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  List of assembly software and tools

  This is a list of assembly software and tools, including software used for assembly language programming, machine code generation, disassembly, debugging, binary analysis, reverse engineering, and instruction-set simulation. == Assemblers and machine-code generators == == Disassemblers and binary-analysis tools == == Debuggers with assembly-level features == == Educational IDEs, simulators and emulators == == Portable and intermediate assembly-like languages == == Assembly language families == Assembly language is not a single programming language, but a family of low-level languages associated with particular instruction set architectures and processor families. Examples include:

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  Tanagra (machine learning)

  Tanagra is a free suite of machine learning software for research and academic purposes developed by Ricco Rakotomalala at the Lumière University Lyon 2, France. Tanagra supports several standard data mining tasks such as: Visualization, Descriptive statistics, Instance selection, feature selection, feature construction, regression, factor analysis, clustering, classification and association rule learning. Tanagra is an academic project. It is widely used in French-speaking universities. Tanagra is frequently used in real studies and in software comparison papers. == History == The development of Tanagra was started in June 2003. The first version was distributed in December 2003. Tanagra is the successor of Sipina, another free data mining tool which is intended only for supervised learning tasks (classification), especially the interactive and visual construction of decision trees. Sipina is still available online and is maintained. Tanagra is an "open source project" as every researcher can access the source code and add their own algorithms, as long as they agree and conform to the software distribution license. The main purpose of the Tanagra project is to give researchers and students a user-friendly data mining software, conforming to the present norms of the software development in this domain (especially in the design of its GUI and the way to use it), and allowing the analyzation of either real or synthetic data. From 2006, Ricco Rakotomalala made an important documentation effort. A large number of tutorials are published on a dedicated website. They describe the statistical and machine learning methods and their implementation with Tanagra on real case studies. The use of other free data mining tools on the same problems is also widely described. The comparison of the tools enables readers to understand the possible differences in the presentation of results. == Description == Tanagra works similarly to current data mining tools. The user can design visually a data mining process in a diagram. Each node is a statistical or machine learning technique, the connection between two nodes represents the data transfer. But unlike the majority of tools which are based on the workflow paradigm, Tanagra is very simplified. The treatments are represented in a tree diagram. The results are displayed in an HTML format. This makes it is easy to export the outputs in order to visualize the results in a browser. It is also possible to copy the result tables to a spreadsheet. Tanagra makes a good compromise between statistical approaches (e.g. parametric and nonparametric statistical tests), multivariate analysis methods (e.g. factor analysis, correspondence analysis, cluster analysis, regression) and machine learning techniques (e.g. neural network, support vector machine, decision trees, random forest).

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  Memtransistor

  The memtransistor (a blend word from Memory Transfer Resistor) is an experimental multi-terminal passive electronic component that might be used in the construction of artificial neural networks. It is a combination of the memristor and transistor technology. This technology is different from the 1T-1R approach since the devices are merged into one single entity. Multiple memristors can be embedded with a single transistor, enabling it to more accurately model a neuron with its multiple synaptic connections. A neural network produced from these would provide hardware-based artificial intelligence with a good foundation. == Applications == These types of devices would allow for a synapse model that could realise a learning rule, by which the synaptic efficacy is altered by voltages applied to the terminals of the device. An example of such a learning rule is spike-timing-dependant-plasticty by which the weight of the synapse, in this case the conductivity, could be modulated based on the timing of pre and post synaptic spikes arriving at each terminal. The advantage of this approach over two terminal memristive devices is that read and write protocols have the possibility to occur simultaneously and distinctly.

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  Automated Pain Recognition

