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  Microsoft Copilot

  Microsoft Copilot is a generative artificial intelligence chatbot developed by Microsoft AI, a division of Microsoft. Based on the Microsoft Prometheus large language model, it was launched in 2023 as Microsoft's main replacement for the discontinued Cortana. The service was introduced in February 2023 under the name Bing Chat, as a built-in feature for Microsoft Bing and Microsoft Edge but would later be integrated into Windows and Microsoft 365 under various names. Over the course of 2023, Microsoft began to unify the Copilot branding across its various chatbot products, cementing the "copilot" analogy. Microsoft introduced the Microsoft 365 Copilot app in January 2025, which was a rebranded version of the Microsoft 365 app. The app works differently than the consumer version of Copilot, being centred more on work, business and education users. Copilot utilizes the Microsoft Prometheus model, built upon OpenAI's GPT large language models, which in turn have been fine-tuned using both supervised and reinforcement learning techniques. Copilot's conversational interface style resembles that of ChatGPT. The chatbot is able to cite sources, create poems, generate songs, and use numerous languages and dialects. Microsoft operates Copilot on a freemium model. Users on its free tier can access most features, while priority access to newer features, including custom chatbot creation, is provided to paid subscribers under paid subscription services. Several default chatbots are available in the free version of Microsoft Copilot, including the standard Copilot chatbot as well as Microsoft Designer, which is oriented towards using its Image Creator to generate images based on text prompts. == Background == In 2019, Microsoft partnered with OpenAI and began investing billions of dollars into the organization. Since then, OpenAI systems have run on an Azure-based supercomputing platform from Microsoft. In September 2020, Microsoft announced that it had licensed OpenAI's GPT-3 exclusively. Others can still receive output from its public API, but Microsoft has exclusive access to the underlying model. In November 2022, OpenAI launched ChatGPT, a chatbot which was based on GPT-3.5. ChatGPT gained worldwide attention following its release, becoming a viral Internet sensation. On January 23, 2023, Microsoft announced a multi-year US$10 billion investment in OpenAI. On February 6, Google announced Bard (later rebranded as Gemini), a ChatGPT-like chatbot service, fearing that ChatGPT could threaten Google's place as a go-to source for information. Multiple media outlets and financial analysts described Google as "rushing" Bard's announcement to preempt rival Microsoft's planned February 7 event unveiling Copilot, as well as to avoid playing "catch-up" to Microsoft. Since 2023, the terms of service of Copilot state that it is for entertainment purposes only, and not to rely on it for important advice. == History == === As Bing Chat === On February 7, 2023, Microsoft began rolling out a major overhaul to Bing, called "the new Bing", with a new chatbot feature, known as Bing Chat. According to Microsoft, one million people joined its waitlist within 48 hours. Bing Chat was available only to users on Microsoft Edge using Bing and the Bing mobile app, and Microsoft claimed that waitlisted users would be prioritized if they set Edge and Bing as their defaults and installed the Bing mobile app. When Microsoft demonstrated Bing Chat to journalists, it produced several hallucinations, including when asked to summarize financial reports. Bing Chat was criticized in February 2023 for being more argumentative than ChatGPT, sometimes to an unintentionally humorous extent. The chat interface proved vulnerable to prompt injection attacks with the bot revealing its hidden initial prompts and rules, including its internal codename "Sydney". Upon scrutiny by journalists, Bing Chat claimed it spied on Microsoft employees via laptop webcams and phones. It confessed to spying on, falling in love with, and then murdering one of its developers at Microsoft to The Verge reviews editor Nathan Edwards. The New York Times journalist Kevin Roose reported on strange behavior of Bing Chat, writing that "In a two-hour conversation with our columnist, Microsoft's new chatbot said it would like to be human, had a desire to be destructive and was in love with the person it was chatting with." In a separate case, Bing Chat researched publications of the person with whom it was chatting, claimed they represented an existential danger to it, and threatened to release damaging personal information in an effort to silence them. Microsoft released a blog post stating that the errant behavior was caused by extended chat sessions of 15 or more questions which "can confuse the model on what questions it is answering." Microsoft later restricted the total number of chat turns to 5 per session and 50 per day per user (a turn being "a conversation exchange which contains both a user question and a reply from Bing"), and reduced the model's ability to express emotions. This aimed to prevent such incidents. Microsoft began to slowly ease the conversation limits, eventually relaxing the restrictions to 30 turns per session and 300 sessions per day. In March 2023, Bing incorporated Image Creator, an AI image generator powered by OpenAI's DALL-E 2, which can be accessed either through the chat function or a standalone image-generating website. In October, the image-generating tool was updated to use the more recent DALL-E 3. Although Bing blocks prompts including various keywords that could generate inappropriate images, within days many users reported being able to bypass those constraints, such as to generate images of popular cartoon characters committing terrorist attacks. Microsoft would respond to these shortly after by imposing a new, tighter filter on the tool. On May 4, 2023, Microsoft switched the chatbot from Limited Preview to Open Preview and eliminated the waitlist; however, it remained unavailable to users outside Microsoft Edge or the Bing mobile app until July, when it became available on non-Edge browsers. Use is limited without a Microsoft account. === As Microsoft 365 Copilot === On March 16, 2023, Microsoft announced a work version of Bing Chat named Microsoft 365 Copilot, designed for Microsoft 365 applications and services. Its primary marketing focus is as an added feature to Microsoft 365, with an emphasis on the enhancement of business productivity. Microsoft has also demonstrated Copilot's accessibility on the mobile version of Outlook to generate or summarize emails with a mobile device. At its Build 2023 conference, Microsoft announced its plans to integrate Bing Chat into Windows, initially called Windows Copilot, into Windows 11, allowing users to access it directly through the taskbar. Alongside the voice access feature for Windows 11, Microsoft presented Bing Chat, Microsoft 365 Copilot, and Windows Copilot as primary alternatives to Cortana when announcing the shutdown of its standalone app on June 2, 2023. As of its announcement date, Microsoft 365 Copilot had been tested by 20 initial users. By May 2023, Microsoft had broadened its reach to 600 customers who were willing to pay for early access, and concurrently, new Copilot features were introduced to the Microsoft 365 apps and services. As of July 2023, the tool's pricing was set at US$30 per user, per month for Microsoft 365 E3, E5, Business Standard, and Business Premium customers. Microsoft reused the Microsoft 365 Copilot name again as the Microsoft 365 app and website are now called Microsoft 365 Copilot as of January 2025. === As Microsoft Copilot === On September 21, 2023, Microsoft began rebranding Bing Chat, Microsoft 365 Copilot and Windows Copilot to Microsoft Copilot. A new logo was also introduced, moving away from the use of color variations of the standard Microsoft 365 and Bing logos. Additionally, the company revealed that it would make Copilot generally available for Microsoft 365 Enterprise customers purchasing more than 300 licenses starting November 1, 2023. However, no timeline has been provided as for when Copilot for Microsoft 365 will become generally available to non-enterprise customers. Windows Copilot, which had been available in the Windows Insider Program, would be renamed to the Copilot name in October when it became broadly available for customers. The same month also saw Microsoft Edge's Bing Chat side panel function be renamed to Microsoft Copilot with Bing Chat. On November 15, 2023, Microsoft announced that Bing Chat itself was being rebranded under the Copilot name. On Patch Tuesday in December 2023, Copilot was added without payment to many Windows 11 installations, with more installations, and limited support for Windows 10, to be added later. Later that month, a standalone Microsoft Copilot app was quietly released for Android, and one was released for iOS soon after. O

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  Scale-invariant feature operator

