AI Coding Neovim

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  Generative art

  Generative art is post-conceptual art that has been created (in whole or in part) with the use of an autonomous system. An autonomous system in this context is generally one that is non-human and can independently determine features of an artwork that would otherwise require decisions made directly by the artist. In some cases the human creator may claim that the generative system represents their own artistic idea, and in others that the system takes on the role of the creator. "Generative art" often refers to algorithmic art (algorithmically determined computer generated artwork) and synthetic media (general term for any algorithmically generated media), but artists can also make generative art using systems of chemistry, biology, mechanics and robotics, smart materials, manual randomization, mathematics, data mapping, symmetry, and tiling. Generative algorithms, algorithms programmed to produce artistic works through predefined rules, stochastic methods, or procedural logic, often yielding dynamic, unique, and contextually adaptable outputs—are central to many of these practices. == History == The use of the word "generative" in the discussion of art has developed over time. The use of "Artificial DNA" defines a generative approach to art focused on the construction of a system able to generate unpredictable events, all with a recognizable common character. The use of autonomous systems, required by some contemporary definitions, focuses a generative approach where the controls are strongly reduced. This approach is also named "emergent". Margaret Boden and Ernest Edmonds have noted the use of the term "generative art" in the broad context of automated computer graphics in the 1960s, beginning with artwork exhibited by Georg Nees and Frieder Nake in 1965: A. Michael Noll did his initial computer art, combining randomness with order, in 1962, and exhibited it along with works by Bell Julesz in 1965. The terms "generative art" and "computer art" have been used in tandem, and more or less interchangeably, since the very earliest days. The first such exhibition showed the work of Nees in February 1965, which some claim was titled "Generative Computergrafik". While Nees does not himself remember, this was the title of his doctoral thesis published a few years later. The correct title of the first exhibition and catalog was "computer-grafik". "Generative art" and related terms was in common use by several other early computer artists around this time, including Manfred Mohr and Ken Knowlton. Vera Molnár (born 1924) is a French media artist of Hungarian origin. Molnar is widely considered to be a pioneer of generative art, and is also one of the first women to use computers in her art practice. The term "Generative Art" with the meaning of dynamic artwork-systems able to generate multiple artwork-events was clearly used the first time for the "Generative Art" conference in Milan in 1998. The term has also been used to describe geometric abstract art where simple elements are repeated, transformed, or varied to generate more complex forms. Thus defined, generative art was practiced by the Argentinian artists Eduardo Mac Entyre and Miguel Ángel Vidal in the late 1960s. In 1972 the Romanian-born Paul Neagu created the Generative Art Group in Britain. It was populated exclusively by Neagu using aliases such as "Hunsy Belmood" and "Edward Larsocchi". In 1972 Neagu gave a lecture titled 'Generative Art Forms' at the Queen's University, Belfast Festival. In 1970 the School of the Art Institute of Chicago created a department called Generative Systems. As described by Sonia Landy Sheridan the focus was on art practices using the then new technologies for the capture, inter-machine transfer, printing and transmission of images, as well as the exploration of the aspect of time in the transformation of image information. Also noteworthy is John Dunn, first a student and then a collaborator of Sheridan. In 1988 Clauser identified the aspect of systemic autonomy as a critical element in generative art: It should be evident from the above description of the evolution of generative art that process (or structuring) and change (or transformation) are among its most definitive features, and that these features and the very term 'generative' imply dynamic development and motion. (the result) is not a creation by the artist but rather the product of the generative process - a self-precipitating structure. In 1989 Celestino Soddu defined the Generative Design approach to Architecture and Town Design in his book Citta' Aleatorie. In 1989 Franke referred to "generative mathematics" as "the study of mathematical operations suitable for generating artistic images." From the mid-1990s Brian Eno popularized the terms generative music and generative systems, making a connection with earlier experimental music by Terry Riley, Steve Reich and Philip Glass. From the end of the 20th century, communities of generative artists, designers, musicians and theoreticians began to meet, forming cross-disciplinary perspectives. The first meeting about generative Art was in 1998, at the inaugural International Generative Art conference at Politecnico di Milano University, Italy. In Australia, the Iterate conference on generative systems in the electronic arts followed in 1999. On-line discussion has centered around the eu-gene mailing list, which began late 1999, and has hosted much of the debate which has defined the field. These activities have more recently been joined by the Generator.x conference in Berlin starting in 2005. In 2012 the new journal GASATHJ, Generative Art Science and Technology Hard Journal was founded by Celestino Soddu and Enrica Colabella jointing several generative artists and scientists in the editorial board. Some have argued that as a result of this engagement across disciplinary boundaries, the community has converged on a shared meaning of the term. As Boden and Edmonds put it in 2011: Today, the term "Generative Art" is still current within the relevant artistic community. Since 1998 a series of conferences have been held in Milan with that title (Generativeart.com), and Brian Eno has been influential in promoting and using generative art methods (Eno, 1996). Both in music and in visual art, the use of the term has now converged on work that has been produced by the activation of a set of rules and where the artist lets a computer system take over at least some of the decision-making (although, of course, the artist determines the rules). In the call of the Generative Art conferences in Milan (annually starting from 1998), the definition of Generative Art by Celestino Soddu: Generative Art is the idea realized as genetic code of artificial events, as construction of dynamic complex systems able to generate endless variations. Each Generative Project is a concept-software that works producing unique and non-repeatable events, like music or 3D Objects, as possible and manifold expressions of the generating idea strongly recognizable as a vision belonging to an artist / designer / musician / architect /mathematician. Discussion on the eu-gene mailing list was framed by the following definition by Adrian Ward from 1999: Generative art is a term given to work which stems from concentrating on the processes involved in producing an artwork, usually (although not strictly) automated by the use of a machine or computer, or by using mathematic or pragmatic instructions to define the rules by which such artworks are executed. A similar definition is provided by Philip Galanter: Generative art refers to any art practice where the artist creates a process, such as a set of natural language rules, a computer program, a machine, or other procedural invention, which is then set into motion with some degree of autonomy contributing to or resulting in a completed work of art. Around the 2020s, generative AI models learned to imitate the distinct style of particular authors. For example, a generative image model such as Stable Diffusion is able to model the stylistic characteristics of an artist like Pablo Picasso (including his particular brush strokes, use of colour, perspective, and so on), and a user can engineer a prompt such as "an astronaut riding a horse, by Picasso" to cause the model to generate a novel image applying the artist's style to an arbitrary subject. Generative image models have received significant backlash from artists who object to their style being imitated without their permission, arguing that this harms their ability to profit from their own work. The emergence of text-to-image generative AI systems has expanded debates over authorship, copyright, and artistic labor. The main issues in these debates include the eligibility of AI-generated outputs for copyright protection and the legal and ethical questions of using existing copyrighted works as training data for generative AI systems. == Types == === Music === Johann Kirnberger's Mu