  Automated Pain Recognition (APR) is a method for objectively measuring pain and at the same time represents an interdisciplinary research area that comprises elements of medicine, psychology, psychobiology, and computer science. The focus is on computer-aided objective recognition of pain, implemented on the basis of machine learning. Automated pain recognition allows for the valid, reliable detection and monitoring of pain in people who are unable to communicate verbally. The underlying machine learning processes are trained and validated in advance by means of unimodal or multimodal body signals. Signals used to detect pain may include facial expressions or gestures and may also be of a (psycho-)physiological or paralinguistic nature. To date, the focus has been on identifying pain intensity, but visionary efforts are also being made to recognize the quality, site, and temporal course of pain. However, the clinical implementation of this approach is a controversial topic in the field of pain research. Critics of automated pain recognition argue that pain diagnosis can only be performed subjectively by humans. == Background == Pain diagnosis under conditions where verbal reporting is restricted - such as in verbally and/or cognitively impaired people or in patients who are sedated or mechanically ventilated - is based on behavioral observations by trained professionals. However, all known observation procedures (e.g., Zurich Observation Pain Assessment (ZOPA)); Pain Assessment in Advanced Dementia Scale (PAINAD) require a great deal of specialist expertise. These procedures can be made more difficult by perception- and interpretation-related misjudgments on the part of the observer. With regard to the differences in design, methodology, evaluation sample, and conceptualization of the phenomenon of pain, it is difficult to compare the quality criteria of the various tools. Even if trained personnel could theoretically record pain intensity several times a day using observation instruments, it would not be possible to measure it every minute or second. In this respect, the goal of automated pain recognition is to use valid, robust pain response patterns that can be recorded multimodally for a temporally dynamic, high-resolution, automated pain intensity recognition system. == Procedure == For automated pain recognition, pain-relevant parameters are usually recorded using non-invasive sensor technology, which captures data on the (physical) responses of the person in pain. This can be achieved with camera technology that captures facial expressions, gestures, or posture, while audio sensors record paralinguistic features. (Psycho-)physiological information such as muscle tone and heart rate can be collected via biopotential sensors (electrodes). Pain recognition requires the extraction of meaningful characteristics or patterns from the data collected. This is achieved using machine learning techniques that are able to provide an assessment of the pain after training (learning), e.g., "no pain," "mild pain," or "severe pain." == Parameters == Although the phenomenon of pain comprises different components (sensory discriminative, affective (emotional), cognitive, vegetative, and (psycho-)motor), automated pain recognition currently relies on the measurable parameters of pain responses. These can be divided roughly into the two main categories of "physiological responses" and "behavioral responses". === Physiological responses === In humans, pain almost always initiates autonomic nervous processes that are reflected measurably in various physiological signals. ==== Physiological signals ==== Measurements can include electrodermal activity (EDA, also skin conductance), electromyography (EMG), electrocardiogram (ECG), blood volume pulse (BVP), electroencephalogram (EEG), respiration, and body temperature, which are regulatory mechanisms of the sympathetic and parasympathetic systems. Physiological signals are mainly recorded using special non-invasive surface electrodes (for EDA, EMG, ECG, and EEG), a blood volume pulse sensor (BVP), a respiratory belt (respiration), and a thermal sensor (body temperature). Endocrinological and immunological parameters can also be recorded, but this requires measures that are somewhat invasive (e.g., blood sampling). === Behavioral responses === Behavioral responses to pain fulfil two functions: protection of the body (e.g., through protective reflexes) and external communication of the pain (e.g., as a cry for help). The responses are particularly evident in facial expressions, gestures, and paralinguistic features. ==== Facial expressions ==== Behavioral signals captured comprise facial expression patterns (expressive behavior), which are measured with the aid of video signals. Facial expression recognition is based on the everyday clinical observation that pain often manifests itself in the patient's facial expressions but that this is not necessarily always the case, since facial expressions can be inhibited through self-control. Despite the possibility that facial expressions may be influenced consciously, facial expression behavior represents an essential source of information for pain diagnosis and is thus also a source of information for automatic pain recognition. One advantage of video-based facial expression recognition is the contact-free measurement of the face, provided that it can be captured on video, which is not possible in every position (e.g., lying face down) or may be limited by bandages covering the face. Facial expression analysis relies on rapid, spontaneous, and temporary changes in neuromuscular activity that lead to visually detectable changes in the face. ==== Gestures ==== Gestures are also captured predominantly using non-contact camera technology. Motor pain responses vary and are strongly dependent on the type and cause of the pain. They range from abrupt protective reflexes (e.g., spontaneous retraction of extremities or doubling up) to agitation (pathological restlessness) and avoidance behavior (hesitant, cautious movements). ==== Paralinguistic features of language ==== Among other things, pain leads to nonverbal linguistic behavior that manifests itself in sounds such as sighing, gasping, moaning, whining, etc. Paralinguistic features are usually recorded using highly sensitive microphones. == Algorithms == After the recording, pre-processing (e.g., filtering), and extraction of relevant features, an optional information fusion can be performed. During this process, modalities from different signal sources are merged to generate new or more precise knowledge. The pain is classified using machine learning processes. The method chosen has a significant influence on the recognition rate and depends greatly on the quality and granularity of the underlying data. Similar to the field of affective computing, the following classifiers are currently being used: Support Vector Machine (SVM): The goal of an SVM is to find a clearly defined optimal hyperplane with the greatest minimal distance to two (or more) classes to be separated. The hyperplane acts as a decision function for classifying an unknown pattern. Random Forest (RF): RF is based on the composition of random, uncorrelated decision trees. An unknown pattern is judged individually by each tree and assigned to a class. The final classification of the patterns by the RF is then based on a majority decision. k-Nearest Neighbors (k-NN): The k-NN algorithm classifies an unknown object using the class label that most commonly classifies the k neighbors closest to it. Its neighbors are determined using a selected similarity measure (e.g., Euclidean distance, Jaccard coefficient, etc.). Artificial neural networks (ANNs): ANNs are inspired by biological neural networks and model their organizational principles and processes in a very simplified manner. Class patterns are learned by adjusting the weights of the individual neuronal connections. == Databases == In order to classify pain in a valid manner, it is necessary to create representative, reliable, and valid pain databases that are available to the machine learner for training. An ideal database would be sufficiently large and would consist of natural (not experimental), high-quality pain responses. However, natural responses are difficult to record and can only be obtained to a limited extent; in most cases they are characterized by suboptimal quality. The databases currently available therefore contain experimental or quasi-experimental pain responses, and each database is based on a different pain model. The following list shows a selection of the most relevant pain databases (last updated: April 2020): UNBC-McMaster Shoulder Pain BioVid Heat Pain EmoPain SenseEmotion X-ITE Pain

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  Ed (chatbot)