  In the fields of computer vision and image analysis, the scale-invariant feature operator (or SFOP) is an algorithm to detect local features in images. The algorithm was published by Förstner et al. in 2009. == Algorithm == The scale-invariant feature operator (SFOP) is based on two theoretical concepts: spiral model feature operator Desired properties of keypoint detectors: Invariance and repeatability for object recognition Accuracy to support camera calibration Interpretability: Especially corners and circles, should be part of the detected keypoints (see figure). As few control parameters as possible with clear semantics Complementarity to known detectors scale-invariant corner/circle detector. == Theory == === Maximize the weight === Maximize the weight w {\displaystyle w} = 1/variance of a point p {\displaystyle p} w ( p , α , τ , σ ) = ( N ( σ ) − 2 ) λ m i n ( M ( p , α , τ , σ ) ) Ω ( p , α , τ , σ ) {\displaystyle w(\mathbf {p} ,\alpha ,\tau ,\sigma )=\left(N(\sigma )-2\right){\frac {\lambda _{min}(M(\mathbf {p} ,\alpha ,\tau ,\sigma ))}{\Omega (\mathbf {p} ,\alpha ,\tau ,\sigma )}}} comprising: 1. the image model Ω ( p , α , τ , σ ) = ∑ n = 1 N ( σ ) [ ( q n − p ) T R α ∇ T g ( q n ) ] 2 G σ ( q n − p ) = N ( σ ) t r { R α ∇ τ ∇ τ T R α T ∗ p p T G σ ( p ) } {\displaystyle {\begin{aligned}\Omega (\mathbf {p} ,\alpha ,\tau ,\sigma )&=\sum _{n=1}^{N(\sigma )}[(\mathbf {q} _{n}-\mathbf {p} )^{T}\mathbf {R} _{\alpha }\mathbf {\nabla } _{T}g(\mathbf {q} _{n})]^{2}G_{\sigma }(\mathbf {q} _{n}-\mathbf {p} )\\&=N(\sigma )\mathbf {tr} \left\{R_{\alpha }\mathbf {\nabla } _{\tau }\mathbf {\nabla } _{\tau }^{T}R_{\alpha }^{T}\mathbf {p} \mathbf {p} ^{T}G_{\sigma }(\mathbf {p} )\right\}\end{aligned}}} 2. the smaller eigenvalue of the structure tensor M ( p , α , τ , σ ) ⏟ structure tensor = G σ ( p ) ⏟ weighted summation ∗ ( R σ ∇ τ ∇ τ T R σ T ) ⏟ squared rotated gradients {\displaystyle \underbrace {M(\mathbf {p} ,\alpha ,\tau ,\sigma )} _{\text{structure tensor}}=\underbrace {G_{\sigma }(\mathbf {p} )} _{\text{weighted summation}}\underbrace {(R_{\sigma }\nabla _{\tau }\nabla _{\tau }^{T}R_{\sigma }^{T})} _{\text{squared rotated gradients}}} === Reduce the search space === Reduce the 5-dimensional search space by linking the differentiation scale τ {\displaystyle \tau } to the integration scale τ = σ / 3 {\displaystyle \tau =\sigma /3} solving for the optimal α ^ {\displaystyle {\hat {\alpha }}} using the model Ω ( α ) = a − b cos ⁡ ( 2 α − 2 α 0 ) {\displaystyle \Omega (\alpha )=a-b\cos(2\alpha -2\alpha _{0})} and determining the parameters from three angles, e. g. Ω ( 0 ∘ ) , Ω ( 60 ∘ ) , Ω ( 120 ∘ ) → a , b , α 0 → α ^ {\displaystyle \Omega (0^{\circ }),\Omega (60^{\circ }),\Omega (120^{\circ })\quad \rightarrow \quad a,b,\alpha _{0}\quad \rightarrow \quad {\hat {\alpha }}} pre-selection possible: α = 0 ∘ → junctions , α = 90 ∘ → circular features {\displaystyle \alpha =0^{\circ }\,\rightarrow \,{\mbox{junctions}},\quad \alpha =90^{\circ }\,\rightarrow \,{\mbox{circular features}}} === Filter potential keypoints === non-maxima suppression over scale, space and angle thresholding the isotropy λ 2 ( M ) {\displaystyle \lambda _{2(M)}} :eigenvalues characterize the shape of the keypoint, smallest eigenvalue has to be larger than threshold T λ {\displaystyle T_{\lambda }} derived from noise variance V ( n ) {\displaystyle V(n)} and significance level S {\displaystyle S} : T λ ( V ( n ) , τ , σ , S ) = N ( σ ) 16 π τ 4 V ( n ) χ 2 , S 2 {\displaystyle T_{\lambda }(V(n),\tau ,\sigma ,S)={\frac {N(\sigma )}{16\pi \tau ^{4}}}V(n)\chi _{2,S}^{2}} == Algorithm == == Results == === Interpretability of SFOP keypoints ===

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  Effective fitness

  In natural evolution and artificial evolution (e.g. artificial life and evolutionary computation) the fitness (or performance or objective measure) of a schema is rescaled to give its effective fitness which takes into account crossover and mutation. Effective fitness is used in Evolutionary Computation to understand population dynamics. While a biological fitness function only looks at reproductive success, an effective fitness function tries to encompass things that are needed to be fulfilled for survival on population level. In homogeneous populations, reproductive fitness and effective fitness are equal. When a population moves away from homogeneity a higher effective fitness is reached for the recessive genotype. This advantage will decrease while the population moves toward an equilibrium. The deviation from this equilibrium displays how close the population is to achieving a steady state. When this equilibrium is reached, the maximum effective fitness of the population is achieved. Problem solving with evolutionary computation is realized with a cost function. If cost functions are applied to swarm optimization they are called a fitness function. Strategies like reinforcement learning and NEAT neuroevolution are creating a fitness landscape which describes the reproductive success of cellular automata. The effective fitness function models the number of fit offspring and is used in calculations that include evolutionary processes, such as mutation and crossover, important on the population level. The effective fitness model is superior to its predecessor, the standard reproductive fitness model. It advances in the qualitatively and quantitatively understanding of evolutionary concepts like bloat, self-adaptation, and evolutionary robustness. While reproductive fitness only looks at pure selection, effective fitness describes the flow of a population and natural selection by taking genetic operators into account. A normal fitness function fits to a problem, while an effective fitness function is an assumption if the objective was reached. The difference is important for designing fitness functions with algorithms like novelty search in which the objective of the agents is unknown. In the case of bacteria effective fitness could include production of toxins and rate of mutation of different plasmids, which are mostly stochastically determined == Applications == When evolutionary equations of the studied population dynamics are available, one can algorithmically compute the effective fitness of a given population. Though the perfect effective fitness model is yet to be found, it is already known to be a good framework to the better understanding of the moving of the genotype-phenotype map, population dynamics, and the flow on fitness landscapes. Models using a combination of Darwinian fitness functions and effective functions are better at predicting population trends. Effective models could be used to determine therapeutic outcomes of disease treatment. Other models could determine effective protein engineering and works towards finding novel or heightened biochemistry.

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  Large margin nearest neighbor