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  TabPFN

  TabPFN (Tabular Prior-data Fitted Network) is a machine learning model for tabular datasets proposed in 2022. It uses a transformer architecture. It is intended for supervised classification and regression analysis on tabular datasets, particularly focusing on small- to medium-sized datasets. The latest version, TabPFN-3, was released in May 2026 and supports datasets with up to one million rows and 200 features. == History == TabPFN was first introduced in a 2022 pre-print and presented at ICLR 2023. TabPFN v2 was published in 2025 in Nature by Hollmann and co-authors. The source code is published on GitHub under a modified Apache License and on PyPi. Writing for ICLR blogs, McCarter states that the model has attracted attention due to its performance on small dataset benchmarks. TabPFN v2.5 was released on November 6, 2025. TabPFN-3 was released on May 12, 2026. Prior Labs, founded in 2024, aims to commercialize TabPFN. As of April 2026, the open-source TabPFN repository had more than 6,000 stars on GitHub. == Overview and pre-training == TabPFN supports classification, regression and generative tasks. It leverages "Prior-Data Fitted Networks" models to model tabular data. By using a transformer pre-trained on synthetic tabular datasets, TabPFN avoids benchmark contamination and costs of curating real-world data. TabPFN v2 was pre-trained on approximately 130 million such datasets. Synthetic datasets are generated using causal models or Bayesian neural networks; this can include simulating missing values, imbalanced data, and noise. Random inputs are passed through these models to generate outputs, with a bias towards simpler causal structures. During pre-training, TabPFN predicts the masked target values of new data points given training data points and their known targets, effectively learning a generic learning algorithm that is executed by running a neural network forward pass. The new dataset is then processed in a single forward pass without retraining. The model's transformer encoder processes features and labels by alternating attention across rows and columns. TabPFN v2 handles numerical and categorical features, missing values, and supports tasks like regression and synthetic data generation, while TabPFN-2.5 scales this approach to datasets with up to 50,000 rows and 2,000 features. TabPFN-3 introduced a redesigned architecture with row-compression, an attention-based many-class decoder, native missing-value handling, and inference optimizations such as row chunking and a reduced key-value cache, with benchmark-validated regimes of up to 1 million rows with 200 features, 100,000 rows with 2,000 features, or 1,000 rows with 20,000 features. Since TabPFN is pre-trained, in contrast to other deep learning methods, it does not require costly hyperparameter optimization. == Research == TabPFN is the subject of on-going research. Applications for TabPFN have been investigated for domains such as chemoproteomics, insurance risk classification, and metagenomics. In clinical research, TabPFN was used in a study on the early detection of pancreatic cancer from blood samples, where it was combined with metabolomic data and reported high diagnostic performance. == Applications == TabPFN has been used in industrial and biomedical contexts. Hitachi Ltd. has been reported to use the model for predictive maintenance in rail networks, with its use described as helping to identify track issues earlier and reduce manual inspections. In the biomedical domain, Oxford Cancer Analytics has used TabPFN in the analysis of proteomic data in lung disease research. A 2025 ML Contests report noted that the winners of DrivenData's PREPARE challenge used TabPFN to generate features for gradient-boosted decision tree models. == Limitations == TabPFN has been criticized for its "one large neural network is all you need" approach to modeling problems. Further, its performance is limited in high-dimensional and large-scale datasets. == Scaling Mode == In late November 2025, Prior Labs introduced ‘‘Scaling Mode’’, an operating mode for TabPFN designed to remove the fixed upper bound on training set size. Earlier versions of TabPFN had been optimized and validated primarily for datasets of up to 100,000 rows, whereas Scaling Mode was reported to extend support to substantially larger datasets, with benchmarked experiments on datasets containing up to 10 million rows. According to Prior Labs, Scaling Mode preserves the existing TabPFN architecture, including its alternating row-attention and column-attention design, as well as the same feature-count limits as prior releases. Inference remains based on a single forward pass rather than dataset-specific gradient-descent training, while scalability is described as being constrained primarily by available compute and memory resources. Prior Labs reported benchmark results on an internal collection of datasets ranging from 1 million to 10 million rows across industry and scientific applications. In these benchmarks, Scaling Mode was compared with CatBoost, XGBoost, LightGBM, and TabPFN 2.5 using 50,000-row subsampling. The company stated that predictive performance improved monotonically with increasing training set size and that no diminishing returns from scaling were observed within the tested range. Prior Labs also announced the release through company and executive social media channels. TabPFN-3 later incorporated related scaling improvements, including row chunking and a reduced key-value cache, into the model architecture and inference pipeline.