  Ed was a chatbot co-developed by the Los Angeles Unified School District and AllHere Education. Described as a learning acceleration platform, it was the first personal assistant for students in the United States. Part of the district's Individual Acceleration Plan, it was able to interact with students both verbally and visually, offering support in 100 languages. The chatbot was launched on March 20, 2024, as part of the district's plan for academic recovery from the COVID-19 pandemic and to improve overall academic performance. Utilizing artificial intelligence, Ed organizes data and reports on grades, test scores, and attendance, creating individualized plans for each student. After the company behind it, AllHere, collapsed, the district shuttered operations of the chatbot on June 14, 2024. The firm is under investigation by the US Federal Bureau of Investigation. == History == On February 14, 2022, Alberto M. Carvalho became the Superintendent of the Los Angeles Unified School District, pledging to give the district a full academic recovery from the COVID-19 pandemic. In December 2022, he announced the Individual Acceleration Plan for the district, which aimed to provide each student with a unique progress report and help them determine if they were on track to graduate. The district faced criticism from disability advocates for its management of Individualized Education Programs, and in April 2022, the United States Department of Education announced that the district had failed to provide appropriate educational services to students with disabilities during the pandemic. The district had been grappling with significant absenteeism issues since the pandemic, which led to declining academic performance and disengagement among students. On February 17, 2023, the district issued a request for proposals to develop a fully integrated portal system. Later that year, they signed a $6 million, five-year contract with AllHere Education, a Boston-based company founded in 2016. The introduction of Ed follows the public launch of ChatGPT, which has been utilized by both teachers and students in educational settings. On August 4, 2023, during an annual address at the Walt Disney Concert Hall, Carvalho and the Los Angeles Unified School District announced the launch of Ed. The district invested $4 million into the chatbot, with Carvalho noting that this cost would be halved thanks to donor and grant funding. The chatbot was launched on March 20, 2024. Following its launch, a press conference was held to address security and technology concerns. Carvalho stated that the district had collaborated with security companies and incorporated filters to screen for threatening language. Months after its launch, AllHere Education furloughed most of its staff on June 14, citing their “current financial position” on its website as the reason. After learning about the furlough, the district terminated its dealings with AllHere Education. However, it stated its intention to bring the chatbot back in the future once officials determine the best course of action. Carvalho announced that he would appoint an independent task force to review what went wrong with AllHere Education and the chatbot. On February 25, 2026, the FBI served a search warrant on Carvalho’s home and office in connection with AllHere. The FBI also raided the LAUSD's headquarters. == Service == The chatbot was described as a personal assistant and a "one-stop shop for parents and students" who want to see information about a student's attendance and grades, as well as other resources from the district. Additionally, the application can function as an alarm clock, provide daily lunch menus from the school cafeteria, and offer updates on the location of school buses. The chatbot also helps students and parents who do not speak English as their first language by translating displayed information into approximately 100 different languages. The application can also help with submitting applications and give updates on progress and upcoming assignments. The district stated that the primary goal of Ed was to actively motivate students to complete homework and other tasks. == Reception == The chatbot received a mostly positive reception among parents and observers upon its launch. Some parents and teachers expressed caution about the technology, voicing concerns that the district's push for its implementation lacked public accountability. Rob Nelson from the University of Pennsylvania described the district's strategy as risky, saying that the release felt "like the beginning of a Clippy-level disaster". After the chatbot's shutdown, The 74 criticized it for misusing student data. Chris Whiteley, a former software engineer at AllHere Education, alleged that the data collected by the chatbot likely violated the district's data privacy rules.

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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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  Kernel principal component analysis

  In the field of multivariate statistics, kernel principal component analysis (kernel PCA) is an extension of principal component analysis (PCA) using techniques of kernel methods. Using a kernel, the originally linear operations of PCA are performed in a reproducing kernel Hilbert space. == Background: Linear PCA == Recall that conventional PCA operates on zero-centered data; that is, 1 N ∑ i = 1 N x i = 0 {\displaystyle {\frac {1}{N}}\sum _{i=1}^{N}\mathbf {x} _{i}=\mathbf {0} } , where x i {\displaystyle \mathbf {x} _{i}} is one of the N {\displaystyle N} multivariate observations. It operates by diagonalizing the covariance matrix, C = 1 N ∑ i = 1 N x i x i ⊤ {\displaystyle C={\frac {1}{N}}\sum _{i=1}^{N}\mathbf {x} _{i}\mathbf {x} _{i}^{\top }} in other words, it gives an eigendecomposition of the covariance matrix: λ v = C v {\displaystyle \lambda \mathbf {v} =C\mathbf {v} } which can be rewritten as λ x i ⊤ v = x i ⊤ C v for i = 1 , … , N {\displaystyle \lambda \mathbf {x} _{i}^{\top }\mathbf {v} =\mathbf {x} _{i}^{\top }C\mathbf {v} \quad {\textrm {for}}~i=1,\ldots ,N} . (See also: Covariance matrix as a linear operator) == Introduction of the Kernel to PCA == To understand the utility of kernel PCA, particularly for clustering, observe that, while N points cannot, in general, be linearly separated in d < N {\displaystyle d Read more →

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  Sample exclusion dimension

  In computational learning theory, sample exclusion dimensions arise in the study of exact concept learning with queries. In algorithmic learning theory, a concept over a domain X is a Boolean function over X. Here we only consider finite domains. A partial approximation S of a concept c is a Boolean function over Y ⊆ X {\displaystyle Y\subseteq X} such that c is an extension to S. Let C be a class of concepts and c be a concept (not necessarily in C). Then a specifying set for c w.r.t. C, denoted by S is a partial approximation S of c such that C contains at most one extension to S. If we have observed a specifying set for some concept w.r.t. C, then we have enough information to verify a concept in C with at most one more mind change. The exclusion dimension, denoted by XD(C), of a concept class is the maximum of the size of the minimum specifying set of c' with respect to C, where c' is a concept not in C.