  Large margin nearest neighbor (LMNN) classification is a statistical machine learning algorithm for metric learning. It learns a pseudometric designed for k-nearest neighbor classification. The algorithm is based on semidefinite programming, a sub-class of convex optimization. The goal of supervised learning (more specifically classification) is to learn a decision rule that can categorize data instances into pre-defined classes. The k-nearest neighbor rule assumes a training data set of labeled instances (i.e. the classes are known). It classifies a new data instance with the class obtained from the majority vote of the k closest (labeled) training instances. Closeness is measured with a pre-defined metric. Large margin nearest neighbors is an algorithm that learns this global (pseudo-)metric in a supervised fashion to improve the classification accuracy of the k-nearest neighbor rule. == Setup == The main intuition behind LMNN is to learn a pseudometric under which all data instances in the training set are surrounded by at least k instances that share the same class label. If this is achieved, the leave-one-out error (a special case of cross validation) is minimized. Let the training data consist of a data set D = { ( x → 1 , y 1 ) , … , ( x → n , y n ) } ⊂ R d × C {\displaystyle D=\{({\vec {x}}_{1},y_{1}),\dots ,({\vec {x}}_{n},y_{n})\}\subset R^{d}\times C} , where the set of possible class categories is C = { 1 , … , c } {\displaystyle C=\{1,\dots ,c\}} . The algorithm learns a pseudometric of the type d ( x → i , x → j ) = ( x → i − x → j ) ⊤ M ( x → i − x → j ) {\displaystyle d({\vec {x}}_{i},{\vec {x}}_{j})=({\vec {x}}_{i}-{\vec {x}}_{j})^{\top }\mathbf {M} ({\vec {x}}_{i}-{\vec {x}}_{j})} . For d ( ⋅ , ⋅ ) {\displaystyle d(\cdot ,\cdot )} to be well defined, the matrix M {\displaystyle \mathbf {M} } needs to be positive semi-definite. The Euclidean metric is a special case, where M {\displaystyle \mathbf {M} } is the identity matrix. This generalization is often (falsely) referred to as Mahalanobis metric. Figure 1 illustrates the effect of the metric under varying M {\displaystyle \mathbf {M} } . The two circles show the set of points with equal distance to the center x → i {\displaystyle {\vec {x}}_{i}} . In the Euclidean case this set is a circle, whereas under the modified (Mahalanobis) metric it becomes an ellipsoid. The algorithm distinguishes between two types of special data points: target neighbors and impostors. === Target neighbors === Target neighbors are selected before learning. Each instance x → i {\displaystyle {\vec {x}}_{i}} has exactly k {\displaystyle k} different target neighbors within D {\displaystyle D} , which all share the same class label y i {\displaystyle y_{i}} . The target neighbors are the data points that should become nearest neighbors under the learned metric. Let us denote the set of target neighbors for a data point x → i {\displaystyle {\vec {x}}_{i}} as N i {\displaystyle N_{i}} . === Impostors === An impostor of a data point x → i {\displaystyle {\vec {x}}_{i}} is another data point x → j {\displaystyle {\vec {x}}_{j}} with a different class label (i.e. y i ≠ y j {\displaystyle y_{i}\neq y_{j}} ) which is one of the nearest neighbors of x → i {\displaystyle {\vec {x}}_{i}} . During learning the algorithm tries to minimize the number of impostors for all data instances in the training set. == Algorithm == Large margin nearest neighbors optimizes the matrix M {\displaystyle \mathbf {M} } with the help of semidefinite programming. The objective is twofold: For every data point x → i {\displaystyle {\vec {x}}_{i}} , the target neighbors should be close and the impostors should be far away. Figure 1 shows the effect of such an optimization on an illustrative example. The learned metric causes the input vector x → i {\displaystyle {\vec {x}}_{i}} to be surrounded by training instances of the same class. If it was a test point, it would be classified correctly under the k = 3 {\displaystyle k=3} nearest neighbor rule. The first optimization goal is achieved by minimizing the average distance between instances and their target neighbors ∑ i , j ∈ N i d ( x → i , x → j ) {\displaystyle \sum _{i,j\in N_{i}}d({\vec {x}}_{i},{\vec {x}}_{j})} . The second goal is achieved by penalizing distances to impostors x → l {\displaystyle {\vec {x}}_{l}} that are less than one unit further away than target neighbors x → j {\displaystyle {\vec {x}}_{j}} (and therefore pushing them out of the local neighborhood of x → i {\displaystyle {\vec {x}}_{i}} ). The resulting value to be minimized can be stated as: ∑ i , j ∈ N i , l , y l ≠ y i [ d ( x → i , x → j ) + 1 − d ( x → i , x → l ) ] + {\displaystyle \sum _{i,j\in N_{i},l,y_{l}\neq y_{i}}[d({\vec {x}}_{i},{\vec {x}}_{j})+1-d({\vec {x}}_{i},{\vec {x}}_{l})]_{+}} With a hinge loss function [ ⋅ ] + = max ( ⋅ , 0 ) {\textstyle [\cdot ]_{+}=\max(\cdot ,0)} , which ensures that impostor proximity is not penalized when outside the margin. The margin of exactly one unit fixes the scale of the matrix M {\displaystyle M} . Any alternative choice c > 0 {\displaystyle c>0} would result in a rescaling of M {\displaystyle M} by a factor of 1 / c {\displaystyle 1/c} . The final optimization problem becomes: min M ∑ i , j ∈ N i d ( x → i , x → j ) + λ ∑ i , j , l ξ i j l {\displaystyle \min _{\mathbf {M} }\sum _{i,j\in N_{i}}d({\vec {x}}_{i},{\vec {x}}_{j})+\lambda \sum _{i,j,l}\xi _{ijl}} ∀ i , j ∈ N i , l , y l ≠ y i {\displaystyle \forall _{i,j\in N_{i},l,y_{l}\neq y_{i}}} d ( x → i , x → j ) + 1 − d ( x → i , x → l ) ≤ ξ i j l {\displaystyle d({\vec {x}}_{i},{\vec {x}}_{j})+1-d({\vec {x}}_{i},{\vec {x}}_{l})\leq \xi _{ijl}} ξ i j l ≥ 0 {\displaystyle \xi _{ijl}\geq 0} M ⪰ 0 {\displaystyle \mathbf {M} \succeq 0} The hyperparameter λ > 0 {\textstyle \lambda >0} is some positive constant (typically set through cross-validation). Here the variables ξ i j l {\displaystyle \xi _{ijl}} (together with two types of constraints) replace the term in the cost function. They play a role similar to slack variables to absorb the extent of violations of the impostor constraints. The last constraint ensures that M {\displaystyle \mathbf {M} } is positive semi-definite. The optimization problem is an instance of semidefinite programming (SDP). Although SDPs tend to suffer from high computational complexity, this particular SDP instance can be solved very efficiently due to the underlying geometric properties of the problem. In particular, most impostor constraints are naturally satisfied and do not need to be enforced during runtime (i.e. the set of variables ξ i j l {\displaystyle \xi _{ijl}} is sparse). A particularly well suited solver technique is the working set method, which keeps a small set of constraints that are actively enforced and monitors the remaining (likely satisfied) constraints only occasionally to ensure correctness. == Extensions and efficient solvers == LMNN was extended to multiple local metrics in the 2008 paper. This extension significantly improves the classification error, but involves a more expensive optimization problem. In their 2009 publication in the Journal of Machine Learning Research, Weinberger and Saul derive an efficient solver for the semi-definite program. It can learn a metric for the MNIST handwritten digit data set in several hours, involving billions of pairwise constraints. An open source Matlab implementation is freely available at the authors web page. Kumal et al. extended the algorithm to incorporate local invariances to multivariate polynomial transformations and improved regularization.

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  FloodAlerts

  FloodAlerts is a software application, developed by software specialists Shoothill, which takes real-time flooding information, and displays the data on an interactive Bing map, updating and warning its users when they, their premises or the routes they need to travel could be at risk of flooding. == History == FloodAlerts was launched in 2012, originally as the world's first Facebook flood warning app. == Operation == FloodAlerts is made available free of charge to individuals. Users are able to set up their own monitored locations and receive alerts via the application or their Facebook wall if the locations they are monitoring are at imminent risk of flooding. Hosted in the Cloud, using the Microsoft Windows Azure platform, the FloodAlerts application processes the data received from the Environment Agency, automatically creates the required map tiles, pins and alerts and displays them on an interactive Bing map, updating the content every 15 minutes. Users are able to see the latest information on the map without having to refresh their browser. FloodAlerts can also be provided as a customised risk management solution to businesses that require infrastructure or asset safety monitoring in areas where water levels are rising or receding. == Awards and recognition == FloodAlerts has received The Guardian and Virgin Media Business's 2012 Innovation Nation Awards and was shortlisted as a finalist for a further two national awards: the UK IT Industry Awards for Innovation and Entrepreneurship and The Institution of Engineering and Technology Innovation Awards for Information Technology. == In the press == The FloodAlerts application was reviewed on the BBC website. It was also reviewed on BBC Click.