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  AdaBoost

  AdaBoost (short for Adaptive Boosting) is a statistical classification meta-algorithm formulated by Yoav Freund and Robert Schapire in 1995, who won the 2003 Gödel Prize for their work. It can be used in conjunction with many types of learning algorithm to improve performance. The output of multiple weak learners is combined into a weighted sum that represents the final output of the boosted classifier. Usually, AdaBoost is presented for binary classification, although it can be generalized to multiple classes or bounded intervals of real values. AdaBoost is adaptive in the sense that subsequent weak learners (models) are adjusted in favor of instances misclassified by previous models. In some problems, it can be less susceptible to overfitting than other learning algorithms. The individual learners can be weak, but as long as the performance of each one is slightly better than random guessing, the final model can be proven to converge to a strong learner. Although AdaBoost is typically used to combine weak base learners (such as decision stumps), it has been shown to also effectively combine strong base learners (such as deeper decision trees), producing an even more accurate model. Every learning algorithm tends to suit some problem types better than others, and typically has many different parameters and configurations to adjust before it achieves optimal performance on a dataset. AdaBoost (with decision trees as the weak learners) is often referred to as the best out-of-the-box classifier. When used with decision tree learning, information gathered at each stage of the AdaBoost algorithm about the relative 'hardness' of each training sample is fed into the tree-growing algorithm such that later trees tend to focus on harder-to-classify examples. == Training == AdaBoost refers to a particular method of training a boosted classifier. A boosted classifier is a classifier of the form F T ( x ) = ∑ t = 1 T f t ( x ) {\displaystyle F_{T}(x)=\sum _{t=1}^{T}f_{t}(x)} where each f t {\displaystyle f_{t}} is a weak learner that takes an object x {\displaystyle x} as input and returns a value indicating the class of the object. For example, in the two-class problem, the sign of the weak learner's output identifies the predicted object class and the absolute value gives the confidence in that classification. Each weak learner produces an output hypothesis h {\displaystyle h} which fixes a prediction h ( x i ) {\displaystyle h(x_{i})} for each sample in the training set. At each iteration t {\displaystyle t} , a weak learner is selected and assigned a coefficient α t {\displaystyle \alpha _{t}} such that the total training error E t {\displaystyle E_{t}} of the resulting t {\displaystyle t} -stage boosted classifier is minimized. E t = ∑ i E [ F t − 1 ( x i ) + α t h ( x i ) ] {\displaystyle E_{t}=\sum _{i}E[F_{t-1}(x_{i})+\alpha _{t}h(x_{i})]} Here F t − 1 ( x ) {\displaystyle F_{t-1}(x)} is the boosted classifier that has been built up to the previous stage of training and f t ( x ) = α t h ( x ) {\displaystyle f_{t}(x)=\alpha _{t}h(x)} is the weak learner that is being considered for addition to the final classifier. === Weighting === At each iteration of the training process, a weight w i , t {\displaystyle w_{i,t}} is assigned to each sample in the training set equal to the current error E ( F t − 1 ( x i ) ) {\displaystyle E(F_{t-1}(x_{i}))} on that sample. These weights can be used in the training of the weak learner. For instance, decision trees can be grown which favor the splitting of sets of samples with large weights. == Derivation == This derivation follows Rojas (2009): Suppose we have a data set { ( x 1 , y 1 ) , … , ( x N , y N ) } {\displaystyle \{(x_{1},y_{1}),\ldots ,(x_{N},y_{N})\}} where each item x i {\displaystyle x_{i}} has an associated class y i ∈ { − 1 , 1 } {\displaystyle y_{i}\in \{-1,1\}} , and a set of weak classifiers { k 1 , … , k L } {\displaystyle \{k_{1},\ldots ,k_{L}\}} each of which outputs a classification k j ( x i ) ∈ { − 1 , 1 } {\displaystyle k_{j}(x_{i})\in \{-1,1\}} for each item. After the ( m − 1 ) {\displaystyle (m-1)} -th iteration our boosted classifier is a linear combination of the weak classifiers of the form: C ( m − 1 ) ( x i ) = α 1 k 1 ( x i ) + ⋯ + α m − 1 k m − 1 ( x i ) , {\displaystyle C_{(m-1)}(x_{i})=\alpha _{1}k_{1}(x_{i})+\cdots +\alpha _{m-1}k_{m-1}(x_{i}),} where the class will be the sign of C ( m − 1 ) ( x i ) {\displaystyle C_{(m-1)}(x_{i})} . At the m {\displaystyle m} -th iteration we want to extend this to a better boosted classifier by adding another weak classifier k m {\displaystyle k_{m}} , with another weight α m {\displaystyle \alpha _{m}} : C m ( x i ) = C ( m − 1 ) ( x i ) + α m k m ( x i ) {\displaystyle C_{m}(x_{i})=C_{(m-1)}(x_{i})+\alpha _{m}k_{m}(x_{i})} So it remains to determine which weak classifier is the best choice for k m {\displaystyle k_{m}} , and what its weight α m {\displaystyle \alpha _{m}} should be. We define the total error E {\displaystyle E} of C m {\displaystyle C_{m}} as the sum of its exponential loss on each data point, given as follows: E = ∑ i = 1 N e − y i C m ( x i ) = ∑ i = 1 N e − y i C ( m − 1 ) ( x i ) e − y i α m k m ( x i ) {\displaystyle E=\sum _{i=1}^{N}e^{-y_{i}C_{m}(x_{i})}=\sum _{i=1}^{N}e^{-y_{i}C_{(m-1)}(x_{i})}e^{-y_{i}\alpha _{m}k_{m}(x_{i})}} Letting w i ( 1 ) = 1 {\displaystyle w_{i}^{(1)}=1} and w i ( m ) = e − y i C m − 1 ( x i ) {\displaystyle w_{i}^{(m)}=e^{-y_{i}C_{m-1}(x_{i})}} for m > 1 {\displaystyle m>1} , we have: E = ∑ i = 1 N w i ( m ) e − y i α m k m ( x i ) {\displaystyle E=\sum _{i=1}^{N}w_{i}^{(m)}e^{-y_{i}\alpha _{m}k_{m}(x_{i})}} We can split this summation between those data points that are correctly classified by k m {\displaystyle k_{m}} (so y i k m ( x i ) = 1 {\displaystyle y_{i}k_{m}(x_{i})=1} ) and those that are misclassified (so y i k m ( x i ) = − 1 {\displaystyle y_{i}k_{m}(x_{i})=-1} ): E = ∑ y i = k m ( x i ) w i ( m ) e − α m + ∑ y i ≠ k m ( x i ) w i ( m ) e α m = ∑ i = 1 N w i ( m ) e − α m + ∑ y i ≠ k m ( x i ) w i ( m ) ( e α m − e − α m ) {\displaystyle {\begin{aligned}E&=\sum _{y_{i}=k_{m}(x_{i})}w_{i}^{(m)}e^{-\alpha _{m}}+\sum _{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}e^{\alpha _{m}}\\&=\sum _{i=1}^{N}w_{i}^{(m)}e^{-\alpha _{m}}+\sum _{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}\left(e^{\alpha _{m}}-e^{-\alpha _{m}}\right)\end{aligned}}} Since the only part of the right-hand side of this equation that depends on k m {\displaystyle k_{m}} is ∑ y i ≠ k m ( x i ) w i ( m ) {\textstyle \sum _{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}} , we see that the k m {\displaystyle k_{m}} that minimizes E {\displaystyle E} is the one in the set { k 1 , … , k L } {\displaystyle \{k_{1},\ldots ,k_{L}\}} that minimizes ∑ y i ≠ k m ( x i ) w i ( m ) {\textstyle \sum _{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}} [assuming that α m > 0 {\displaystyle \alpha _{m}>0} ], i.e. the weak classifier with the lowest weighted error (with weights w i ( m ) = e − y i C m − 1 ( x i ) {\displaystyle w_{i}^{(m)}=e^{-y_{i}C_{m-1}(x_{i})}} ). To determine the desired weight α m {\displaystyle \alpha _{m}} that minimizes E {\displaystyle E} with the k m {\displaystyle k_{m}} that we just determined, we differentiate: d E d α m = d ( ∑ y i = k m ( x i ) w i ( m ) e − α m + ∑ y i ≠ k m ( x i ) w i ( m ) e α m ) d α m {\displaystyle {\frac {dE}{d\alpha _{m}}}={\frac {d(\sum _{y_{i}=k_{m}(x_{i})}w_{i}^{(m)}e^{-\alpha _{m}}+\sum _{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}e^{\alpha _{m}})}{d\alpha _{m}}}} The value of α m {\displaystyle \alpha _{m}} that minimizes the above expression is: α m = 1 2 ln ⁡ ( ∑ y i = k m ( x i ) w i ( m ) ∑ y i ≠ k m ( x i ) w i ( m ) ) {\displaystyle \alpha _{m}={\frac {1}{2}}\ln \left({\frac {\sum _{y_{i}=k_{m}(x_{i})}w_{i}^{(m)}}{\sum _{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}}}\right)} We calculate the weighted error rate of the weak classifier to be ϵ m = ∑ y i ≠ k m ( x i ) w i ( m ) ∑ i = 1 N w i ( m ) {\displaystyle \epsilon _{m}={\frac {\sum _{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}}{\sum _{i=1}^{N}w_{i}^{(m)}}}} , so it follows that: α m = 1 2 ln ⁡ ( 1 − ϵ m ϵ m ) {\displaystyle \alpha _{m}={\frac {1}{2}}\ln \left({\frac {1-\epsilon _{m}}{\epsilon _{m}}}\right)} which is the negative logit function multiplied by 0.5. Due to the convexity of E {\displaystyle E} as a function of α m {\displaystyle \alpha _{m}} , this new expression for α m {\displaystyle \alpha _{m}} gives the global minimum of the loss function. Note: This derivation only applies when k m ( x i ) ∈ { − 1 , 1 } {\displaystyle k_{m}(x_{i})\in \{-1,1\}} , though it can be a good starting guess in other cases, such as when the weak learner is biased ( k m ( x ) ∈ { a , b } , a ≠ − b {\displaystyle k_{m}(x)\in \{a,b\},a\neq -b} ), has multiple leaves ( k m ( x ) ∈ { a , b , … , n } {\displaystyle k_{m}(x)\in \{a,b,\dots ,n\}} ) or is some other function k m ( x ) ∈ R {\displaystyle k_{m}(x)\in \mathbb {R} } . Thus we have derived the AdaBoost algorithm: At each