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  Boris FX

  Boris FX is a visual effects, video editing, photography, and audio software plug-in developer based in Miami, Florida, USA. The developer is known for its flagship products, Continuum (formerly Boris Continuum Complete/BCC), Sapphire, Mocha, and Silhouette. Boris FX creates plug-in tools for feature film, broadcast television, and multimedia post-production workflows. The plug-ins are compatible with various NLEs, including Adobe After Effects and Premiere Pro, Avid Media Composer, Apple Final Cut Pro, and OFX hosts such as Autodesk Flame, Foundry Nuke, Blackmagic Design DaVinci Resolve and Fusion, and VEGAS Pro. Boris FX has incorporated artificial intelligence into its software, introducing features for noise reduction, rotoscoping, upscaling, and masking. The company has acquired technologies via mergers and acquisitions from Imagineer Systems, GenArts, Silhouette FX, Digital Film Tools, CrumplePop and Andersson Technologies to expand its visual effects, editing, photography, and audio tools. == History == Boris FX was founded in 1995 by Boris Yamnitsky. The former Media 100 engineer (a member of the original Media 100 launch team in 1993) released “Boris FX,” the first plug-in-based digital video effects (DVE) for Adobe Premiere and Media 100, in 1995. The plug-in won Best of Show at Apple Macworld in Boston, MA that same year. The Boris FX Suite includes a range of visual effects and post-production tools, such as Sapphire, Continuum, Mocha Pro, Silhouette, SynthEyes, CrumplePop, Optics, and Particle Illusion. == Media 100 == In October 2005, Yamnitsky acquired Media 100 the company that launched his plug-in career. Boris FX had a long relationship with Media 100 which bundled Boris RED software as its main titling and compositing solution. Media 100's video editing software is available as freeware for macOS. == Continuum == Continuum is a visual effect and compositing plugin suite that includes a library of over 300 effects and more than 40 transitions, including tools for image restoration, compositing, titling, particle generation, and stylized effects, along with features such as lens flares, lighting effects, and cinematic color grading presets. A key component of Continuum is its integration with the Mocha planar tracking and masking system, enabling advanced tracking and rotoscoping within the effects. The suite also includes Particle Illusion, a real-time particle generator used for creating visual effects such as explosions, smoke, and abstract motion graphics, as well as Primatte Studio, a chroma keying and compositing toolset for green screen and blue screen workflows. Continuum supports GPU acceleration and offers compatibility with HDR and 360/VR content. Regular updates introduce new effects, presets, and performance enhancements to expand its capabilities. In October 2018, Continuum relaunched Particle Illusion, a Mocha Essentials workflow with magnetic edge-snapping, and updates to Title Studio. In October 2019, Continuum introduced Corner Pin Studio with built-in Mocha tracking for quick screen replacement and inserts, 6 stylized transitions, and 4 creative effects. In October 2020, Continuum released an update that included over 80 GPU-accelerated effects such as film stocks, color grades, optical filter simulations, and a digital gobo library. The update also introduced a custom FX Editor interface, real-time particles, and more than 1,000 drag-and-drop presets. In November 2021, it added multi-frame rendering for After Effects, native Apple M1 support, fluid dynamics in Particle Illusion, and 60 color-grade presets. In October 2022, the software introduced 10 additional transitions, a revised Particle Illusion workflow, an atmospheric glow effect, and more than 250 curated presets. Continuum plugins have been used in television, streaming, and film projects, including A Black Lady Sketch Show (HBO/HBO Max), Star Trek: Discovery (CBS), Andor (Disney+), The Curse of Oak Island (History Channel), Keeping up with the Kardashians (E!), This Old House (PBS), Ms. Marvel (Disney+), MasterChef (Fox), WipeOut (TBS), The Boys (Prime Video), and The Today Show (NBC). == Mocha Pro == In December 2014, Boris FX merged with Imagineer Systems, the UK-based developer of the Academy Award-winning planar motion tracking software, Mocha Pro. Mocha Pro's features include planar tracking (motion tracking), rotoscoping, image stabilization, 3D camera tracking, and object removal. In June 2016, Mocha released (v5) which introduced Mocha Pro's tools as plug-ins for Adobe After Effects and Premiere Pro, Avid Media Composer, and OFX hosts Foundry's NUKE, Blackmagic Design Fusion, VEGAS Pro, and HitFilm. A simplified version, Mocha AE, is included with Adobe After Effects Creative Cloud and has been bundled with the software since CS4. A similar version is also available with HitFilm Pro from FXhome and VEGAS Pro. Mocha's tracking SDK is integrated into other visual effects tools, including SAM Quantel Pablo Rio, Silhouette FX, CoreMelt, and Motion VFX. Mocha Pro has been used in various film and television productions, including Birdman, Black Swan, the Harry Potter series, The Hobbit, Star Wars, The Mandalorian, Star Trek: Discovery, and The Umbrella Academy. It has also been employed in projects such as Gone Girl, The Hunger Games: Mockingjay – Part 1, Game of Thrones, and House of Cards. == Sapphire == GenArts, founded by Karl Sims in 1996, developed visual effects plug-ins that were used by studios and post-production facilities. In September 2016, Boris FX merged with former competitor, GenArts, Inc., developer of Sapphire high-end visual effects plug-ins, to expand its suite of motion graphics and VFX tools. The merger brought Sapphire alongside Boris Continuum Complete (BCC) and Mocha Pro, integrating these tools for film and television post-production. The Sapphire suite includes a library of over 270 effects and transitions, organized into categories such as lighting, stylization, distortions, textures, and transitions. Commonly used effects include glows, lens flares, film looks, and blurs. The plug-ins are designed to be GPU-accelerated, allowing for improved rendering performance and real-time previews in supported host applications. A central feature of Sapphire is the Builder tool, a node-based workspace that allows users to create custom effects and transitions by combining multiple Sapphire plug-ins. This enables a high level of creative flexibility and reusability, making it a popular tool for both editors and VFX artists. Sapphire also integrates with Mocha, Boris FX's planar tracking and masking system, allowing for advanced control of visual elements within an effect. In October 2017, Boris FX released its first new version of Sapphire since the GenArts acquisition. Sapphire (v11) now includes integrated Mocha tracking and masking tools. Sapphire is available for Adobe, Avid, the Autodesk Flame family, and OFX hosts including Blackmagic DaVinci Resolve and Fusion, and Foundry's NUKE. As part of the merger, Boris FX acquired the rights to Particle Illusion. In 2018, Boris FX reintroduced the product to the larger NLE/Compositing market. Sapphire's plug-ins transitioned from C to C++ to improve performance and support higher-resolution visual effects. This update enhanced floating-point calculations, compatibility with film editing APIs, and integration with NVIDIA's CUDA for faster rendering. The plug-ins have been used in various films, including Avatar, the Harry Potter and the Prisoner of Azkaban, Iron Man, The Lord of the Rings, The Matrix trilogy, Titanic, and X-Men. == Particle Illusion == As part of the merger with GenArts in 2016, Boris FX acquired the rights to the Particle Illusion (formerly particleIllusion) product, a storied particle system from the original developer Alan Lorence, the founder of Wondertouch. In 2018, Boris FX released a redesigned version of the product to a larger NLE/compositing market as part of Continuum (2019). The new Particle Illusion plug-in supports Adobe, Avid, and many OFX hosts. == Silhouette == In September 2019, Boris FX merged with SilhouetteFX, Academy Award-winning developer of Silhouette, a high-end digital paint, advanced rotoscoping, motion tracking, and node-based compositing application for visual effects in film post-production. The acquisition integrated Silhouette's advanced rotoscoping and paint technology, recognized by the Academy of Motion Pictures, into Boris FX's suite of products, alongside Sapphire, Continuum, and Mocha Pro. In May 2021, Boris FX released Silhouette 2021, the first version of Silhouette released by Boris FX to function both as a standalone application and as a plug-in for Adobe, Autodesk, Nuke, and other OFX hosts. Silhouette has been used in the visual effects of films such as Avatar, Avengers: Infinity War, Blade Runner 2049, Ex Machina, and Interstellar. == Optics == In June 2020, Boris FX launched Optics, its first plugin deve