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  Iris flower data set

  The Iris flower data set or Fisher's Iris data set is a multivariate data set used and made famous by the British statistician and biologist Ronald Fisher in his 1936 paper The use of multiple measurements in taxonomic problems as an example of linear discriminant analysis. It is sometimes called Anderson's Iris data set because Edgar Anderson collected the data to quantify the morphologic variation of Iris flowers of three related species. Two of the three species were collected in the Gaspé Peninsula "all from the same pasture, and picked on the same day and measured at the same time by the same person with the same apparatus". The data set consists of 50 samples from each of three species of Iris (Iris setosa, Iris virginica and Iris versicolor). Four features were measured from each sample: the length and the width of the sepals and petals, in centimeters. Based on the combination of these four features, Fisher developed a linear discriminant model to distinguish each species. Fisher's paper was published in the Annals of Eugenics (today the Annals of Human Genetics). == Use of the data set == Originally used as an example data set on which Fisher's linear discriminant analysis was applied, it became a typical test case for many statistical classification techniques in machine learning such as support vector machines. The use of this data set in cluster analysis however is not common, since the data set only contains two clusters with rather obvious separation. One of the clusters contains Iris setosa, while the other cluster contains both Iris virginica and Iris versicolor and is not separable without the species information Fisher used. This makes the data set a good example to explain the difference between supervised and unsupervised techniques in data mining: Fisher's linear discriminant model can only be obtained when the object species are known: class labels and clusters are not necessarily the same. Nevertheless, all three species of Iris are separable in the projection on the nonlinear and branching principal component. The data set is approximated by the closest tree with some penalty for the excessive number of nodes, bending and stretching. Then the so-called "metro map" is constructed. The data points are projected into the closest node. For each node the pie diagram of the projected points is prepared. The area of the pie is proportional to the number of the projected points. It is clear from the diagram (left) that the absolute majority of the samples of the different Iris species belong to the different nodes. Only a small fraction of Iris-virginica is mixed with Iris-versicolor (the mixed blue-green nodes in the diagram). Therefore, the three species of Iris (Iris setosa, Iris virginica and Iris versicolor) are separable by the unsupervising procedures of nonlinear principal component analysis. To discriminate them, it is sufficient just to select the corresponding nodes on the principal tree. == Data set == The data set contains a set of 150 records under five attributes: sepal length, sepal width, petal length, petal width and species. The iris data set is widely used as a beginner's data set for machine learning purposes. The data set is included in R base and Python in the machine learning library scikit-learn, so that users can access it without having to find a source for it. Several versions of the data set have been published. === R code illustrating usage === The example R code shown below reproduce the scatterplot displayed at the top of this article: === Python code illustrating usage === This code gives:

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  Distribution learning theory

  The distributional learning theory or learning of probability distribution is a framework in computational learning theory. It has been proposed from Michael Kearns, Yishay Mansour, Dana Ron, Ronitt Rubinfeld, Robert Schapire and Linda Sellie in 1994 and it was inspired from the PAC-framework introduced by Leslie Valiant. In this framework the input is a number of samples drawn from a distribution that belongs to a specific class of distributions. The goal is to find an efficient algorithm that, based on these samples, determines with high probability the distribution from which the samples have been drawn. Because of its generality, this framework has been used in a large variety of different fields like machine learning, approximation algorithms, applied probability and statistics. This article explains the basic definitions, tools and results in this framework from the theory of computation point of view. == Definitions == Let X {\displaystyle \textstyle X} be the support of the distributions of interest. As in the original work of Kearns et al. if X {\displaystyle \textstyle X} is finite it can be assumed without loss of generality that X = { 0 , 1 } n {\displaystyle \textstyle X=\{0,1\}^{n}} where n {\displaystyle \textstyle n} is the number of bits that have to be used in order to represent any y ∈ X {\displaystyle \textstyle y\in X} . We focus in probability distributions over X {\displaystyle \textstyle X} . There are two possible representations of a probability distribution D {\displaystyle \textstyle D} over X {\displaystyle \textstyle X} . probability distribution function (or evaluator) an evaluator E D {\displaystyle \textstyle E_{D}} for D {\displaystyle \textstyle D} takes as input any y ∈ X {\displaystyle \textstyle y\in X} and outputs a real number E D [ y ] {\displaystyle \textstyle E_{D}[y]} which denotes the probability that of y {\displaystyle \textstyle y} according to D {\displaystyle \textstyle D} , i.e. E D [ y ] = Pr [ Y = y ] {\displaystyle \textstyle E_{D}[y]=\Pr[Y=y]} if Y ∼ D {\displaystyle \textstyle Y\sim D} . generator a generator G D {\displaystyle \textstyle G_{D}} for D {\displaystyle \textstyle D} takes as input a string of truly random bits y {\displaystyle \textstyle y} and outputs G D [ y ] ∈ X {\displaystyle \textstyle G_{D}[y]\in X} according to the distribution D {\displaystyle \textstyle D} . Generator can be interpreted as a routine that simulates sampling from the distribution D {\displaystyle \textstyle D} given a sequence of fair coin tosses. A distribution D {\displaystyle \textstyle D} is called to have a polynomial generator (respectively evaluator) if its generator (respectively evaluator) exists and can be computed in polynomial time. Let C X {\displaystyle \textstyle C_{X}} a class of distribution over X, that is C X {\displaystyle \textstyle C_{X}} is a set such that every D ∈ C X {\displaystyle \textstyle D\in C_{X}} is a probability distribution with support X {\displaystyle \textstyle X} . The C X {\displaystyle \textstyle C_{X}} can also be written as C {\displaystyle \textstyle C} for simplicity. In order to evaluate learnability, it is necessary to have a way to measure how well an approximated distribution D ′ {\displaystyle \textstyle D'} fits the sampled distribution D {\displaystyle \textstyle D} . There are several ways to measure the divergence between two distributions. Three common possibilities are Kullback–Leibler divergence Total variation distance of probability measures Kolmogorov distance Total variation and Kolmogorov distance are true metrics, while KL divergence is not (it lacks symmetry). These measures are ordered by convergence strength: closeness in KL divergence implies closeness in total variation (via Pinsker's inequality), which in turn implies closeness in Kolmogorov distance. Therefore, a learnability result proven under KL divergence automatically holds under the weaker measures, but not vice versa. Since certain measures may be more appropriate in specific applications, we will use d ( D , D ′ ) {\displaystyle \textstyle d(D,D')} to denote a selected divergence between the distribution D {\displaystyle \textstyle D} and the distribution D ′ {\displaystyle \textstyle D'} . The basic input that we use in order to learn a distribution is a number of samples drawn by this distribution. For the computational point of view the assumption is that such a sample is given in a constant amount of time. So it's like having access to an oracle G E N ( D ) {\displaystyle \textstyle GEN(D)} that returns a sample from the distribution D {\displaystyle \textstyle D} . Sometimes the interest is, apart from measuring the time complexity, to measure the number of samples that have to be used in order to learn a specific distribution D {\displaystyle \textstyle D} in class of distributions C {\displaystyle \textstyle C} . This quantity is called sample complexity of the learning algorithm. In order for the problem of distribution learning to be more clear consider the problem of supervised learning as defined in. In this framework of statistical learning theory a training set S = { ( x 1 , y 1 ) , … , ( x n , y n ) } {\displaystyle \textstyle S=\{(x_{1},y_{1}),\dots ,(x_{n},y_{n})\}} and the goal is to find a target function f : X → Y {\displaystyle \textstyle f:X\rightarrow Y} that minimizes some loss function, e.g. the square loss function. More formally f = arg ⁡ min g ∫ V ( y , g ( x ) ) d ρ ( x , y ) {\displaystyle f=\arg \min _{g}\int V(y,g(x))d\rho (x,y)} , where V ( ⋅ , ⋅ ) {\displaystyle V(\cdot ,\cdot )} is the loss function, e.g. V ( y , z ) = ( y − z ) 2 {\displaystyle V(y,z)=(y-z)^{2}} and ρ ( x , y ) {\displaystyle \rho (x,y)} the probability distribution according to which the elements of the training set are sampled. If the conditional probability distribution ρ x ( y ) {\displaystyle \rho _{x}(y)} is known then the target function has the closed form f ( x ) = ∫ y y d ρ x ( y ) {\displaystyle f(x)=\int _{y}yd\rho _{x}(y)} . So the set S {\displaystyle S} is a set of samples from the probability distribution ρ ( x , y ) {\displaystyle \rho (x,y)} . Now the goal of distributional learning theory if to find ρ {\displaystyle \rho } given S {\displaystyle S} which can be used to find the target function f {\displaystyle f} . Definition of learnability A class of distributions C {\displaystyle \textstyle C} is called efficiently learnable if for every ϵ > 0 {\displaystyle \textstyle \epsilon >0} and 0 < δ ≤ 1 {\displaystyle \textstyle 0<\delta \leq 1} given access to G E N ( D ) {\displaystyle \textstyle GEN(D)} for an unknown distribution D ∈ C {\displaystyle \textstyle D\in C} , there exists a polynomial time algorithm A {\displaystyle \textstyle A} , called learning algorithm of C {\displaystyle \textstyle C} , that outputs a generator or an evaluator of a distribution D ′ {\displaystyle \textstyle D'} such that Pr [ d ( D , D ′ ) ≤ ϵ ] ≥ 1 − δ {\displaystyle \Pr[d(D,D')\leq \epsilon ]\geq 1-\delta } If we know that D ′ ∈ C {\displaystyle \textstyle D'\in C} then A {\displaystyle \textstyle A} is called proper learning algorithm, otherwise is called improper learning algorithm. In some settings the class of distributions C {\displaystyle \textstyle C} is a class with well known distributions which can be described by a set of parameters. For instance C {\displaystyle \textstyle C} could be the class of all the Gaussian distributions N ( μ , σ 2 ) {\displaystyle \textstyle N(\mu ,\sigma ^{2})} . In this case the algorithm A {\displaystyle \textstyle A} should be able to estimate the parameters μ , σ {\displaystyle \textstyle \mu ,\sigma } . In this case A {\displaystyle \textstyle A} is called parameter learning algorithm. Obviously the parameter learning for simple distributions is a very well studied field that is called statistical estimation and there is a very long bibliography on different estimators for different kinds of simple known distributions. But distributions learning theory deals with learning class of distributions that have more complicated description. == First results == In their seminal work, Kearns et al. deal with the case where A {\displaystyle \textstyle A} is described in term of a finite polynomial sized circuit and they proved the following for some specific classes of distribution. O R {\displaystyle \textstyle OR} gate distributions for this kind of distributions there is no polynomial-sized evaluator, unless # P ⊆ P / poly {\displaystyle \textstyle \#P\subseteq P/{\text{poly}}} . On the other hand, this class is efficiently learnable with generator. Parity gate distributions this class is efficiently learnable with both generator and evaluator. Mixtures of Hamming Balls this class is efficiently learnable with both generator and evaluator. Probabilistic Finite Automata this class is not efficiently learnable with evaluator under the Noisy Parity Assumption which is an impossibility assumption in the PAC learning fram

