entropy numbers. Uniform covering numbers can be controlled by the notion of Vapnik–Chervonenkis classes of sets – or shortly VC sets. Consider a collection C {\displaystyle {\mathcal {C}}} of subsets of the sample space X {\displaystyle

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  Analogical modeling

  Analogical modeling (AM) is a formal theory of exemplar based analogical reasoning, proposed by Royal Skousen, professor of Linguistics and English language at Brigham Young University in Provo, Utah. It is applicable to language modeling and other categorization tasks. Analogical modeling is related to connectionism and nearest neighbor approaches, in that it is data-based rather than abstraction-based; but it is distinguished by its ability to cope with imperfect datasets (such as caused by simulated short term memory limits) and to base predictions on all relevant segments of the dataset, whether near or far. In language modeling, AM has successfully predicted empirically valid forms for which no theoretical explanation was known (see the discussion of Finnish morphology in Skousen et al. 2002). == Implementation == === Overview === An exemplar-based model consists of a general-purpose modeling engine and a problem-specific dataset. Within the dataset, each exemplar (a case to be reasoned from, or an informative past experience) appears as a feature vector: a row of values for the set of parameters that define the problem. For example, in a spelling-to-sound task, the feature vector might consist of the letters of a word. Each exemplar in the dataset is stored with an outcome, such as a phoneme or phone to be generated. When the model is presented with a novel situation (in the form of an outcome-less feature vector), the engine algorithmically sorts the dataset to find exemplars that helpfully resemble it, and selects one, whose outcome is the model's prediction. The particulars of the algorithm distinguish one exemplar-based modeling system from another. In AM, we think of the feature values as characterizing a context, and the outcome as a behavior that occurs within that context. Accordingly, the novel situation is known as the given context. Given the known features of the context, the AM engine systematically generates all contexts that include it (all of its supracontexts), and extracts from the dataset the exemplars that belong to each. The engine then discards those supracontexts whose outcomes are inconsistent (this measure of consistency will be discussed further below), leaving an analogical set of supracontexts, and probabilistically selects an exemplar from the analogical set with a bias toward those in large supracontexts. This multilevel search exponentially magnifies the likelihood of a behavior's being predicted as it occurs reliably in settings that specifically resemble the given context. === Analogical modeling in detail === AM performs the same process for each case it is asked to evaluate. The given context, consisting of n variables, is used as a template to generate 2 n {\displaystyle 2^{n}} supracontexts. Each supracontext is a set of exemplars in which one or more variables have the same values that they do in the given context, and the other variables are ignored. In effect, each is a view of the data, created by filtering for some criteria of similarity to the given context, and the total set of supracontexts exhausts all such views. Alternatively, each supracontext is a theory of the task or a proposed rule whose predictive power needs to be evaluated. It is important to note that the supracontexts are not equal peers one with another; they are arranged by their distance from the given context, forming a hierarchy. If a supracontext specifies all of the variables that another one does and more, it is a subcontext of that other one, and it lies closer to the given context. (The hierarchy is not strictly branching; each supracontext can itself be a subcontext of several others, and can have several subcontexts.) This hierarchy becomes significant in the next step of the algorithm. The engine now chooses the analogical set from among the supracontexts. A supracontext may contain exemplars that only exhibit one behavior; it is deterministically homogeneous and is included. It is a view of the data that displays regularity, or a relevant theory that has never yet been disproven. A supracontext may exhibit several behaviors, but contain no exemplars that occur in any more specific supracontext (that is, in any of its subcontexts); in this case it is non-deterministically homogeneous and is included. Here there is no great evidence that a systematic behavior occurs, but also no counterargument. Finally, a supracontext may be heterogeneous, meaning that it exhibits behaviors that are found in a subcontext (closer to the given context), and also behaviors that are not. Where the ambiguous behavior of the nondeterministically homogeneous supracontext was accepted, this is rejected because the intervening subcontext demonstrates that there is a better theory to be found. The heterogeneous supracontext is therefore excluded. This guarantees that we see an increase in meaningfully consistent behavior in the analogical set as we approach the given context. With the analogical set chosen, each appearance of an exemplar (for a given exemplar may