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  Variational autoencoder

  In machine learning, a variational autoencoder (VAE) is an artificial neural network architecture introduced by Diederik P. Kingma and Max Welling in 2013. It is part of the families of probabilistic graphical models and variational Bayesian methods. In addition to being seen as an autoencoder neural network architecture, variational autoencoders can also be studied within the mathematical formulation of variational Bayesian methods, connecting a neural encoder network to its decoder through a probabilistic latent space (for example, as a multivariate Gaussian distribution) that corresponds to the parameters of a variational distribution. Thus, the encoder maps each point (such as an image) from a large complex dataset into a distribution within the latent space, rather than to a single point in that space. The decoder has the opposite function, which is to map from the latent space to the input space, again according to a distribution (although in practice, noise is rarely added during the decoding stage). By mapping a point to a distribution instead of a single point, the network can avoid overfitting the training data. Both networks are typically trained together with the usage of the reparameterization trick, although the variance of the noise model can be learned separately. Although this type of model was initially designed for unsupervised learning, its effectiveness has been proven for semi-supervised learning and supervised learning. == Overview of architecture and operation == A variational autoencoder is a generative model with a prior and noise distribution respectively. Usually such models are trained using the expectation-maximization meta-algorithm (e.g. probabilistic PCA, (spike & slab) sparse coding). Such a scheme optimizes a lower bound of the data likelihood, which is usually computationally intractable, and in doing so requires the discovery of q-distributions, or variational posteriors. These q-distributions are normally parameterized for each individual data point in a separate optimization process. However, variational autoencoders use a neural network as an amortized approach to jointly optimize across data points. In that way, the same parameters are reused for multiple data points, which can result in massive memory savings. The first neural network takes as input the data points themselves, and outputs parameters for the variational distribution. As it maps from a known input space to the low-dimensional latent space, it is called the encoder. The decoder is the second neural network of this model. It is a function that maps from the latent space to the input space, e.g. as the means of the noise distribution. It is possible to use another neural network that maps to the variance, however this can be omitted for simplicity. In such a case, the variance can be optimized with gradient descent. To optimize this model, one needs to know two terms: the "reconstruction error", and the Kullback–Leibler divergence (KL-D). Both terms are derived from the free energy expression of the probabilistic model, and therefore differ depending on the noise distribution and the assumed prior of the data, here referred to as p-distribution. For example, a standard VAE task such as IMAGENET is typically assumed to have a gaussianly distributed noise; however, tasks such as binarized MNIST require a Bernoulli noise. The KL-D from the free energy expression maximizes the probability mass of the q-distribution that overlaps with the p-distribution, which unfortunately can result in mode-seeking behaviour. The "reconstruction" term is the remainder of the free energy expression, and requires a sampling approximation to compute its expectation value. More recent approaches replace Kullback–Leibler divergence (KL-D) with various statistical distances, see "Statistical distance VAE variants" below. == Formulation == From the point of view of probabilistic modeling, one wants to maximize the likelihood of the data x {\displaystyle x} by their chosen parameterized probability distribution p θ ( x ) = p ( x | θ ) {\displaystyle p_{\theta }(x)=p(x|\theta )} . This distribution is usually chosen to be a Gaussian N ( x | μ , σ ) {\displaystyle N(x|\mu ,\sigma )} which is parameterized by μ {\displaystyle \mu } and σ {\displaystyle \sigma } respectively, and as a member of the exponential family it is easy to work with as a noise distribution. Simple distributions are easy enough to