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  Persian Speech Corpus

  The Persian Speech Corpus is a Modern Persian speech corpus for speech synthesis. The corpus contains phonetic and orthographic transcriptions of about 2.5 hours of Persian speech aligned with recorded speech on the phoneme level, including annotations of word boundaries. Previous spoken corpora of Persian include FARSDAT, which consists of read aloud speech from newspaper texts from 100 Persian speakers and the Telephone FARsi Spoken language DATabase (TFARSDAT) which comprises seven hours of read and spontaneous speech produced by 60 native speakers of Persian from ten regions of Iran. The Persian Speech Corpus was built using the same methodologies laid out in the doctoral project on Modern Standard Arabic of Nawar Halabi at the University of Southampton. The work was funded by MicroLinkPC, who own an exclusive license to commercialise the corpus, though the corpus is available for non-commercial use through the corpus' website. It is distributed under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. The corpus was built for speech synthesis purposes, but has been used for building HMM based voices in Persian. It can also be used to automatically align other speech corpora with their phonetic transcript and could be used as part of a larger corpus for training speech recognition systems. == Contents == The corpus is downloadable from its website, and contains the following: 396 .wav files containing spoken utterances 396 .lab files containing text utterances 396 .TextGrid files containing the phoneme labels with time stamps of the boundaries where these occur in the .wav files. phonetic-transcript.txt which has the form "[wav_filename]" "[Phoneme Sequence]" in every line orthographic-transcript.txt which has the form "[wav_filename]" "[Orthographic Transcript]" in every line

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  GazoPa

  GazoPa was an image search engine that used features from an image to search for and identify similar images which closed in 2011. GazoPa began in TechCrunch50 in 2008 before launching into a state of open beta in 2009. GazoPa branched out and released a flower photo community site called "GazoPa Bloom" in 2010. This site was for exploring flower images and, if users need help identifying a flower, uploading images for other people try to identify them. Both sites closed to the public in 2011 when the company decided to focus on other areas of their business.

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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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  CN2 algorithm

  The CN2 induction algorithm is a learning algorithm for rule induction. It is designed to work even when the training data is imperfect. It is based on ideas from the AQ algorithm and the ID3 algorithm. As a consequence it creates a rule set like that created by AQ but is able to handle noisy data like ID3. == Description of algorithm == The algorithm must be given a set of examples, TrainingSet, which have already been classified in order to generate a list of classification rules. A set of conditions, SimpleConditionSet, which can be applied, alone or in combination, to any set of examples is predefined to be used for the classification. routine CN2(TrainingSet) let the ClassificationRuleList be empty repeat let the BestConditionExpression be Find_BestConditionExpression(TrainingSet) if the BestConditionExpression is not nil then let the TrainingSubset be the examples covered by the BestConditionExpression remove from the TrainingSet the examples in the TrainingSubset let the MostCommonClass be the most common class of examples in the TrainingSubset append to the ClassificationRuleList the rule 'if ' the BestConditionExpression ' then the class is ' the MostCommonClass until the TrainingSet is empty or the BestConditionExpression is nil return the ClassificationRuleList routine Find_BestConditionExpression(TrainingSet) let the ConditionalExpressionSet be empty let the BestConditionExpression be nil repeat let the TrialConditionalExpressionSet be the set of conditional expressions, {x and y where x belongs to the ConditionalExpressionSet and y belongs to the SimpleConditionSet}. remove all formulae in the TrialConditionalExpressionSet that are either in the ConditionalExpressionSet (i.e., the unspecialized ones) or null (e.g., big = y and big = n) for every expression, F, in the TrialConditionalExpressionSet if F is statistically significant and F is better than the BestConditionExpression by user-defined criteria when tested on the TrainingSet then replace the current value of the BestConditionExpression by F while the number of expressions in the TrialConditionalExpressionSet > user-defined maximum remove the worst expression from the TrialConditionalExpressionSet let the ConditionalExpressionSet be the TrialConditionalExpressionSet until the ConditionalExpressionSet is empty return the BestConditionExpression

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  Accumulated local effects

  Accumulated local effects (ALE) is a machine learning interpretability method. == Concepts == ALE uses a conditional feature distribution as an input and generates augmented data, creating more realistic data than a marginal distribution. It ignores far out-of-distribution (outlier) values. Unlike partial dependence plots and marginal plots, ALE is not defeated in the presence of correlated predictors. It analyzes differences in predictions instead of averaging them by calculating the average of the differences in model predictions over the augmented data, instead of the average of the predictions themselves. == Example == Given a model that predicts house prices based on its distance from city center and size of the building area, ALE compares the differences of predictions of houses of different sizes. The result separates the impact of the size from otherwise correlated features. == Limitations == Defining evaluation windows is subjective. High correlations between features can defeat the technique. ALE requires more and more uniformly distributed observations than PDP so that the conditional distribution can be reliably determined. The technique may produce inadequate results if the data is highly sparse, which is more common with high-dimensional data (curse of dimensionality).

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  Conversational user interface

  A conversational user interface (CUI) is a user interface for computers that emulates a conversation with a human. Historically, computers have relied on text-based user interfaces and graphical user interfaces (GUIs) (such as the user pressing a "back" button) to translate the user's desired action into commands the computer understands. While an effective mechanism of completing computing actions, there is a learning curve for the user associated with GUI. Instead, CUIs provide opportunity for the user to communicate with the computer in their natural language rather than in a syntax specific commands.

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  Least-squares support vector machine