appear in several of the analogical supracontexts) is given a pointer to every other appearance of an exemplar within its supracontexts. One of these pointers is then selected at random and followed, and the exemplar to which it points provides the outcome. This gives each supracontext an importance proportional to the square of its size, and makes each exemplar likely to be selected in direct proportion to the sum of the sizes of all analogically consistent supracontexts in which it appears. Then, of course, the probability of predicting a particular outcome is proportional to the summed probabilities of all the exemplars that support it. (Skousen 2002, in Skousen et al. 2002, pp. 11–25, and Skousen 2003, both passim) === Formulas === Given a context with n {\displaystyle n} elements: total number of pairings: n 2 {\displaystyle n^{2}} number of agreements for outcome i: n i 2 {\displaystyle n_{i}^{2}} number of disagreements for outcome i: n i ( n − n i ) {\displaystyle n_{i}(n-n_{i})} total number of agreements: ∑ n i 2 {\displaystyle \sum {n_{i}^{2}}} total number of disagreements: ∑ n i ( n − n i ) = n 2 − ∑ n i 2 {\displaystyle \sum {n_{i}(n-n_{i})}=n^{2}-\sum {n_{i}^{2}}} === Example === This terminology is best understood through an example. In the example used in the second chapter of Skousen (1989), each context consists of three variables with potential values 0-3 Variable 1: 0,1,2,3 Variable 2: 0,1,2,3 Variable 3: 0,1,2,3 The two outcomes for the dataset are e and r, and the exemplars are: 3 1 0 e 0 3 2 r 2 1 0 r 2 1 2 r 3 1 1 r We define a network of pointers like so: The solid lines represent pointers between exemplars with matching outcomes; the dotted lines represent pointers between exemplars with non-matching outcomes. The statistics for this example are as follows: n = 5 {\displaystyle n=5} n r = 4 {\displaystyle n_{r}=4} n e = 1 {\displaystyle n_{e}=1} total number of pairings: n 2 = 25 {\displaystyle n^{2}=25} number of agreements for outcome r: n r 2 = 16 {\displaystyle n_{r}^{2}=16} number of agreements for outcome e: n e 2 = 1 {\displaystyle n_{e}^{2}=1} number of disagreements for outcome r: n r ( n − n r ) = 4 {\displaystyle n_{r}(n-n_{r})=4} number of disagreements for outcome e: n e ( n − n e ) = 4 {\displaystyle n_{e}(n-n_{e})=4} total number of agreements: n r 2 + n e 2 = 17 {\displaystyle n_{r}^{2}+n_{e}^{2}=17} total number of disagreements: n r ( n − n r ) + n e ( n − n e ) = n 2 − ( n r 2 + n e 2 ) = 8 {\displaystyle n_{r}(n-n_{r})+n_{e}(n-n_{e})=n^{2}-(n_{r}^{2}+n_{e}^{2})=8} uncertainty or fraction of disagreement: 8 / 25 = .32 {\displaystyle 8/25=.32} Behavior can only be predicted for a given context; in this example, let us predict the outcome for the context "3 1 2". To do this, we first find all of the contexts containing the given context; these contexts are called supracontexts. We find the supracontexts by systematically eliminating the variables in the given context; with m variables, there will generally be 2 m {\displaystyle 2^{m}} supracontexts. The following table lists each of the sub- and supracontexts; x means "not x", and - means "anything". These contexts are shown in the venn diagram below: The next step is to determine which exemplars belong to which contexts in order to determine which of the contexts are homogeneous. The table below shows each of the subcontexts, their behavior in terms of the given exemplars, and the number of disagreements within the behavior: Analyzing the subcontexts in the table above, we see that there is only 1 subcontext with any disagreements: "3 1 2", which in the dataset consists of "3 1 0 e" and "3 1 1 r". There are 2 disagreements in this subcontext; 1 pointing from each of the exemplars to the other (see the pointer network pictured above). Therefore, only supracontexts containing this subcontext will contain any disagreements. We use a simple rule to identify the homogeneous supraco

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  Boosting (machine learning)

  In machine learning (ML), boosting is an ensemble learning method that combines a set of less accurate models (called "weak learners") to create a single, highly accurate model (a "strong learner"). Unlike other ensemble methods that build models in parallel (such as bagging), boosting algorithms build models sequentially. Each new model in the sequence is trained to correct the errors made by its predecessors. This iterative process allows the overall model to improve its accuracy, particularly by reducing bias. Boosting is a popular and effective technique used in supervised learning for both classification and regression tasks. The theoretical foundation for boosting came from a question posed by Kearns and Valiant (1988, 1989): "Can a set of weak learners create a single strong learner?" A weak learner is defined as a classifier that performs only slightly better than random guessing, whereas a strong learner is a classifier that is highly correlated with the true classification. Robert Schapire's affirmative answer to this question in a 1990 paper led to the development of practical boosting algorithms. The first such algorithm was developed by Schapire, with Freund and Schapire later developing AdaBoost, which remains a foundational example of boosting. == Algorithms == While boosting is not algorithmically constrained, most