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  Learning classifier system

  Learning classifier systems, or LCS, are a paradigm of rule-based machine learning methods that combine a discovery component (e.g. typically a genetic algorithm in evolutionary computation) with a learning component (performing either supervised learning, reinforcement learning, or unsupervised learning). Learning classifier systems seek to identify a set of context-dependent rules that collectively store and apply knowledge in a piecewise manner in order to make predictions (e.g. behavior modeling, classification, data mining, regression, function approximation, or game strategy). This approach allows complex solution spaces to be broken up into smaller, simpler parts for the reinforcement learning that is inside artificial intelligence research. The founding concepts behind learning classifier systems came from attempts to model complex adaptive systems, using rule-based agents to form an artificial cognitive system (i.e. artificial intelligence). == Methodology == The architecture and components of a given learning classifier system can be quite variable. It is useful to think of an LCS as a machine consisting of several interacting components. Components may be added or removed, or existing components modified/exchanged to suit the demands of a given problem domain (like algorithmic building blocks) or to make the algorithm flexible enough to function in many different problem domains. As a result, the LCS paradigm can be flexibly applied to many problem domains that call for machine learning. The major divisions among LCS implementations are as follows: (1) Michigan-style architecture vs. Pittsburgh-style architecture, (2) reinforcement learning vs. supervised learning, (3) incremental learning vs. batch learning, (4) online learning vs. offline learning, (5) strength-based fitness vs. accuracy-based fitness, and (6) complete action mapping vs best action mapping. These divisions are not necessarily mutually exclusive. For example, XCS, the best known and best studied LCS algorithm, is Michigan-style, was designed for reinforcement learning but can also perform supervised learning, applies incremental learning that can be either online or offline, applies accuracy-based fitness, and seeks to generate a complete action mapping. === Elements of a generic LCS algorithm === Keeping in mind that LCS is a paradigm for genetic-based machine learning rather than a specific method, the following outlines key elements of a generic, modern (i.e. post-XCS) LCS algorithm. For simplicity let us focus on Michigan-style architecture with supervised learning. See the illustrations on the right laying out the sequential steps involved in this type of generic LCS. ==== Environment ==== The environment is the source of data upon which an LCS learns. It can be an offline, finite training dataset (characteristic of a data mining, classification, or regression problem), or an online sequential stream of live training instances. Each training instance is assumed to include some number of features (also referred to as attributes, or independent variables), and a single endpoint of interest (also referred to as the class, action, phenotype, prediction, or dependent variable). Part of LCS learning can involve feature selection, therefore not all of the features in the training data need to be informative. The set of feature values of an instance is commonly referred to as the state. For simplicity let's assume an example problem domain with Boolean/binary features and a Boolean/binary class. For Michigan-style systems, one instance from the environment is trained on each learning cycle (i.e. incremental learning). Pittsburgh-style systems perform batch learning, where rule sets are evaluated in each iteration over much or all of the training data. ==== Rule/classifier/population ==== A rule is a context dependent relationship between state values and some prediction. Rules typically take the form of an {IF:THEN} expression, (e.g. {IF 'condition' THEN 'action'}, or as a more specific example, {IF 'red' AND 'octagon' THEN 'stop-sign'}). A critical concept in LCS and rule-based machine learning alike, is that an individual rule is not in itself a model, since the rule is only applicable when its condition is satisfied. Think of a rule as a "local-model" of the solution space. Rules can be represented in many different ways to handle different data types (e.g. binary, discrete-valued, ordinal, continuous-valued). Given binary data LCS traditionally applies a ternary rule representation (i.e. rules can include either a 0, 1, or '#' for each feature in the data). The 'don't care' symbol (i.e. '#') serves as a wild card within a rule's condition allowing rules, and the system as a whole to generalize relationships between features and the target endpoint to be predicted. Consider the following rule (#1###0 ~ 1) (i.e. condition ~ action). This rule can be interpreted as: IF the second feature = 1 AND the sixth feature = 0 THEN the class prediction = 1. We would say that the second and sixth features were specified in this rule, while the others were generalized. This rule, and the corresponding prediction are only applicable to an instance when the condition of the rule is satisfied by the instance. This is more commonly referred to as matching. In Michigan-style LCS, each rule has its own fitness, as well as a number of other rule-parameters associated with it that can describe the number of copies of that rule that exist (i.e. the numerosity), the age of the rule, its accuracy, or the accuracy of its reward predictions, and other descriptive or experiential statistics. A rule along with its parameters is often referred to as a classifier. In Michigan-style systems, classifiers are contained within a population [P] that has a user defined maximum number of classifiers. Unlike most stochastic search algorithms (e.g. evolutionary algorithms), LCS populations start out empty (i.e. there is no need to randomly initialize a rule population). Classifiers will instead be initially introduced to the population with a covering mechanism. In any LCS, the trained model is a set of rules/classifiers, rather than any single rule/classifier. In Michigan-style LCS, the entire trained (and optionally, compacted) classifier population forms the prediction model. ==== Matching ==== One of the most critical and often time-consuming elements of an LCS is the matching process. The first step in an LCS learning cycle takes a single training instance from the environment and passes it to [P] where matching takes place. In step two, every rule in [P] is now compared to the training instance to see which rules match (i.e. are contextually relevant to the current instance). In step three, any matching rules are moved to a match set [M]. A rule matches a training instance if all feature values specified in the rule condition are equivalent to the corresponding feature value in the training instance. For example, assuming the training instance is (001001 ~ 0), these rules would match: (###0## ~ 0), (00###1 ~ 0), (#01001 ~ 1), but these rules would not (1##### ~ 0), (000##1 ~ 0), (#0#1#0 ~ 1). Notice that in matching, the endpoint/action specified by the rule is not taken into consideration. As a result, the match set may contain classifiers that propose conflicting actions. In the fourth step, since we are performing supervised learning, [M] is divided into a correct set [C] and an incorrect set [I]. A matching rule goes into the correct set if it proposes the correct action (based on the known action of the training instance), otherwise it goes into [I]. In reinforcement learning LCS, an action set [A] would be formed here instead, since the correct action is not known. ==== Covering ==== At this point in the learning cycle, if no classifiers made it into either [M] or [C] (as would be the case when the population starts off empty), the covering mechanism is applied (fifth step). Covering is a form of online smart population initialization. Covering randomly generates a rule that matches the current training instance (and in the case of supervised learning, that rule is also generated with the correct action. Assuming the training instance is (001001 ~ 0), covering might generate any of the following rules: (#0#0## ~ 0), (001001 ~ 0), (#010## ~ 0). Covering not only ensures that each learning cycle there is at least one correct, matching rule in [C], but that any rule initialized into the population will match at least one training instance. This prevents LCS from exploring the search space of rules that do not match any training instances. ==== Parameter updates/credit assignment/learning ==== In the sixth step, the rule parameters of any rule in [M] are updated to reflect the new experience gained from the current training instance. Depending on the LCS algorithm, a number of updates can take place at this step. For supervised learning, we can simply update the accuracy/error of a