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

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  Correlation clustering

  Clustering is the problem of partitioning data points into groups based on similarity or dissimilarity. Correlation clustering is a clustering framework in which a set of objects is partitioned into clusters based on pairwise similarity and dissimilarity information, without requiring the number of clusters to be specified in advance. == Description of the problem == In machine learning, correlation clustering (also known as cluster editing) considers settings in which pairwise similarity or dissimilarity relationships between objects are known. A standard formulation models the input as an unweighted complete graph G = ( V , E ) {\displaystyle G=(V,E)} , where each edge is labeled either + {\displaystyle +} or − {\displaystyle -} (that is, the graph is a signed graph), indicating whether the corresponding endpoints are similar or dissimilar. The goal is to find a clustering (that is, a partition of V {\displaystyle V} ) that either maximizes the number of agreements—the sum of positive edges whose endpoints lie in the same cluster and negative edges whose endpoints lie in different clusters—or minimizes the number of disagreements—the sum of positive edges whose endpoints are separated and negative edges whose endpoints lie in the same cluster. Unlike other clustering methods such as k-means, correlation clustering does not require choosing the number of clusters k {\displaystyle k} in advance. It is not always possible to find a clustering with zero disagreements. For example, consider a triangle graph containing two positive edges and one negative edge. In this case, every clustering incurs at least one disagreement. Such configurations are referred to in the literature as bad triangles. From a computational perspective, optimizing the correlation clustering objective is challenging. The (decision version of the) problem is NP-complete. A large body of subsequent work has developed approximation algorithms for correlation clustering under various assumptions, including complete or general graphs and unweighted or weighted graphs, for both minimization and maximization objectives. This problem is considered one of the fundamental combinatorial optimization problems, and many algorithmic techniques have been developed to address it. The problem has also been studied extensively across multiple disciplines. A comprehensive literature review of early correlation clustering research is provided by Wahid and Hassini. == Formal Definitions == Let G = ( V , E ) {\displaystyle G=(V,E)} be a graph with nodes V {\displaystyle V} and edges E {\displaystyle E} . A clustering of G {\displaystyle G} is a partition of its node set Π = { π 1 , … , π k } {\displaystyle \Pi =\{\pi _{1},\dots ,\pi _{k}\}} with V = π 1 ∪ ⋯ ∪ π k {\displaystyle V=\pi _{1}\cup \dots \cup \pi _{k}} and π i ∩ π j = ∅ {\displaystyle \pi _{i}\cap \pi _{j}=\emptyset } for i ≠ j {\displaystyle i\neq j} . For a given clustering Π {\displaystyle \Pi } , let δ ( Π ) = { { u , v } ∈ E ∣ { u , v } ⊈ π ∀ π ∈ Π } {\displaystyle \delta (\Pi )=\{\{u,v\}\in E\mid \{u,v\}\not \subseteq \pi \;\forall \pi \in \Pi \}} denote the subset of edges of G {\displaystyle G} whose endpoints are in different subsets of the clustering Π {\displaystyle \Pi } . Now, let w : E → R ≥ 0 {\displaystyle w\colon E\to \mathbb {R} _{\geq 0}} be a function that assigns a non-negative weight to each edge of the graph and let E = E + ∪ E − {\displaystyle E=E^{+}\cup E^{-}} be a partition of the edges into attractive ( E + {\displaystyle E^{+}} ) and repulsive ( E − {\displaystyle E^{-}} ) edges; that is, the edges are signed. The minimum disagreement correlation clustering problem is the following optimization problem: minimize Π ∑ e ∈ E + ∩ δ ( Π ) w e + ∑ e ∈ E − ∖ δ ( Π ) w e . {\displaystyle {\begin{aligned}&{\underset {\Pi }{\operatorname {minimize} }}&&\sum _{e\in E^{+}\cap \delta (\Pi )}w_{e}+\sum _{e\in E^{-}\setminus \delta (\Pi )}w_{e}\;.\end{aligned}}} Here, the set E + ∩ δ ( Π ) {\displaystyle E^{+}\cap \delta (\Pi )} contains the attractive edges whose endpoints are in different components with respect to the clustering Π {\displaystyle \Pi } and the set E − ∖ δ ( Π ) {\displaystyle E^{-}\setminus \delta (\Pi )} contains the repulsive edges whose endpoints are in the same component with respect to the clustering Π {\displaystyle \Pi } . Together these two sets contain all edges that disagree with the clustering Π {\displaystyle \Pi } . Similarly to the minimum disagreement correlation clustering problem, the maximum agreement correlation clustering problem is defined as maximize Π ∑ e ∈ E + ∖ δ ( Π ) w e + ∑ e ∈ E − ∩ δ ( Π ) w e . {\displaystyle {\begin{aligned}&{\underset {\Pi }{\operatorname {maximize} }}&&\sum _{e\in E^{+}\setminus \delta (\Pi )}w_{e}+\sum _{e\in E^{-}\cap \delta (\Pi )}w_{e}\;.\end{aligned}}} Here, the set E + ∖ δ ( Π ) {\displaystyle E^{+}\setminus \delta (\Pi )} contains the attractive edges whose endpoints are in the same component with respect to the clustering Π {\displaystyle \Pi } and the set E − ∩ δ ( Π ) {\displaystyle E^{-}\cap \delta (\Pi )} contains the repulsive edges whose endpoints are in different components with respect to the clustering Π {\displaystyle \Pi } . Together these two sets contain all edges that agree with the clustering Π {\displaystyle \Pi } . Instead of formulating the correlation clustering problem in terms of non-negative edge weights and a partition of the edges into attractive and repulsive edges the problem is also formulated in terms of positive and negative edge costs without partitioning the set of edges explicitly. For given weights w : E → R ≥ 0 {\displaystyle w\colon E\to \mathbb {R} _{\geq 0}} and a given partition E = E + ∪ E − {\displaystyle E=E^{+}\cup E^{-}} of the edges into attractive and repulsive edges, the edge costs can be defined by c e = { w e if e ∈ E + − w e if e ∈ E − {\displaystyle {\begin{aligned}c_{e}={\begin{cases}\;\;w_{e}&{\text{if }}e\in E^{+}\\-w_{e}&{\text{if }}e\in E^{-}\end{cases}}\end{aligned}}} for all e ∈ E {\displaystyle e\in E} . An edge whose endpoints are in different clusters is said to be cut. The set δ ( Π ) {\displaystyle \delta (\Pi )} of all edges that are cut is often called a multicut of G {\displaystyle G} . The minimum cost multicut problem is the problem of finding a clustering Π {\displaystyle \Pi } of G {\displaystyle G} such that the sum of the costs of the edges whose endpoints are in different clusters is minimal: minimize Π ∑ e ∈ δ ( Π ) c e . {\displaystyle {\begin{aligned}&{\underset {\Pi }{\operatorname {minimize} }}&&\sum _{e\in \delta (\Pi )}c_{e}\;.\end{aligned}}} Similar to the minimum cost multicut problem, coalition structure generation in weighted graph games is the problem of finding a clustering such that the sum of the costs of the edges that are not cut is maximal: maximize Π ∑ e ∈ E ∖ δ ( Π ) c e . {\displaystyle {\begin{aligned}&{\underset {\Pi }{\operatorname {maximize} }}&&\sum _{e\in E\setminus \delta (\Pi )}c_{e}\;.\end{aligned}}} This formulation is also known as the clique partitioning problem. It can be shown that all four problems that are formulated above are equivalent. This means that a clustering that is optimal with respect to any of the four objectives is optimal for all of the four objectives. == Algorithms == If the graph admits a clustering with zero disagreements, then deleting all negative edges and computing the connected components of the remaining graph yields an optimal clustering. A necessary and sufficient condition for the existence of such a clustering was given by Davis: no cycle in the graph may contain exactly one negative edge. Bansal et al. discuss the NP-completeness proof and also present both a constant factor approximation algorithm and polynomial-time approximation scheme to find the clusters in this setting. Ailon et al. propose a randomized 3-approximation algorithm for the same problem. CC-Pivot(G=(V,E+,E−)) Pick random pivot i ∈ V Set C = { i } {\displaystyle C=\{i\}} , V'=Ø For all j ∈ V, j ≠ i; If (i,j) ∈ E+ then Add j to C Else (If (i,j) ∈ E−) Add j to V' Let G' be the subgraph induced by V' Return clustering C,CC-Pivot(G') The authors show that the above algorithm is a 3-approximation algorithm for correlation clustering. The best polynomial-time approximation algorithm known at the moment for this problem achieves a ~2.06 approximation by rounding a linear program, as shown by Chawla, Makarychev, Schramm, and Yaroslavtsev. Karpinski and Schudy proved existence of a polynomial time approximation scheme (PTAS) for that problem on complete graphs and fixed number of clusters. == Optimal number of clusters == In 2011, it was shown by Bagon and Galun that the optimization of the correlation clustering functional is closely related to well known discrete optimization methods. In their work they proposed a probabilistic analysis of the underlying implicit model that allows the correlation clustering functional to estimate the