boosting algorithms consist of iteratively learning weak classifiers with respect to a distribution and adding them to a final strong classifier. When they are added, they are weighted in a way that is related to the weak learners' accuracy. After a weak learner is added, the data weights are readjusted, known as "re-weighting". Misclassified input data gain a higher weight and examples that are classified correctly lose weight. Thus, future weak learners focus more on the examples that previous weak learners misclassified. There are many boosting algorithms. The original ones, proposed by Robert Schapire (a recursive majority gate formulation), and Yoav Freund (boost by majority), were not adaptive and could not take full advantage of the weak learners. Schapire and Freund then developed AdaBoost, an adaptive boosting algorithm that won the prestigious Gödel Prize. Only algorithms that are provable boosting algorithms in the probably approximately correct learning formulation can accurately be called boosting algorithms. Other algorithms that are similar in spirit to boosting algorithms are sometimes called "leveraging algorithms", although they are also sometimes incorrectly called boosting algorithms. The main variation between many boosting algorithms is their method of weighting training data points and hypotheses. AdaBoost is very popular and the most significant historically as it was the first algorithm that could adapt to the weak learners. It is often the basis of introductory coverage of boosting in university machine learning courses. There are many more recent algorithms such as LPBoost, TotalBoost, BrownBoost, xgboost, MadaBoost, LogitBoost, CatBoost and others. Many boosting algorithms fit into the AnyBoost framework, which shows that boosting performs gradient descent in a function space using a convex cost function. == Object categorization in computer vision == Given images containing various known objects in the world, a classifier can be learned from them to automatically classify the objects in future images. Simple classifiers built based on some image feature of the object tend to be weak in categorization performance. Using boosting methods for object categorization is a way to unify the weak classifiers in a special way to boost the overall ability of categorization. === Problem of object categorization === Object categorization is a typical task of computer vision that involves determining whether or not an image contains some specific category of object. The idea is closely related with recognition, identification, and detection. Appearance based object categorization typically contains feature extraction, learning a classifier, and applying the classifier to new examples. There are many ways to represent a category of objects, e.g. from shape analysis, bag of words models, or local descriptors such as SIFT, etc. Examples of supervised classifiers are Naive Bayes classifiers, support vector machines, mixtures of Gaussians, and neural networks. However, research has shown that object categories and their locations in images can be discovered in an unsupervised manner as well. === Status quo for object categorization === The recognition of object categories in images is a challenging problem in computer vision, especially when the number of categories is large. This is due to high intra class variability and the need for generalization across variations of objects within the same category. Objects within one category may look quite different. Even the same object may appear unalike under different viewpoint, scale, and illumination. Background clutter and partial occlusion add difficulties to recognition as well. Humans are able to recognize thousands of object types, whereas most of the existing object recognition systems are trained to recognize only a few, e.g. human faces, cars, simple objects, etc. Research has been very active on dealing with more categories and enabling incremental additions of new categories, and although the general problem remains unsolved, several multi-category objects detectors (for up to hundreds or thousands of categories) have been developed. One means is by feature sharing and boosting. === Boosting for binary categorization === AdaBoost can be used for face detection as an example of binary categorization. The two categories are faces versus background. The general algorithm is as follows: Form a large set of simple features Initialize weights for training images For T rounds Normalize the weights For available features from the set, train a classifier using a single feature and evaluate the training error Choose the classifier with the lowest error Update the weights of the training images: increase if classified wrongly by this classifier, decrease if correctly Form the final strong classifier as the linear combination of the T classifiers (coefficient larger if training error is small) After boosting, a classifier constructed from 200 features could yield a 95% detection rate under a 10 − 5 {\displaystyle 10^{-5}} false positive rate. Another application of boosting for binary categorization is a system that detects pedestrians using patterns of motion and appearance. This work is the first to combine both motion information and appearance information as features to detect a walking person. It takes a