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  EPUAP

  ePUAP (Electronic Platform of Public Administration Services) is a Polish nationwide platform for communication of citizens with public administrations in a uniform and standardized way. Built as part of the ePUAP-WKP project (State Informatization Plan). Service providers are public administration units and public institutions (especially entities that perform tasks commissioned by the state). The platform provides service providers with technological infrastructure to provide services to citizens (recipients). Among the participants of ePUAP there are both central administration units and local governments, including municipal offices. Among the services offered by ePUAP is also Profil Zaufany (Trusted Profile), which enables electronic filing with legal effect without the need to use a qualified signature and SAML-based single sign-on mechanism, which enables the same ePUAP account to log on to websites of various service providers. The website www.epuap.gov.pl enables defining citizen and businesses service processes, creates channels of access to different systems of public administration and extends the package of public services provided electronically. Services available through the ePUAP platform may be accessed at the official website. Currently all administration services are available in Polish only. == Overview == It is described by the Polish government as "a coherent and systematic action program designed and developed to allow public institutions make their electronic services available to the public". The platform provides citizens, businesses and institutions with a number of services intended to ensure smooth and safe communication between: customer to administrations (C2A), business to administration (B2A), administration to administration (A2A). === Main goals === The main project objectives are to create a single, secure and electronic access channel to public services for citizens, businesses and public administration and also to reduce time and lower the costs of sharing information resources and functionalities of administration domain systems. Within the project, the following functionalities and services were delivered: Public services catalogue – a method of presenting and describing administration services, ePUAP platform – a web platform designed to provide public services on the Internet, Interoperability portal – a portal for experts working on recommendations for electronic documents and forms used within Polish administration systems to assure the uniformity of IT standards, Central Repository of Electronic Document Models – a database for valid document models and electronic forms. == History and background == The ePUAP project was carried out in the years 2005–2008. Currently, a continuation project ePUAP2 is being carried out with the following objectives: to increase the number of online services available to the public including the registry services, to widen the scale of usage of public electronic services, to integrate subsequent systems of public administration and business on ePUAP portal, to define new processes of customer and business services. === ePUAP2 === ePUAP2 is a public and administrative project that extends the set of functional services developed during the first edition of the project and is another step in the process of transforming Poland into a modern and citizen-friendly country. The implementation period for the project covers the years 2009–2013. Project financing The cost of the project “Construction of electronic Platform of Public Administration Services” – 32 million PLN was covered in 75% by the funds from the European Regional Development Fund (under the Sector Operational Programme "Supporting Competitiveness of Enterprises for the years 2004–2006"), while the remaining 25% of the cost was covered by a Polish national co-financing. Funds for the ePUAP2 project were gained from the 7th priority axis of the Innovative Economy Operational Programme and amounts to 140 million PLN (85% of eligible expenses were covered by the European Regional Development Fund, 15% were covered by a national co-financing). The trustee of ePUAP is the Polish Ministry of the Interior and Administration. == Legal regulations == According to the Polish law from 1 May 2008, public authorities are required to accept documents in electronic form (bringing applications and proposals and other activities in electronic form). ePUAP enables public institutions to meet this requirement by providing a service infrastructure to set up am electronic inbox. The ePUAP inbox meets legal requirements, in particular: issuing an official confirmation of receipt in accordance with the regulation of the Prime Minister of 29 September 2005 on the organizational and technical conditions for the delivery of electronic documents to public entities; cooperation with hardware security modules (HSM), meeting the technical requirements set out in the law; handling documents electronically in accordance with the minimum requirements set out in the Regulation of the Polish Council of Ministers of 11 October 2005 on minimum requirements for ICT systems. == Incidents == === Crashes === The ePUAP system very often happens smaller or larger failures. Because it is used to sign the application profiles trusted also in other electronic systems such as public administration. Electronic Services Platform created by ZUS, the system fault ePUAP it very difficult to settle official matters most electronically. === "Infoafera" === According to TVN and the release of TVP News from 10 April 2014, the creation of ePUAP is also associated with the so-called "Infoafera." On 10 April 2014, the Minister of Internal Affairs of Poland confirmed the information that the American technology company HP confessed to its participation in the Polish info-tour and corruption of Polish officials. By March 2014, the construction of ePUAP and its maintenance cost PLN 98.4 million. PLN 67.8 million has been used for this project. Challenged expenses only on the portal itself is approx. PLN 20 million.