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  World Programming System

  The World Programming System, also known as WPS Analytics or WPS, is a software product developed by a company called World Programming (acquired by Altair Engineering). WPS Analytics supports users of mixed ability to access and process data and to perform data science tasks. It has interactive visual programming tools using data workflows, and it has coding tools supporting the use of the SAS language mixed with Python, R and SQL. == About == WPS can use programs written in the language of SAS without the need for translating them into any other language. In this regard WPS is compatible with the SAS system. WPS has a built-in language interpreter able to process the language of SAS and produce similar results. WPS is available to run on z/OS, Windows, macOS, Linux (x86, Armv8 64-bit, IBM Power LE, IBM Z), and AIX. On all supported platforms, programs written in the language of SAS can be executed from a WPS command line interface, often referred to as running in batch mode. WPS can also be used from a graphical user interface known as the WPS Workbench for managing, editing and running programs written in the language of SAS. The WPS Workbench user interface is based on Eclipse. WPS version 4 (released in March 2018) introduced a drag-and-drop workflow canvas providing interactive blocks for data retrieval, blending and preparation, data discovery and profiling, predictive modelling powered by machine learning algorithms, model performance validation and scorecards. WPS version 3 (released in February 2012) provided a new client/server architecture that allows the WPS Workbench GUI to execute SAS programs on remote server installations of WPS in a network or cloud. The resulting output, data sets, logs, etc., can then all be viewed and manipulated from inside the Workbench as if the workloads had been executed locally. SAS programs do not require any special language statements to use this feature. == Summary of main features == Runs on Windows, macOS, z/OS, Linux (x86, Armv8 64-bit, IBM Power LE, IBM Z), and AIX An integrated development environment based on Eclipse for Linux, macOS and Windows. Support for language of SAS elements. Support for the language of SAS Macros. Matrix Programming support using PROC IML. Support for generating band plots, bar charts, box plots, bubble plots, contour plots, dendrogram plots, ellipse plots, fringe plots, heat maps, high-low plots, histograms, loess plots, needle plots, pie charts, penalised b-spline, radar charts, reference lines, scatter plots, series plots, step plots, regression plots and vector plots. Support for statistical procedures ACECLUS, ASSOCRULES, ANOVA, BIN, BOXPLOT, CANCORR, CANDISC, CLUSTER, CORRESP, DISCRIM, DISTANCE, FACTOR, FASTCLUS, FREQ, GAM, GANNO, GENMOD, GLIMMIX, GLM, GLMMOD, GLMSELECT, ICLIFETEST, KDE, LIFEREG, LIFETEST, LOESS, LOGISTIC, MDS, MEANS, MI, MIANALYSE, MIXED, MODECLUS, NESTED, NLIN, NPAR1WAY, PHREG, PLAN, PLS, POWER, PRINCOMP, PROBIT, QUANTREG, RBF, REG, ROBUSTREG, RSREG, SCORE, SEGMENT, SIMNORMAL, STANDARD, STDSIZE, STDRATE, STEPDISC, SUMMARY, SURVEYMEANS, SURVEYSELECT, TPSPLINE, TRANSREG, TREE, TTEST, UNIVARIATE, VARCLUS, VARCOMP Support for time series procedures ARIMA, AUTOREG, ESM, EXPAND, FORECAST, LOAN, SEVERITY, SPECTRA, TIMESERIES, X12 Support for machine learning procedures DECISIONFOREST, DECISIONTREE, GMM, MLP, OPTIMALBIN, SEGMENT, SVM Support for ODS. Reads and writes SAS datasets (compressed or uncompressed). Access: Actian Matrix (previously known as ParAccel), DASD, DB2, Excel, Greenplum, Hadoop, Informix, Kognitio Archived 2012-08-24 at the Wayback Machine, MariaDB, MySQL, Netezza, ODBC, OLEDB, Oracle, PostgreSQL, SAND, Snowflake, SPSS/PSPP, SQL Server, Sybase, Sybase IQ, Teradata, VSAM, Vertica and XML. Support for SAS Tape Format. Direct output of reports to CSV, PDF and HTML. Support to connect WPS systems programmatically, remote submit parts of a program to execute on connected remote servers, upload and download data between the connected systems. Support for Hadoop Support for R Support for Python == Industry recognition == Gartner recognized World Programming in their Cool Vendors in Data Science, 2014 Report. == Lawsuit == In 2010 World Programming defended its use of the language of SAS in the High Court of England and Wales in SAS Institute Inc. v World Programming Ltd. The software was the subject of a lawsuit by SAS Institute. The EU Court of Justice ruled in favor of World Programming, stating that the copyright protection does not extend to the software functionality, the programming language used and the format of the data files used by the program. It stated that there is no copyright infringement when a company which does not have access to the source code of a program studies, observes and tests that program to create another program with the same functionality.

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  Sprite multiplexing

  Sprite multiplexing is a computer graphics technique where additional sprites (moving images) can be drawn on the screen, beyond the nominal maximum. It is largely historical, applicable principally to older hardware, where limited resources (such as CPU speed and memory) meant only a relatively small number of sprites were supported. On the other hand, it is also true that without multiplexing, the sprite circuitry would be idle much of the time, and limited resources were wasted. == Description == The sprite multiplexing technique is based on the idea that while the hardware may only support a finite number of sprites, it is sometimes possible to re-use the same sprite "slots" more than once per frame or scan line. The program will first use the hardware to draw one or more sprite(s), as normal. Before the next frame (or next scanline) needs to be drawn, the software reprograms the hardware to display additional sprites, in other positions. For example, the Nintendo Entertainment System explicitly supports hardware sprite multiplexing, where it has 64 hardware sprites, but is only capable of rendering 8 of them per scanline. On the older Atari 2600, sprite multiplexing was not intentionally designed in, but programmers discovered they could reset the TIA graphics chip to draw additional sprites on the same scanline. The sprite multiplexing technique relies on the program being able to identify what part of the video screen is being drawn at the moment, or being triggered by the video hardware to run a subroutine at the crucial moment. The programmer must carefully consider the layout of the screen. If the video graphics hardware is not reprogrammed in time for the extra sprites to be displayed, they will not appear, or will be drawn incorrectly. Modern video graphics hardware typically does not use hardware sprites, since modern computer systems do not have the kind of limitations that sprite hardware is designed to circumvent. == Implementations == Systems that allow the programmer to employ the sprite multiplexing technique include: Atari 2600 Atari 8-bit computers Amiga Commodore 64 MSX Nintendo Entertainment System Super Nintendo Entertainment System Master System Sega Genesis/Mega Drive