similar approach to the Viola-Jones object detection framework. === Boosting for multi-class categorization === Compared with binary categorization, multi-class categorization looks for common features that can be shared across the categories at the same time. They turn to be more generic edge like features. During learning, the detectors for each category can be trained jointly. Compared with training separately, it generalizes better, needs less training data, and requires fewer features to achieve the same performance. The main flow of the algorithm is similar to the binary case. What is different is that a measure of the joint training error shall be defined in advance. During each iteration the algorithm chooses a classifier of a single feature (features that can be shared by more categories shall be encouraged). This can be done via converting multi-class classification into a binary one (a set of categories versus the rest), or by introducing a penalty error from the categories that do not have the feature of the classifier. In the paper "Sharing visual features for multiclass and multiview object detection", A. Torralba et al. used GentleBoost for boosting and showed that when training data is limited, learning via sharing features does a much better job than no sharing, given same boosting rounds. Also, for a given performance level, the total number of features required (and therefore the run time cost of the classifier) for the feature sharing detectors, is observed to scale approximately logarithmically with the number of class, i.e., slower than linear growth in the non-sharing case. Similar results are shown in the paper "Incremental learning of object detectors using a visual shape alphabet", yet the authors used AdaBoost for boosting. == Convex vs. non-convex boosting algorithms == Boosting algorithms can be based on convex or non-convex optimization algorithms. Convex algorithms, such as AdaBoost and LogitBoost, can be "defeated" by random noise such that they can't learn basic and learnable combinations of weak hypotheses. This limitation was pointed out by Long & Servedio in 2008. However, by 2009, multiple authors demonstrated that boosting algorithms based on non-convex optimization, such as BrownBoost, can learn from nois

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  SMART Health Card

  The SMART Health Card framework is an open source immunity passport program designed to store and share medical information in paper or digital form. It was initially launched as a vaccine passport during the COVID-19 pandemic, but is envisioned for use for other infectious diseases. SMART Health Cards include a QR code which can be scanned and verified using the official SMART Health Card Verifier mobile app, supported by Apple and Android. It was rolled out by the Vaccination Credential Initiative (VCI) based on technology developed at Boston Children's Hospital, and standards set by Health Level Seven International (HL7) and the World Wide Web Consortium (W3C). It is recognized by the International Organization for Standardization. == History == === Founding === In February 2009, United States president Barack Obama signed an economic stimulus package which included $19 billion in funds for investment in health information technology. The following month, researchers from Boston Children's Hospital and Harvard Medical School, Kenneth Mandl and Isaac Kohane, published an article in The New England Journal of Medicine calling for the modernization of electronic health records through API integrations on mobile devices. In April 2010, the pair secured a $15 million grant through the Office of the National Coordinator for Health Information Technology's Strategic Health IT Advanced Research Projects (SHARP) program. With this federal funding, the researchers began development of an interoperable healthcare IT platform they called "Substitutable Medical Applications and Reusable Technologies" (SMART). The first iteration of the platform API was previewed later that year, and "SMART Classic" was released in 2011. In 2013, SMART adopted the open-source Fast Health Interoperability Resources (FHIR) standard developed by Health Level Seven International (HL7). The newly named SMART on FHIR platform was debuted in February 2014 at the Health Information Management Systems Society conference. === 21st Century Cures Act === According to SMART Health IT, Mandl successfully lobbied for the inclusion of a universal API requirement in the 21st Century Cures Act, signed into law on December 13, 2016. The team also advocated for a federal rule establishing SMART as the universal API. In 2019, the Office of the National Coordinator for Health Information Technology published the "final rule" specifying the SMART framework as the standard to satisfy the requirements of the 21st Century Cures Act; the rule was implemented in June 2020. === COVID-19 === The SMART Health Card framework was deployed as a "de facto standard" for vaccine passports in the COVID-19 pandemic in the United States and other international jurisdictions. On January 14, 2021, the Mitre Corporation announced the launch of a new public–private partnership called the Vaccination Credential Initiative (VCI) alongside the CARIN Alliance, Cerner, Change Healthcare, The Commons Project Foundation, Epic Systems, Evernorth, Mayo