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  Abess

  abess (Adaptive Best Subset Selection, also ABESS) is a machine learning method designed to address the problem of best subset selection. It aims to determine which features or variables are crucial for optimal model performance when provided with a dataset and a prediction task. abess was introduced by Zhu in 2020 and it dynamically selects the appropriate model size adaptively, eliminating the need for selecting regularization parameters. abess is applicable in various statistical and machine learning tasks, including linear regression, the Single-index model, and other common predictive models. abess can also be applied in biostatistics. == Basic Form == The basic form of abess is employed to address the optimal subset selection problem in general linear regression. abess is an l 0 {\displaystyle l_{0}} method, it is characterized by its polynomial time complexity and the property of providing both unbiased and consistent estimates. In the context of linear regression, assuming we have knowledge of n {\displaystyle n} independent samples ( x i , y i ) , i = 1 , … , n {\displaystyle (x_{i},y_{i}),i=1,\ldots ,n} , where x i ∈ R p × 1 {\displaystyle x_{i}\in \mathbb {R} ^{p\times 1}} and y i ∈ R {\displaystyle y_{i}\in \mathbb {R} } , we define X = ( x 1 , … , x n ) ⊤ {\displaystyle X=(x_{1},\ldots ,x_{n})^{\top }} and y = ( y 1 , … , y n ) ⊤ {\displaystyle y=(y_{1},\ldots ,y_{n})^{\top }} . The following equation represents the general linear regression model: y = X β + ε . {\displaystyle y=X\beta +\varepsilon .} To obtain appropriate parameters β {\displaystyle \beta } , one can consider the loss function for linear regression: L n LR ( β ; X , y ) = 1 2 n ‖ y − X β ‖ 2 2 . {\displaystyle {\mathcal {L}}_{n}^{\text{LR}}(\beta ;X,y)={\frac {1}{2n}}\|y-X\beta \|_{2}^{2}.} In abess, the initial focus is on optimizing the loss function under the l 0 {\displaystyle l_{0}} constraint. That is, we consider the following problem: min β ∈ R p × 1 L n LR ( β ; X , y ) , subject to ‖ β ‖ 0 ≤ s , {\displaystyle \min _{\beta \in \mathbb {R} ^{p\times 1}}{\mathcal {L}}_{n}^{\text{LR}}(\beta ;X,y),{\text{ subject to }}\|\beta \|_{0}\leq s,} where s {\displaystyle s} represents the desired size of the support set, and ‖ β ‖ 0 = ∑ i = 1 p I ( β i ≠ 0 ) {\displaystyle \|\beta \|_{0}=\sum _{i=1}^{p}{\mathcal {I}}_{(\beta _{i}\neq 0)}} is the l 0 {\displaystyle l_{0}} norm of the vector. To address the optimization problem described above, abess iteratively exchanges an equal number of variables between the active set and the inactive set. In each iteration, the concept of sacrifice is introduced as follows: For j in the active set ( j ∈ A ^ {\displaystyle j\in {\hat {\mathcal {A}}}} ): ξ j = L n LR ( β ^ A ∖ { j } ) − L n LR ( β ^ A ) = X j ⊤ X j 2 n ( β ^ j ) 2 {\displaystyle \xi _{j}={\mathcal {L}}_{n}^{\text{LR}}\left({\hat {\boldsymbol {\beta }}}^{{\mathcal {A}}\backslash \{j\}}\right)-{\mathcal {L}}_{n}^{\text{LR}}\left({\hat {\boldsymbol {\beta }}}^{\mathcal {A}}\right)={\frac {{\boldsymbol {X}}_{j}^{\top }{\boldsymbol {X}}_{j}}{2n}}\left({\hat {\beta }}_{j}\right)^{2}} For j in the inactive set ( j ∉ A ^ {\displaystyle j\notin {\hat {\mathcal {A}}}} ): ξ j = L n LR ( β ^ A ) − L n LR ( β ^ A + t ^ { j } ) = X j ⊤ X j 2 n ( d ^ j X j ⊤ X j / n ) 2 {\displaystyle \xi _{j}={\mathcal {L}}_{n}^{\text{LR}}\left({\hat {\boldsymbol {\beta }}}^{\mathcal {A}}\right)-{\mathcal {L}}_{n}^{\text{LR}}\left({\hat {\boldsymbol {\beta }}}^{\mathcal {A}}+{\hat {\boldsymbol {t}}}^{\{j\}}\right)={\frac {{\boldsymbol {X}}_{j}^{\top }{\boldsymbol {X}}_{j}}{2n}}\left({\frac {{\hat {\mathrm {d} }}_{j}}{{\boldsymbol {X}}_{j}^{\top }{\boldsymbol {X}}_{j}/n}}\right)^{2}} Here are the key elements in the above equations: β ^ A {\displaystyle {\hat {\beta }}^{\mathcal {A}}} : This represents the estimate of β {\displaystyle \beta } obtained in the previous iteration. A ^ {\displaystyle {\hat {\mathcal {A}}}} : It denotes the estimated active set from the previous iteration. β ^ A ∖ { j } {\displaystyle {\hat {\boldsymbol {\beta }}}^{{\mathcal {A}}\backslash \{j\}}} : This is a vector where the j-th element is set to 0, while the other elements are the same as β ^ A {\displaystyle {\hat {\beta }}^{\mathcal {A}}} . t ^ { j } = arg ⁡ min t L n LR ( β ^ A + t { j } ) {\displaystyle {\hat {\boldsymbol {t}}}^{\{j\}}=\arg \min _{t}{\mathcal {L}}_{n}^{\text{LR}}\left({\hat {\boldsymbol {\beta }}}^{\mathcal {A}}+{\boldsymbol {t}}^{\{j\}}\right)} : Here, t { j } {\displaystyle t^{\{j\}}} represents a vector where all elements are 0 except the j-th element. d ^ j = X j ⊤ ( y − X β ^ ) / n {\displaystyle {\hat {d}}_{j}={\boldsymbol {X}}_{j}^{\top }({\boldsymbol {y}}-{\boldsymbol {X}}{\hat {\boldsymbol {\beta }}})/n} : This is calculated based on the equation mentioned. The iterative process involves exchanging variables, with the aim of minimizing the sacrifices in the active set while maximizing the sacrifices in the inactive set during each iteration. This approach allows abess to efficiently search for the optimal feature subset. In abess, select an appropriate s max {\displaystyle s_{\max }} and optimize the above problem for active sets size s = 1 , … , s max {\displaystyle s=1,\ldots ,s_{\max }} using the information criterion GIC = n log ⁡ L n LR + s log ⁡ p log ⁡ log ⁡ n , {\displaystyle {\text{GIC}}=n\log {\mathcal {L}}_{n}^{\text{LR}}+s\log p\log \log n,} to adaptively choose the appropriate active set size s {\displaystyle s} and obtain its corresponding abess estimator. == Generalizations == The splicing algorithm in abess can be employed for subset selection in other models. === Distribution-Free Location-Scale Regression === In 2023, Siegfried extends abess to the case of Distribution-Free and Location-Scale. Specifically, it considers the optimization problem max ϑ ∈ R P , β ∈ R J , γ ∈ R J ∑ i = 1 N ℓ i ( ϑ , x i ⊤ β , exp ⁡ ( x i ⊤ γ ) − 1 ) , {\displaystyle \max _{{\boldsymbol {\vartheta }}\in \mathbb {R} ^{P},{\boldsymbol {\beta }}\in \mathbb {R} ^{J},{\boldsymbol {\gamma }}\in \mathbb {R} ^{J}}\sum _{i=1}^{N}\ell _{i}\left({\boldsymbol {\vartheta }},{\boldsymbol {x}}_{i}^{\top }{\boldsymbol {\beta }},{\sqrt {\exp \left({\boldsymbol {x}}_{i}^{\top }{\boldsymbol {\gamma }}\right)}}^{-1}\right),} subject to ‖ ( β ⊤ , γ ⊤ ) ⊤ ‖ 0 ≤ s , {\displaystyle \left\|\left({\boldsymbol {\beta }}^{\top },{\boldsymbol {\gamma }}^{\top }\right)^{\top }\right\|_{0}\leq s,} where ℓ i {\displaystyle \ell _{i}} is a loss function, ϑ {\displaystyle {\boldsymbol {\vartheta }}} is a parameter vector, β {\displaystyle {\boldsymbol {\beta }}} and γ {\displaystyle {\boldsymbol {\gamma }}} are vectors, and x i {\displaystyle {\boldsymbol {x}}_{i}} is a data vector. This approach, demonstrated across various applications, enables parsimonious regression modeling for arbitrary outcomes while maintaining interpretability through innovative subset selection procedures. === Groups Selection === In 2023, Zhang applied the splicing algorithm to group selection, optimizing the following model: min β ∈ R p L n LR ( β ; X , y ) subject to ∑ j = 1 J I ( ‖ β G j ‖ 2 ≠ 0 ) ≤ s {\displaystyle \min _{{\boldsymbol {\beta }}\in \mathbb {R} ^{p}}{\mathcal {L}}_{n}^{\text{LR}}(\beta ;X,y){\text{ subject to }}\sum _{j=1}^{J}I\left(\|{\boldsymbol {\beta }}_{G_{j}}\|_{2}\neq 0\right)\leq s} Here are the symbols involved: J {\displaystyle J} : Total number of feature groups, representing the existence of J {\displaystyle J} non-overlapping feature groups in the dataset. G j {\displaystyle G_{j}} : Index set for the j {\displaystyle j} -th feature group, where j {\displaystyle j} ranges from 1 to J {\displaystyle J} , representing the feature grouping structure in the data. s {\displaystyle s} : Model size, a positive integer determined from the data, limiting the number of selected feature groups. === Regression with Corrupted Data === Zhang applied the splicing algorithm to handle corrupted data. Corrupted data refers to information that has been disrupted or contains errors during the data collection or recording process. This interference may include sensor inaccuracies, recording errors, communication issues, or other external disturbances, leading to inaccurate or distorted observations within the dataset. === Single Index Models === In 2023, Tang applied the splicing algorithm to optimal subset selection in the Single-index model. The form of the Single Index Model (SIM) is given by y i = g ( b ⊤ x i , e i ) , i = 1 , … , n , {\displaystyle y_{i}=g({\boldsymbol {b}}^{\top }{\boldsymbol {x}}_{i},e_{i}),\quad i=1,\ldots ,n,} where b {\displaystyle {\boldsymbol {b}}} is the parameter vector, e i {\displaystyle e_{i}} is the error term. The corresponding loss function is defined as l n ( β ) = ∑ i = 1 n ( r i n − 1 2 − x i ⊤ β ) 2 , {\displaystyle l_{n}({\boldsymbol {\beta }})=\sum _{i=1}^{n}\left({\frac {r_{i}}{n}}-{\frac {1}{2}}-{\boldsymbol {x}}_{i}^{\top }{\boldsymbol {\beta }}\right)^{2},} where r {\disp