maximize, however distributions where a prior is assumed over the latents z {\displaystyle z} results in intractable integrals. Let us find p θ ( x ) {\displaystyle p_{\theta }(x)} via marginalizing over z {\displaystyle z} . p θ ( x ) = ∫ z p θ ( x , z ) d z , {\displaystyle p_{\theta }(x)=\int _{z}p_{\theta }({x,z})\,dz,} where p θ ( x , z ) {\displaystyle p_{\theta }({x,z})} represents the joint distribution under p θ {\displaystyle p_{\theta }} of the observable data x {\displaystyle x} and its latent representation or encoding z {\displaystyle z} . According to the chain rule, the equation can be rewritten as p θ ( x ) = ∫ z p θ ( x | z ) p θ ( z ) d z {\displaystyle p_{\theta }(x)=\int _{z}p_{\theta }({x|z})p_{\theta }(z)\,dz} In the vanilla variational autoencoder, z {\displaystyle z} is usually taken to be a finite-dimensional vector of real numbers, and p θ ( x | z ) {\displaystyle p_{\theta }({x|z})} to be a Gaussian distribution. Then p θ ( x ) {\displaystyle p_{\theta }(x)} is a mixture of Gaussian distributions. It is now possible to define the set of the relationships between the input data and its latent representation as Prior p θ ( z ) {\displaystyle p_{\theta }(z)} Likelihood p θ ( x | z ) {\displaystyle p_{\theta }(x|z)} Posterior p θ ( z | x ) {\displaystyle p_{\theta }(z|x)} Unfortunately, the computation of p θ ( z | x ) {\displaystyle p_{\theta }(z|x)} is expensive and in most cases intractable. To speed up the calculus to make it feasible, it is necessary to introduce a further function to approximate the posterior distribution as q ϕ ( z | x ) ≈ p θ ( z | x ) {\displaystyle q_{\phi }({z|x})\approx p_{\theta }({z|x})} with ϕ {\displaystyle \phi } defined as the set of real values that parametrize q {\displaystyle q} . This is sometimes called amortized inference, since by "investing" in finding a good q ϕ {\displaystyle q_{\phi }} , one can later infer z {\displaystyle z} from x {\displaystyle x} quickly without doing any integrals. In this way, the problem is to find a good probabilistic autoencoder, in which the conditional likelihood distribution p θ ( x | z ) {\displaystyle p_{\theta }(x|z)} is computed by the probabilistic decoder, and the approximated posterior distribution q ϕ ( z | x ) {\displaystyle q_{\phi }(z|x)} is computed by the probabilistic encoder. Parametrize the encoder as E ϕ {\displaystyle E_{\phi }} , and the decoder as D θ {\displaystyle D_{\theta }} . == Evidence lower bound (ELBO) == Like many deep learning approaches that use gradient-based optimization, VAEs require a differentiable loss function to update the network weights through backpropagation. For variational autoencoders, the idea is to jointly optimize the generative model parameters θ {\displaystyle \theta } to reduce the reconstruction error between the input and the output, and ϕ {\displaystyle \phi } to make q ϕ ( z | x ) {\displaystyle q_{\phi }({z|x})} as close as possible to p θ ( z | x ) {\displaystyle p_{\theta }(z|x)} . As reconstruction loss, mean squared error and cross entropy are often used. The Kullback–Leibler divergence D K L ( q ϕ ( z | x ) ∥ p θ ( z | x ) ) {\displaystyle D_{KL}(q_{\phi }({z|x})\parallel p_{\theta }({z|x}))} can be used as a loss function to squeeze q ϕ ( z | x ) {\displaystyle q_{\phi }({z|x})} under p θ ( z | x ) {\displaystyle p_{\theta }(z|x)} . This divergence loss expands to D K L ( q ϕ ( z | x ) ∥ p θ ( z | x ) ) = E z ∼ q ϕ ( ⋅ | x ) [ ln ⁡ q ϕ ( z | x ) p θ ( z | x ) ] = E z ∼ q ϕ ( ⋅ | x ) [ ln ⁡ q ϕ ( z | x ) p θ ( x ) p θ ( x , z ) ] = ln ⁡ p θ ( x ) + E z ∼ q ϕ ( ⋅ | x ) [ ln ⁡ q ϕ ( z | x ) p θ ( x , z ) ] . {\displaystyle {\begin{aligned}D_{KL}(q_{\phi }({z|x})\parallel p_{\theta }({z|x}))&=\mathbb {E} _{z\sim q_{\phi }(\cdot |x)}\left[\ln {\frac {q_{\phi }(z|x)}{p_{\theta }(z|x)}}\right]\\&=\mathbb {E} _{z\sim q_{\phi }(\cdot |x)}\left[\ln {\frac {q_{\phi }({z|x})p_{\theta }(x)}{p_{\theta }(x,z)}}\right]\\&=\ln p_{\theta }(x)+\mathbb {E} _{z\sim q_{\phi }(\cdot |x)}\left[\ln {\frac {q_{\phi }({z|x})}{p_{\theta }(x,z)}}\right].\end{aligned}}} Now, define the evidence lower bound (ELBO): L θ , ϕ ( x ) := E z ∼ q ϕ ( ⋅ | x ) [ ln ⁡ p θ ( x , z ) q ϕ ( z | x ) ] = ln ⁡ p θ ( x ) − D K L ( q ϕ ( ⋅ | x ) ∥ p θ ( ⋅ | x ) ) {\displaystyle L_{\theta ,\phi }(x):=\mathbb {E} _{z\sim q_{\phi }(\cdot |x)}\left[\ln {\frac {p_{\theta }(x,z)}{q_{\phi }({z|x})}}\right]=\ln p_{\theta }(x)-D_{KL}(q_{\phi }({\cdot |x})\parallel p_{\theta }({\cdot |x}))} Maximizing the ELBO θ ∗ , ϕ ∗ = argmax θ , ϕ L θ , ϕ ( x ) {\dis