Clinic, Microsoft, Oracle, Safe Health, and Salesforce. VCI's purpose was to employ the SMART Health Card framework in order to create a unified proof-of-vaccination system for COVID-19 vaccines.The California Department of Public Health introduced a Digital Covid-19 Vaccine Record portal in June 2021, allowing individuals to verify their vaccination status using the SMART Health Card reader. On August 5, 2021, New York Governor Andrew Cuomo announced the introduction of the "Excelsior Pass Plus" which would expand its Excelsior Pass program into other states and internationally by connecting it to the SMART Health Card system. As of August 27, 2021, 415,000 citizens of Louisiana had added their COVID-19 vaccination status to their state-run, SMART Health Card enabled LA Wallet. On September 8, 2021, Hawaii governor David Ige announced the rollout of the state's Hawaiʻi SMART Health Card. County-level health departments across the United States partnered with VaccineCheck to issue SMART Health Cards by verifying vaccine cards provided by the Centers for Disease Control and Prevention. The Government of Canada spent CAD$4.6 million to develop a proof-of-vaccination credential on the SMART Health Card framework, enabling its ArriveCAN travel application to store, recognize and verify credentials from every province, territory and foreign country. Since October 2021, Canadian provinces and territories used the SMART Health Card format as a requirement by the federal government, including British Columbia, Newfoundland and Labrador, the Northwest Territories, Nova Scotia, Nunavut, Ontario, Quebec, Saskatchewan and the Yukon. On October 13, 2021, the American Immunization Registry Association (AIRA) published a statement encouraging adoption of SMART Health Cards as a common standard "where allowed by local law and policy." "SMARTHealth.Cards" was listed as a supporting member of AIRA through the VCI. A SMART Health Cards Global Forum was held on October 28, 2021. The event featured keynote speakers Andy Slavitt (former Senior Pandemic Advisor to President Joe Biden’s COVID-19 pandemic response team) and Mike Leavitt (former United States Secretary of Health and Human Services). On December 20, 2021, Japan's Ministry of Health, Labour and Welfare launched its COVID-19 Vaccination Certificate Application using the SMART Health Card. By January 2022, about 80% of Americans who had received a COVID-19 vaccine had access to a SMART Health Card through their state governments, local businesses, universities and healthcare systems. == Participants == === Developers === SMART Health IT is based out of the Computational Health Informatics Program (CHIP) at the Boston Children's Hospital. CHIP's related projects include Apache cTAKES, Genomic Information Commons, HealthMap, and VaccineFinder. The SMART Health Card's project sponsor is HL7 International's Public Health Work Group, consisting of representatives from Allscripts, the Altarum Institute, Tennessee Department of Health and Washington State Department of Health. === Issuers === Official registries of authorized SMART Health Card issuers are maintained by SMART Health IT, the Vaccination Credential Initiative, and the CommonTrust Network. Authorized issuers include:

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  Facial recognition system

  A facial recognition system is a technology potentially capable of matching a human face from a digital image or a video frame against a database of faces. Such a system is typically employed to authenticate users through ID verification services, and works by pinpointing and measuring facial features from a given image. Development on similar systems began in the 1960s as a form of computer application. Since their inception, facial recognition systems have seen wider uses in recent times on smartphones and in other forms of technology, such as robotics. Because computerized facial recognition involves the measurement of a human's physiological characteristics, facial recognition systems are categorized as biometrics. Although the accuracy of facial recognition systems as a biometric technology is lower than iris recognition, fingerprint image acquisition, palm recognition or voice recognition, it is widely adopted due to its contactless process. Facial recognition systems have been deployed in advanced human–computer interaction, video surveillance, law enforcement, passenger screening, decisions on employment and housing, and automatic indexing of images. Facial recognition systems are employed throughout the world today by governments and private companies. Their effectiveness varies, and some systems have previously been scrapped because of their ineffectiveness. The use of facial recognition systems has also raised controversy, with claims that the systems violate citizens' privacy, commonly make incorrect identifications, encourage gender norms and racial profiling, and do not protect important biometric data. The appearance of synthetic media such as deepfakes has also raised concerns about its security. These claims have led to the ban of facial recognition systems in several cities in the United States. Growing societal concerns led social networking company Meta Platforms to shut down its Facebook facial recognition system in 2021, deleting the face-scan data of