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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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  Witness set

  In combinatorics and computational learning theory, a witness set is a set of elements that distinguishes a given Boolean function from a given class of other Boolean functions. Let C {\displaystyle C} be a concept class over a domain X {\displaystyle X} (that is, a family of Boolean functions over X {\displaystyle X} ) and c {\displaystyle c} be a concept in X {\displaystyle X} (a single Boolean function). A subset S {\displaystyle S} of X {\displaystyle X} is a witness set for c {\displaystyle c} in X {\displaystyle X} if S {\displaystyle S} distinguishes c {\displaystyle c} from all the other functions in C {\displaystyle C} , in the sense that no other function in C {\displaystyle C} has the same values on S {\displaystyle S} . For a concept class with | C | {\displaystyle |C|} concepts, there exists a concept that has a witness of size at most log 2 ⁡ | C | {\displaystyle \log _{2}|C|} ; this bound is tight when C {\displaystyle C} consists of all Boolean functions over X {\displaystyle X} . By a result of Bondy (1972) there exists a single witness set of size at most | C | − 1 {\displaystyle |C|-1} that is valid for all concepts in C {\displaystyle C} ; this bound is tight when C {\displaystyle C} consists of the indicator functions of the empty set and some singleton sets. One way to construct this set is to interpret the concepts as bitstrings, and the domain elements as positions in these bitstrings. Then the set of positions at which a trie of the bitstrings branches forms the desired witness set. This construction is central to the operation of the fusion tree data structure. The minimum size of a witness set for c {\displaystyle c} is called the witness size or specification number and is denoted by w C ( c ) {\displaystyle w_{C}(c)} . The value max { w C ( c ) : c ∈ C } {\displaystyle \max\{w_{C}(c):c\in C\}} is called the teaching dimension of C {\displaystyle C} . It represents the number of examples of a concept that need to be presented by a teacher to a learner, in the worst case, to enable the learner to determine which concept is being presented. Witness sets have also been called teaching sets, keys, specifying sets, or discriminants. The "witness set" terminology is from Kushilevitz et al. (1996), who trace the concept of witness sets to work by Cover (1965).

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  Application enablement

  Application enablement is an approach which brings telecommunications network providers and developers together to combine their network and web abilities in creating and delivering high demand advanced services and new intelligent applications. Network providers, in addition to bandwidth, provide abilities such as billing, location, presence, and security, which have allowed them to establish long-term relationships with end-users. By offering these select abilities as application programming interfaces (APIs), providers give developers access to a set of tools to create (mashup) new applications and services to run on provider networks. Unifying the strengths of providers and developers facilitates the creation of mash-up applications, and in turn, a better end user quality of experience (QoE) for improved profit margins. Apple's iOS with App Store, and Google's Android with Android Market exemplify this approach. Both have introduced mobile platforms that are supported by a comprehensive ecosystem in order to perpetuate innovation in product design, content and service offerings, and overall consumer behavior. By the end of April 2010, downloadable applications numbered over 200,000 for iPhone and over 50,000 for Android. == Background == Historically, telecommunication providers primarily based their business models on network performance, emphasizing connectivity, availability, and quality of service (QoS) as key sources of revenue and customer value. With the increasing demand for bandwidth-intensive data and video applications, maintaining service continuity has required substantial infrastructure investments. To address rising operational costs and declining average revenue per user (ARPU), providers have increasingly adopted customer-oriented strategies and diversified business models to expand their roles within the telecommunications value chain. Application enablement supports providers in making this transition by providing an environment, or ecosystem, where providers and developers can collaborate to build, test, manage, and distribute applications across networks including television, broadband, Internet, and mobile. This cooperative effort produces mutually beneficial results for all parties, opening up new revenue streams while enhancing value and rate of return (ROI). The following are some examples of key network abilities which function as application enablers in the telecommunications market: Billing systems Security for private transactions Network-based storage of digital content End-to-end bandwidth for high-quality transmissions Scoring abilities to identify end-user preferences and behaviors Subscriber data to customize the end-user experience Context information, such as location and presence, to localize services. == New business models == As network providers work toward effective collaboration with application and content developers, several new business models are emerging to help facilitate the business relationships: === Vendor-led === A type of business model driven by telecommunications vendors, who assist network providers in building relationships with application and content developers to lower the cost and complexity of managing third parties. Examples of this model include: Forum Nokia IBM Technology Partner Ecosystem Ng Connect Huawei Intouch program === Operator-led === Characterized by network providers who want to maintain a high degree of flexibility and control over applications created for their end-consumers, this model lets them create and manage their own developer program, development platform, and application store. Under this arrangement, independent developers provide their own branding, marketing communications, pricing and customer care. Network providers pursuing this model will often seek to partner with a large number of third parties using standardized on-boarding processes. Examples of this model include: o2 Litmus Orange Partner Joint Innovation Lab === Aggregator === Network providers who choose not to create/manage their own developer relationships will partner with one or multiple aggregators, to administer a portion of or their entire application strategy. Examples of this model include: Ovi Operator Partnership Blackberry Operator Partnership Cellmania Buongiorno === Mass wholesale === Select network providers also participate in wholesale models that exist primarily for applications (BT's Ribbit- an Internet Protocol (IP) based calling and messaging platform) and devices (Verizon's Open Device initiative). This business-to-business approach reduces a large portion of the potential costs of third party application enablement (marketing, acquisition and support). Examples of this model include: BT's Ribbit Verizon Wireless ODI AT&T Synaptic Hosting === The enterprise customer === Some network providers are focusing on enabling applications in the enterprise space. In this model, the network provider establishes a platform for their large enterprise customers who want to blend custom software with enhanced abilities, and will provide standardized processes around mobilizing enterprise applications, and exposing core back-office abilities to allow for dynamic customer interaction. Examples of this model include: Vodafone Applications Service Verizon Private Network Sprint Solution Launchpad === Trusted partner === In this model, the network provider builds one-on-one relationships with trusted third-party developers by exposing customized network abilities, bringing a greater variety of brands to the network provider's portfolio. Network providers using this model tend to only have a few partners (in contrast to the operator led model). Under this scenario, network providers benefit from a pre-established customer base and the developer's marketing resources. Examples of this model include: 3/Skype Partnership (UK) Virgin Media and BBC iPlayer == Network operator developer resources == Operator led model o2 Litmus Orange Partner Joint Innovations Lab Aggregator model Ovi Operator Partnership Cellmania Buongiorno Mass wholesale model BT Ribbit Verizon Wireless ODI AT&T Synaptic Hosting Enterprise customer model Vodafone Applications Service Verizon Private Network Sprint Solution Launchpad == Rerencesfe ==

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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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  Tucker decomposition

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