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  Kernel principal component analysis

  In the field of multivariate statistics, kernel principal component analysis (kernel PCA) is an extension of principal component analysis (PCA) using techniques of kernel methods. Using a kernel, the originally linear operations of PCA are performed in a reproducing kernel Hilbert space. == Background: Linear PCA == Recall that conventional PCA operates on zero-centered data; that is, 1 N ∑ i = 1 N x i = 0 {\displaystyle {\frac {1}{N}}\sum _{i=1}^{N}\mathbf {x} _{i}=\mathbf {0} } , where x i {\displaystyle \mathbf {x} _{i}} is one of the N {\displaystyle N} multivariate observations. It operates by diagonalizing the covariance matrix, C = 1 N ∑ i = 1 N x i x i ⊤ {\displaystyle C={\frac {1}{N}}\sum _{i=1}^{N}\mathbf {x} _{i}\mathbf {x} _{i}^{\top }} in other words, it gives an eigendecomposition of the covariance matrix: λ v = C v {\displaystyle \lambda \mathbf {v} =C\mathbf {v} } which can be rewritten as λ x i ⊤ v = x i ⊤ C v for i = 1 , … , N {\displaystyle \lambda \mathbf {x} _{i}^{\top }\mathbf {v} =\mathbf {x} _{i}^{\top }C\mathbf {v} \quad {\textrm {for}}~i=1,\ldots ,N} . (See also: Covariance matrix as a linear operator) == Introduction of the Kernel to PCA == To understand the utility of kernel PCA, particularly for clustering, observe that, while N points cannot, in general, be linearly separated in d < N {\displaystyle d Read more →

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  Bayesian hierarchical modeling