more than one billion users. The change represented one of the largest shifts in facial recognition usage in the technology's history. IBM also stopped offering facial recognition technology due to similar concerns. == History of facial recognition technology == Automated facial recognition was pioneered in the 1960s by Woody Bledsoe, Helen Chan Wolf, and Charles Bisson, whose work focused on teaching computers to recognize human faces. Their early facial recognition project was dubbed "man-machine" because a human first needed to establish the coordinates of facial features in a photograph before they could be used by a computer for recognition. Using a graphics tablet, a human would pinpoint facial features coordinates, such as the pupil centers, the inside and outside corners of eyes, and the widows peak in the hairline. The coordinates were used to calculate 20 individual distances, including the width of the mouth and of the eyes. A human could process about 40 pictures an hour, building a database of these computed distances. A computer would then automatically compare the distances for each photograph, calculate the difference between the distances, and return the closed records as a possible match. In 1970, Takeo Kanade publicly demonstrated a face-matching system that located anatomical features such as the chin and calculated the distance ratio between facial features without human intervention. Later tests revealed that the system could not always reliably identify facial features. Nonetheless, interest in the subject grew and in 1977 Kanade published the first detailed book on facial recognition technology. In 1993, the Defense Advanced Research Project Agency (DARPA) and the Army Research Laboratory (ARL) established the face recognition technology program FERET to develop "automatic face recognition capabilities" that could be employed in a productive real life environment "to assist security, intelligence, and law enforcement personnel in the performance of their duties." Face recognition systems that had been trialled in research labs were evaluated. The FERET tests found that while the performance of existing automated facial recognition systems varied, a handful of existing methods could viably be used to recognize faces in still images taken in a controlled environment. The FERET tests spawned three US companies that sold automated facial recognition systems. Vision Corporation and Miros Inc were founded in 1994, by researchers who used the results of the FERET tests as a selling point. Viisage Technology was established by an identification card defense contractor in 1996 to commercially exploit the rights to the facial recognition algorithm developed by Alex Pentland at MIT. Following the 1993 FERET face-recognition vendor test, the Department of Motor Vehicles (DMV) offices in West Virginia and New Mexico became the first DMV offices to use automated facial recognition systems to prevent people from obtaining multiple driving licenses using different names. Driver's licenses in the United States were at that point a commonly accepted form of photo identification. DMV offices across the United States were undergoing a technological upgrade and were in the process of establishing databases of digital ID photographs. This enabled DMV offices to deploy the facial recognition systems on the market to search photographs for new driving licenses against the existing DMV database. DMV offices became one of the first major markets for automated facial recognition technology and introduced US citizens to facial recognition as a standard method of identification. The increase of the US prison population in the 1990s prompted U.S. states to established connected and automated identification systems that incorporated digital biometric databases, in some instances this included facial recognition. In 1999, Minnesota incorporated the facial recognition system FaceIT by Visionics into a mug shot booking system that allowed police, judges and court officers to track criminals across the state. Until the 1990s, facial recognition systems were developed primarily by using photographic portraits of human faces. Research on face recognition to reliably locate a face in an image that contains other objects gained traction in the early 1990s with the principal component analysis (PCA). The PCA method of face detection is also known as Eigenface and was developed by Matthew Turk and Alex Pentland. Turk and Pentland combined the conceptual approach of the Karhunen–Loève theorem and factor analysis, to develop a linear model. Eigenfaces are determined based on global and orthogonal features in human faces. A human face is calculated as a weighted combination of a number of Eigenfaces. Because few Eigenfaces were used to encode human faces of a given population, Turk and Pentland's PCA face detection method greatly reduced the amount of data that had to be processed to detect a face. Pentland in 1994 defined Eigenface features, including eigen eyes, eigen mouths and eigen noses, to advance the use of PCA in facial recognition. In 1997, the PCA Eigenface method of face recognition was improved upon using linear discriminant analysis (LDA) to produce Fisherfaces. LDA Fisherfaces became dominantly used in PCA feature based face recognition. While Eigenfaces were also used for face