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  Word2vec

  Word2vec is a technique in natural language processing for obtaining vector representations of words. These vectors capture information about the meaning of the word based on the surrounding words. The word2vec algorithm estimates these representations by modeling text in a large corpus. Once trained, such a model can detect synonymous words or suggest additional words for a partial sentence. Word2vec was developed by Tomáš Mikolov, Kai Chen, Greg Corrado, Ilya Sutskever and Jeff Dean at Google, and published in 2013. Word2vec represents a word as a high-dimension vector of numbers which capture relationships between words. In particular, words which appear in similar contexts are mapped to vectors which are nearby as measured by cosine similarity. This indicates the level of semantic similarity between the words, so for example the vectors for walk and ran are nearby, as are those for "but" and "however", and "Berlin" and "Germany". == Approach == Word2vec is a group of related models that are used to produce word embeddings. These models are shallow, two-layer neural networks that are trained to reconstruct linguistic contexts of words. Word2vec takes as its input a large corpus of text and produces a mapping of the set of words to a vector space, typically of several hundred dimensions, with each unique word in the corpus being assigned a vector in the space. Word2vec can use either of two model architectures to produce these distributed representations of words: continuous bag of words (CBOW) or continuously sliding skip-gram. In both architectures, word2vec considers both individual words and a sliding context window as it iterates over the corpus. The CBOW can be viewed as a 'fill in the blank' task, where the word embedding represents the way the word influences the relative probabilities of other words in the context window. Words which are semantically similar should influence these probabilities in similar ways, because semantically similar words should be used in similar contexts. The order of context words does not influence prediction (bag of words assumption). In the continuous skip-gram architecture, the model uses the current word to predict the surrounding window of context words. The skip-gram architecture weighs nearby context words more heavily than more distant context words. According to the authors' note, CBOW is faster while skip-gram does a better job for infrequent words. After the model is trained, the learned word embeddings are positioned in the vector space such that words that share common contexts in the corpus — that is, words that are semantically and syntactically similar — are located close to one another in the space. More dissimilar words are located farther from one another in the space. == Mathematical details == This section is based on expositions. A corpus is a sequence of words. Both CBOW and skip-gram are methods to learn one vector per word appearing in the corpus. Let V {\displaystyle V} ("vocabulary") be the set of all words appearing in the corpus C {\displaystyle C} . Our goal is to learn one vector v w ∈ R d {\displaystyle v_{w}\in \mathbb {R} ^{d}} for each word w ∈ V {\displaystyle w\in V} . The idea of skip-gram is that the vector of a word should be close to the vector of each of its neighbors. The idea of CBOW is that the vector-sum of a word's neighbors should be close to the vector of the word. === Continuous bag-of-words (CBOW) === The idea of CBOW is to represent each word with a vector, such that it is possible to predict a word using the sum of the vectors of its neighbors. Specifically, for each word w i {\displaystyle w_{i}} in the corpus, the one-hot encoding of the word is used as the input to the neural network. The output of the neural network is a probability distribution over the dictionary, representing a prediction of individual words in the neighborhood of w i {\displaystyle w_{i}} . The objective of training is to maximize ∑ i ln ⁡ Pr ( w i ∣ w i + j : j ∈ N ) {\displaystyle \sum _{i}\ln \Pr(w_{i}\mid w_{i+j}\colon j\in N)} where N {\displaystyle N} is a set of (non-zero) indices representing the relative locations of nearby words considered to be in w i {\displaystyle w_{i}} 's neighborhood. For example, if we want each word in the corpus to be predicted by every other word in a small span of 4 words. The set of relative indexes of neighbor words will be: N = { − 2 , − 1 , + 1 , + 2 } {\displaystyle N=\{-2,-1,+1,+2\}} , and the objective is to maximize ∑ i ln ⁡ Pr ( w i ∣ w i − 2 , w i − 1 , w i + 1 , w i + 2 ) {\displaystyle \sum _{i}\ln \Pr(w_{i}\mid w_{i-2},w_{i-1},w_{i+1},w_{i+2})} . In standard bag-of-words, a word's context is represented by a word-count (aka a word histogram) of its neighboring words. For example, the "sat" in "the cat sat on the mat" is represented as {"the": 2, "cat": 1, "on": 1}. Note that the last word "mat" is not used to represent "sat", because it is outside the neighborhood N = { − 2 , − 1 , + 1 , + 2 } {\displaystyle N=\{-2,-1,+1,+2\}} . In continuous bag-of-words, the histogram is multiplied by a matrix V {\displaystyle V} to obtain a continuous representation of the word's context. The matrix V {\displaystyle V} is also called a dictionary. Its columns are the word vectors. It has D {\displaystyle D} columns, where D {\displaystyle D} is the size of the dictionary. Let d {\displaystyle d} be the length of each word vector. We have V ∈ R d × D {\displaystyle V\in \mathbb {R} ^{d\times D}} . For example, multiplying the word histogram {"the": 2, "cat": 1, "on": 1} with V {\displaystyle V} , we obtain 2 v the + v cat + v on {\displaystyle 2v_{\text{the}}+v_{\text{cat}}+v_{\text{on}}} . This is then multiplied with another matrix V ′ {\displaystyle V'} of shape R D × d {\displaystyle \mathbb {R} ^{D\times d}} . Each row of it is a word vector v ′ {\displaystyle v'} . This results in a vector of length D {\displaystyle D} , one entry per dictionary entry. Then, apply the softmax to obtain a probability distribution over the dictionary. This system can be visualized as a neural network, similar in spirit to an autoencoder, of architecture linear-linear-softmax, as depicted in the diagram. The system is trained by gradient descent to minimize the cross-entropy loss. In full formula, the cross-entropy loss is: − ∑ i ln ⁡ e v w i ′ ⋅ ( ∑ j ∈ N v w j + i ) ∑ w ′ e v w ′ ′ ⋅ ( ∑ j ∈ N v w j + i ) {\displaystyle -\sum _{i}\ln {\frac {e^{v_{w_{i}}'\cdot (\sum _{j\in N}v_{w_{j+i}})}}{\sum _{w'}e^{v_{w'}'\cdot (\sum _{j\in N}v_{w_{j+i}})}}}} where the outer summation ∑ i {\displaystyle \sum _{i}} is over the words in a corpus, the quantity ∑ j ∈ N v w j + i {\displaystyle \sum _{j\in N}v_{w_{j+i}}} is the sum of a word's neighbors' vectors, etc. Once such a system is trained, we have two trained matrices V , V ′ {\displaystyle V,V'} . Either the column vectors of V {\displaystyle V} or the row vectors of V ′ {\displaystyle V'} can serve as the dictionary. For example, the word "sat" can be represented as either the "sat"-th column of V {\displaystyle V} or the "sat"-th row of V ′ {\displaystyle V'} . It is also possible to simply define V ′ = V ⊤ {\displaystyle V'=V^{\top }} , in which case there would no longer be a choice. === Skip-gram === The idea of skip-gram is to represent each word with a vector, such that it is possible to predict the vectors of its neighbors using the vector of a word. The architecture is still linear-linear-softmax, the same as CBOW, but the input and the output are switched. Specifically, for each word w i {\displaystyle w_{i}} in the corpus, the one-hot encoding of the word is used as the input to the neural network. The output of the neural network is a probability distribution over the dictionary, representing a prediction of individual words in the neighborhood of w i {\displaystyle w_{i}} . The objective of training is to maximize ∑ i ∑ j ∈ N ln ⁡ Pr ( w j + i ∣ w i ) {\displaystyle \sum _{i}\sum _{j\in N}\ln \Pr(w_{j+i}\mid w_{i})} . In full formula, the loss function is − ∑ i ∑ j ∈ N ln ⁡ e v w j + i ′ ⋅ v w i ∑ w ′ e v w ′ ′ ⋅ v w i {\displaystyle -\sum _{i}\sum _{j\in N}\ln {\frac {e^{v_{w_{j+i}}'\cdot v_{w_{i}}}}{\sum _{w'}e^{v_{w'}'\cdot v_{w_{i}}}}}} Same as CBOW, once such a system is trained, we have two trained matrices V , V ′ {\displaystyle V,V'} . Either the column vectors of V {\displaystyle V} or the row vectors of V ′ {\displaystyle V'} can serve as the dictionary. It is also possible to simply define V ′ = V ⊤ {\displaystyle V'=V^{\top }} , in which case there would no longer be a choice. Essentially, skip-gram and CBOW are exactly the same in architecture. They only differ in the objective function during training. == History == During the 1980s, there were some early attempts at using neural networks to represent words and concepts as vectors. In 2010, Tomáš Mikolov (then at Brno University of Technology) with co-authors applied a simple recurrent neural network with a single hidden