reconstruction. In these approaches no global structure of the face is calculated which links the facial features or parts. Purely feature based approaches to facial recognition were overtaken in the late 1990s by the Bochum system, which used Gabor filter to record the face features and computed a grid of the face structure to link the features. Christoph von der Malsburg and his research team at the University of Bochum developed Elastic Bunch Graph Matching in the mid-1990s to extract a face out of an image using skin segmentation. By 1997, the face detection method developed by Malsburg outperformed most other facial detection systems on the market. The so-called "Bochum system" of face detection was sold commercially on the market as ZN-Face to operators of airports and other busy locations. The software was "robust enough to make identifications from less-than-perfect face views. It can also often see through such impediments to identification as mustaches, beards, changed hairstyles and glasses—even sunglasses". Real-time face detection in video footage became possible in 2001 with the Viola–Jones object detection framework for faces. Paul Viola and Michael Jones combined their face detection method with the Haar-like feature approach to object recognition in digital images to launch AdaBoost, the first real-time frontal-view face detector. By 2015, the Viola–Jones algorithm had been implemented using small low power detectors on handheld devices and embedded systems. Therefore, the Viola–Jones algorithm has not only broadened the practical application of face recognition systems but

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  Dynamic time warping

  In time series analysis, dynamic time warping (DTW) is an algorithm for measuring similarity between two temporal sequences, which may vary in speed. For instance, similarities in walking could be detected using DTW, even if one person was walking faster than the other, or if there were accelerations and decelerations during the course of an observation. DTW has been applied to temporal sequences of video, audio, and graphics data — indeed, any data that can be turned into a one-dimensional sequence can be analyzed with DTW. A well-known application has been automatic speech recognition, to cope with different speaking speeds. Other applications include speaker recognition and online signature recognition. It can also be used in partial shape matching applications. In general, DTW is a method that calculates an optimal match between two given sequences (e.g. time series) with certain restriction and rules: Every index from the first sequence must be matched with one or more indices from the other sequence, and vice versa The first index from the first sequence must be matched with the first index from the other sequence (but it does not have to be its only match) The last index from the first sequence must be matched with the last index from the other sequence (but it does not have to be its only match) The mapping of the indices from the first sequence to indices from the other sequence must be monotonically increasing, and vice versa, i.e. if j > i {\displaystyle j>i} are indices from the first sequence, then there must not be two indices l > k {\displaystyle l>k} in the other sequence, such that index i {\displaystyle i} is matched with index l {\displaystyle l} and index j {\displaystyle j} is matched with index k {\displaystyle k} , and vice versa We can plot each match between the sequences 1 : M {\displaystyle 1:M} and 1 : N {\displaystyle 1:N} as a path in a M × N {\displaystyle M\times N} matrix from ( 1 , 1 ) {\displaystyle (1,1)} to ( M , N ) {\displaystyle (M,N)} , such that each step is one of ( 0 , 1 ) , ( 1 , 0 ) , ( 1 , 1 ) {\displaystyle (0,1),(1,0),(1,1)} . In this formulation, we see that the number of possible matches is the Delannoy number. The optimal match is denoted by the match that satisfies all the restrictions and the rules and that has the minimal cost, where the cost is computed as the sum of absolute differences, for each matched pair of indices, between their values. The sequences are "warped" non-linearly in the time dimension to determine a measure of their similarity independent of certain non-linear variations in the time dimension. This sequence alignment method is often used in time series classification. Although DTW measures a distance-like quantity between two given sequences, it doesn't guarantee the triangle inequality to hold. In addition to a similarity measure between the two sequences (a so called "warping path" is produced), by warping according to this path the two signals may be aligned in time. The signal with an original set of points X(original), Y(original) is transformed to X(warped), Y(warped). This finds applications in genetic sequence and audio synchronisation. In a related technique sequences of varying speed may be averaged using this technique see the average sequence section. This is conceptually very similar to the Needleman–Wunsch algorithm. == Implementation == This example illustrates the implementation of the dynamic time warping algorithm when the two sequences s and t are strings of discrete symbols. For two symbols x and y, d ( x , y ) {\displaystyle d(x,y)} is a distance between the symbols, e.g., d ( x , y ) = | x − y | {\displaystyle d(x,y)=|x-y|} . int DTWDistance(s: array [1..n], t: array [1..m]) { DTW := array [0..n, 0..m] for i := 0 to n for j := 0 to m DTW[i, j] := infinity DTW[0, 0] := 0 for i := 1 to n for j := 1 to m cost := d(s[i], t[j]) DTW[i, j] := cost + minimum(DTW[i-1, j ], // insertion DTW[i , j-1], // deletion DTW[i-1, j-1]) // match return DTW[n, m] } where DTW[i, j] is the distance between s[1:i] and t[1:j] with the best alignment. We sometimes want to add a locality constraint. That is, we require that if s[i] is matched with t[j], then | i − j | {\displaystyle |i-j|} is no larger than w, a window parameter. We can easily modify the above algorithm to add a locality constraint (differences marked). However, the above given modification works only if | n − m | {\displaystyle |n-m|} is no larger than w, i.e. the end point is within the window length from diagonal. In order to make the algorithm work, the window parameter w must be adapted so that | n − m | ≤ w {\displaystyle |n-m|\leq w} (see the line marked with () in the code). int DTWDistance(s: array [1..n], t: array [1..m], w: int) { DTW := array [0..n, 0..m] w := max(w, abs(n-m)) // adapt window size () for i := 0 to n for j:= 0 to m DTW[i, j] := infinity DTW[0, 0] := 0 for i := 1 to n for j := max(1, i-w) to min(m, i+w) DTW[i, j] := 0 for i := 1 to n for j := max(1, i-w) to min(m, i+w) cost := d(s[i], t[j]) DTW[i, j] := cost + minimum(DTW[i-1, j ], // insertion DTW[i , j-1], // deletion DTW[i-1, j-1]) // match return DTW[n, m] } == Warping properties == The DTW algorithm produces a discrete matching between existing elements of one series to another. In other words, it does not allow time-scaling of segments within the sequence. Other methods allow continuous warping. For example, Correlation Optimized Warping (COW) divides the sequence into uniform segments that are scaled in time using linear interpolation, to produce the best matching warping. The segment scaling causes potential creation of new elements, by time-scaling segments either down or up, and thus produces a more sensitive warping than DTW's discrete matching of raw elements. == Complexity == The time complexity of the DTW algorithm is O ( N M ) {\displaystyle O(NM)} , where N {\displaystyle N} and M {\displaystyle M} are the lengths of the two input sequences. The 50 years old quadratic time bound was broken in 2016: an algorithm due to Gold and Sharir enables computing DTW in O ( N 2 / log ⁡ log ⁡ N ) {\displaystyle O({N^{2}}/\log \log N)} time and space for two input sequences of length N {\displaystyle N} . This algorithm can also be adapted to sequences of different lengths. Despite this improvement, it was shown that a strongly subquadratic running time of the form O ( N 2 − ϵ ) {\displaystyle O(N^{2-\epsilon })} for some ϵ > 0 {\displaystyle \epsilon >0} cannot exist unless the Strong exponential time hypothesis fails. While the dynamic programming algorithm for DTW requires O ( N M ) {\displaystyle O(NM)} space in a naive implementation, the space consumption can be reduced to O ( min ( N , M ) ) {\displaystyle O(\min(N,M))} using Hirschberg's algorithm. == Fast computation == Fast techniques for computing DTW include PrunedDTW, SparseDTW, FastDTW, and the MultiscaleDTW. A common task, retrieval of similar time series, can be accelerated by using lower bounds such as LB_Keogh, LB_Improved, or LB_Petitjean. However, the Early Abandon and Pruned DTW algorithm reduces the degree of acceleration that lower bounding provides and sometimes renders it ineffective. In a survey, Wang et al. reported slightly better results with the LB_Improved lower bound than the LB_Keogh bound, and found that other techniques were inefficient. Subsequent to this survey, the LB_Enhanced bound was developed that is always tighter than LB_Keogh while also being more efficient to compute. LB_Petitjean is the tightest known lower bound that can be computed in linear time. == Average sequence == Averaging for dynamic time warping is the problem of finding an average sequence for a set of sequences. NLAAF is an exact method to average two sequences using DTW. For more than two sequences, the problem is related to that of multiple alignment and requires heuristics. DBA is currently a reference method to average a set of sequences consistently with DTW. COMASA efficiently randomizes the search for the average sequence, using DBA as a local optimization process. == Supervised learning == A nearest-neighbour classifier can achieve state-of-the-art performance when using dynamic time warping as a distance measure. == Amerced Dynamic Time Warping == Amerced Dynamic Time Warping (ADTW) is a variant of DTW designed to better control DTW's permissiveness in the alignments that it allows. The windows that classical DTW uses to constrain alignments introduce a step function. Any warping of the path is allowed within the window and none beyond it. In contrast, ADTW employs an additive penalty that is incurred each time that the path is warped. Any amount of warping is allowed, but each warping action incurs a direct penalty. ADTW significantly outperforms DTW with windowing when applied as a nearest neighbor classifier on a set of benchmark time series classification tasks. == Alternative approaches == In functional data analysis, time series are regarde

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