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  Rectified linear unit

  In the context of artificial neural networks, the rectifier or ReLU (rectified linear unit) activation function is an activation function defined as the non-negative part of its argument, i.e., the ramp function: ReLU ⁡ ( x ) = x + = max ( 0 , x ) = x + | x | 2 = { x if x > 0 , 0 x ≤ 0 {\displaystyle \operatorname {ReLU} (x)=x^{+}=\max(0,x)={\frac {x+|x|}{2}}={\begin{cases}x&{\text{if }}x>0,\\0&x\leq 0\end{cases}}} where x {\displaystyle x} is the input to a neuron. This is analogous to half-wave rectification in electrical engineering. ReLU is one of the most popular activation functions for artificial neural networks, and finds application in computer vision and speech recognition using deep neural nets and computational neuroscience. == History == The ReLU was first used by Alston Householder in 1941 as a mathematical abstraction of biological neural networks. Kunihiko Fukushima in 1969 used ReLU in the context of visual feature extraction in hierarchical neural networks. In 1998, Gregory Woodbury demonstrated that the rectified linear function could account for a broad range of emergent properties in the visual cortex. His work showed that a single unified model could drive the joint development of refined retinotopic maps, ocular dominance columns, and orientation selectivity. By utilizing the rectifier's "cutoff" property, Woodbury achieved a close quantitative fit to biological data, matching the spatial periodicities and topographic refinement patterns observed in macaque and cat cortical maps. Furthermore, he extended this framework to adult plasticity, accurately replicating the spatial and temporal dynamics of lesion-induced cortical reorganization. This research established that the rectified linear response was a necessary mechanism for the stable self-organisation and maintenance of complex, multi-feature neural maps. In 2000, Hahnloser et al. argued that ReLU approximates the biological relationship between neural firing rates and input current, in addition to enabling recurrent neural network dynamics to stabilise under weaker criteria. Prior to 2010, most activation functions used were the logistic sigmoid (which is inspired by probability theory; see logistic regression) and its more numerically efficient counterpart, the hyperbolic tangent. Around 2010, the use of ReLU became common again. Jarrett et al. (2009) noted that rectification by either absolute or ReLU (which they called "positive part") was critical for object recognition in convolutional neural networks (CNNs), specifically because it allows average pooling without neighboring filter outputs cancelling each other out. They hypothesized that the use of sigmoid or tanh was responsible for poor performance in previous CNNs. Nair and Hinton (2010) made a theoretical argument that the softplus activation function should be used, in that the softplus function numerically approximates the sum of an exponential number of linear models that share parameters. They then proposed ReLU as a good approximation to it. Specifically, they began by considering a single binary neuron in a Boltzmann machine that takes x {\displaystyle x} as input, and produces 1 as output with probability σ ( x ) = 1 1 + e − x {\displaystyle \sigma (x)={\frac {1}{1+e^{-x}}}} . They then considered extending its range of output by making infinitely many copies of it X 1 , X 2 , X 3 , … {\displaystyle X_{1},X_{2},X_{3},\dots } , that all take the same input, offset by an amount 0.5 , 1.5 , 2.5 , … {\displaystyle 0.5,1.5,2.5,\dots } , then their outputs are added together as ∑ i = 1 ∞ X i {\displaystyle \sum _{i=1}^{\infty }X_{i}} . They then demonstrated that ∑ i = 1 ∞ X i {\displaystyle \sum _{i=1}^{\infty }X_{i}} is approximately equal to N ( log ⁡ ( 1 + e x ) , σ ( x ) ) {\displaystyle {\mathcal {N}}(\log(1+e^{x}),\sigma (x))} , which is also approximately equal to ReLU ⁡ ( N ( x , σ ( x ) ) ) {\displaystyle \operatorname {ReLU} ({\mathcal {N}}(x,\sigma (x)))} , where N {\displaystyle {\mathcal {N}}} stands for the gaussian distribution. They also argued for another reason for using ReLU: that it allows "intensity equivariance" in image recognition. That is, multiplying input image by a constant k {\displaystyle k} multiplies the output also. In contrast, this is false for other activation functions like sigmoid or tanh. They found that ReLU activation allowed good empirical performance in restricted Boltzmann machines. Glorot et al (2011) argued that ReLU has the following advantages over sigmoid or tanh: ReLU is more similar to biological neurons' responses in their main operating regime. ReLU avoids vanishing gradients. ReLU is cheaper to compute. ReLU creates sparse representation naturally, because many hidden units output exactly zero for a given input. They also found empirically that deep networks trained with ReLU can achieve strong performance without unsupervised pre-training, especially on large, purely supervised tasks. In 2017, the rectified linear function became a central component of the transformer architecture introduced in the Vaswani et al paper "Attention Is All You Need". Within every transformer layer, ReLU is utilized in the position-wise feed-forward networks (FFN), defined by Equation 2 of their paper: FFN ⁡ ( x ) = max ( 0 , x W 1 + b 1 ) W 2 + b 2 {\displaystyle \operatorname {FFN} (x)=\max(0,xW_{1}+b_{1})W_{2}+b_{2}} This equation is foundational to the model's capacity; while the attention mechanism determines the relationships between tokens, the ReLU-based FFN performs the majority of the numerical computation and houses the bulk of the model's parameters. The efficiency and scalability of this rectified framework triggered a global technological revolution, enabling the development of Large Language Models that have had a profound economic impact. The industrial response to this architecture—including the massive expansion of AI-specific hardware and the birth of the generative AI sector—has positioned the Transformer as a cornerstone of 21st-century infrastructure. During the post 2017 period of rapid AI advancement, the rectified linear unit function has been key to achieving increased model performance and scaling due to the fact that it zeros out responses that are immaterial for a given stimuli, preventing them from accumulating in massive scale models. It is the complete silencing of the parts of the model found to be stimuli-irrelevant during learning that allows for scaling. As the stimuli-irrelevant proportion of the model becomes more massive, these highly numerous connections within the model would inevitably accumulate during scaling no matter how small each individual response is. Therefore, the rectified linear unit function, with its absolute zeroing property, enabled the scaling to hundred billion parameter models and beyond. Early Transformer scaling giants like GPT-3 (2020) and Falcon-180B (2023) relied on the rectified linear unit function explicitly, while successors such as GPT-4 (2023) and Llama 3 (2024) utilized smoother variants like GELU or SwiGLU. These variants were used to improve training stability while fundamentally preserving the rectified principle of zeroing low responses. At the centre of modern artificial intelligence ReLU and its variants maintain absolute zero response across the bulk of the model at any one time, while maintaining approximately linear reponses for stimuli-relevant connections enabling high performance on each specific cognitive task. This feature of activation sparsity has been critical for massive scaling and performance gains of AI models right up to the present day. == Advantages == Advantages of ReLU include: Sparse activation: for example, in a randomly initialized network, only about 50% of hidden units are activated (i.e. have a non-zero output). Better gradient propagation: fewer vanishing gradient problems compared to sigmoidal activation functions that saturate in both directions. Efficiency: only requires comparison and addition. Scale-invariant (homogeneous, or "intensity equivariance"): max ( 0 , a x ) = a max ( 0 , x ) for a ≥ 0 {\displaystyle \max(0,ax)=a\max(0,x){\text{ for }}a\geq 0} . == Potential problems == Possible downsides can include: Non-differentiability at zero (however, it is differentiable anywhere else, and the value of the derivative at zero can be chosen to be 0 or 1 arbitrarily). Not zero-centered: ReLU outputs are always non-negative. This can make it harder for the network to learn during backpropagation, because gradient updates tend to push weights in one direction (positive or negative). Batch normalization can help address this. ReLU is unbounded. Redundancy of the parametrization: Because ReLU is scale-invariant, the network computes the exact same function by scaling the weights and biases in front of a ReLU activation by k {\displaystyle k} , and the weights after by 1 / k {\displaystyle 1/k} . Dying ReLU: ReLU neurons can sometimes be pushed into states

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