  Bayesian hierarchical modelling is a statistical model written in multiple levels (hierarchical form) that estimates the posterior distribution of model parameters using the Bayesian method. The sub-models combine to form the hierarchical model, and Bayes' theorem is used to integrate them with the observed data and account for all the uncertainty that is present. This integration enables calculation of updated posterior over the (hyper)parameters, effectively updating prior beliefs in light of the observed data. Frequentist statistics may yield conclusions seemingly incompatible with those offered by Bayesian statistics due to the Bayesian treatment of the parameters as random variables and its use of subjective information in establishing assumptions on these parameters. As the approaches answer different questions the formal results are not technically contradictory but the two approaches disagree over which answer is relevant to particular applications. Bayesians argue that relevant information regarding decision-making and updating beliefs cannot be ignored and that hierarchical modeling has the potential to overrule classical methods in applications where respondents give multiple observational data. Moreover, the model has proven to be robust, with the posterior distribution less sensitive to the more flexible hierarchical priors. Hierarchical modeling, as its name implies, retains nested data structure, and is used when information is available at several different levels of observational units. For example, in epidemiological modeling to describe infection trajectories for multiple countries, observational units are countries, and each country has its own time-based profile of daily infected cases. In decline curve analysis to describe oil or gas production decline curve for multiple wells, observational units are oil or gas wells in a reservoir region, and each well has each own time-based profile of oil or gas production rates (usually, barrels per month). Hierarchical modeling is used to devise computation based strategies for multiparameter problems. == Philosophy == Statistical methods and models commonly involve multiple parameters that can be regarded as related or connected in such a way that the problem implies a dependence of the joint probability model for these parameters. Individual degrees of belief, expressed in the form of probabilities, come with uncertainty. Amidst this is the change of the degrees of belief over time. As was stated by Professor José M. Bernardo and Professor Adrian F. Smith, "The actuality of the learning process consists in the evolution of individual and subjective beliefs about the reality." These subjective probabilities are more directly involved in the mind rather than the physical probabilities. Hence, it is with this need of updating beliefs that Bayesians have formulated an alternative statistical model which takes into account the prior occurrence of a particular event. == Bayes' theorem == The assumed occurrence of a real-world event will typically modify preferences between certain options. This is done by modifying the degrees of belief attached, by an individual, to the events defining the options. Suppose in a study of the effectiveness of cardiac treatments, with the patients in hospital j having survival probability θ j {\displaystyle \theta _{j}} , the survival probability will be updated with the occurrence of y, the event in which a controversial serum is created which, as believed by some, increases survival in cardiac patients. In order to make updated probability statements about θ j {\displaystyle \theta _{j}} , given the occurrence of event y, we must begin with a model providing a joint probability distribution for θ j {\displaystyle \theta _{j}} and y. This can be written as a product of the two distributions that are often referred to as the prior distribution P ( θ ) {\displaystyle P(\theta )} and the sampling distribution P ( y ∣ θ ) {\displaystyle P(y\mid \theta )} respectively: P ( θ , y ) = P ( θ ) P ( y ∣ θ ) {\displaystyle P(\theta ,y)=P(\theta )P(y\mid \theta )} Using the basic property of conditional probability, the posterior distribution will yield: P ( θ ∣ y ) = P ( θ , y ) P ( y ) = P ( y ∣ θ ) P ( θ ) P ( y ) {\displaystyle P(\theta \mid y)={\frac {P(\theta ,y)}{P(y)}}={\frac {P(y\mid \theta )P(\theta )}{P(y)}}} This equation, showing the relationship between the conditional probability and the individual events, is known as Bayes' theorem. This simple expression encapsulates the technical core of Bayesian inference which aims to deconstruct the probability, P ( θ ∣ y ) {\displaystyle P(\theta \mid y)} , relative to solvable subsets of its supportive evidence. == Exchangeability == The usual starting point of a statistical analysis is the assumption that the n values y 1 , y 2 , … , y n {\displaystyle y_{1},y_{2},\ldots ,y_{n}} are exchangeable. If no information – other than data y – is available to distinguish any of the θ j {\displaystyle \theta _{j}} 's from any others, and no ordering or grouping of the parameters can be made, one must assume symmetry of prior distribution parameters. This symmetry is represented probabilistically by exchangeability. Generally, it is useful and appropriate to model data from an exchangeable distribution as independently and identically distributed, given some unknown parameter vector θ {\displaystyle \theta } , with distribution P ( θ ) {\displaystyle P(\theta )} . === Finite exchangeability === For a fixed number n, the set y 1 , y 2 , … , y n {\displaystyle y_{1},y_{2},\ldots ,y_{n}} is exchangeable if the joint probability P ( y 1 , y 2 , … , y n ) {\displaystyle P(y_{1},y_{2},\ldots ,y_{n})} is invariant under permutations of the indices. That is, for every permutation π {\displaystyle \pi } or ( π 1 , π 2 , … , π n ) {\displaystyle (\pi _{1},\pi _{2},\ldots ,\pi _{n})} of (1, 2, …, n), P ( y 1 , y 2 , … , y n ) = P ( y π 1 , y π 2 , … , y π n ) . {\displaystyle P(y_{1},y_{2},\ldots ,y_{n})=P(y_{\pi _{1}},y_{\pi _{2}},\ldots ,y_{\pi _{n}}).} The following is an exchangeable, but not independent and identical (iid), example: Consider an urn with a red ball and a blue ball inside, with probability 1 2 {\displaystyle {\frac {1}{2}}} of drawing either. Balls are drawn without replacement, i.e. after one ball is drawn from the n {\displaystyle n} balls, there will be n − 1 {\displaystyle n-1} remaining balls left for the next draw. Let Y i = { 1 , if the i th ball is red , 0 , otherwise . {\displaystyle {\text{Let }}Y_{i}={\begin{cases}1,&{\text{if the }}i{\text{th ball is red}},\\0,&{\text{otherwise}}.\end{cases}}} The probability of selecting a red ball in the first draw and a blue ball in the second draw is equal to the probability of selecting a blue ball on the first draw and a red on the second, both of which are 1/2: P ( y 1 = 1 , y 2 = 0 ) = P ( y 1 = 0 , y 2 = 1 ) = 1 2 {\displaystyle P(y_{1}=1,y_{2}=0)=P(y_{1}=0,y_{2}=1)={\frac {1}{2}}} . This makes y 1 {\displaystyle y_{1}} and y 2 {\displaystyle y_{2}} exchangeable. But the probability of selecting a red ball on the second draw given that the red ball has already been selected in the first is 0. This is not equal to the probability that the red ball is selected in the second draw, which is 1/2: P ( y 2 = 1 ∣ y 1 = 1 ) = 0 ≠ P ( y 2 = 1 ) = 1 2 {\displaystyle P(y_{2}=1\mid y_{1}=1)=0\neq P(y_{2}=1)={\frac {1}{2}}} . Thus, y 1 {\displaystyle y_{1}} and y 2 {\displaystyle y_{2}} are not independent. If x 1 , … , x n {\displaystyle x_{1},\ldots ,x_{n}} are independent and identically distributed, then they are exchangeable, but the converse is not necessarily true. === Infinite exchangeability === Infinite exchangeability is the property that every finite subset of an infinite sequence y 1 {\displaystyle y_{1}} , y 2 , … {\displaystyle y_{2},\ldots } is exchangeable. For any n, the sequence y 1 , y 2 , … , y n {\displaystyle y_{1},y_{2},\ldots ,y_{n}} is exchangeable. == Hierarchical models == === Components === Bayesian hierarchical modeling makes use of two important concepts in deriving the posterior distribution, namely: Hyperparameters: parameters of the prior distribution Hyperpriors: distributions of Hyperparameters Suppose a random variable Y follows a normal distribution with parameter θ {\displaystyle \theta } as the mean and 1 as the variance, that is Y ∣ θ ∼ N ( θ , 1 ) {\displaystyle Y\mid \theta \sim N(\theta ,1)} . The tilde relation ∼ {\displaystyle \sim } can be read as "has the distribution of" or "is distributed as". Suppose also that the parameter θ {\displaystyle \theta } has a distribution given by a normal distribution with mean μ {\displaystyle \mu } and variance 1, i.e. θ ∣ μ ∼ N ( μ , 1 ) {\displaystyle \theta \mid \mu \sim N(\mu ,1)} . Furthermore, μ {\displaystyle \mu } follows another distribution given, for example, by the standard normal distribution, N ( 0 , 1 ) {\displaystyle {\text{N}}(0,1)} . The parameter μ {\dis

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  Three-factor learning

  In neuroscience and machine learning, three-factor learning is the combination of Hebbian plasticity with a third modulatory factor to stabilise and enhance synaptic learning. This third factor can represent various signals such as reward, punishment, error, surprise, or novelty, often implemented through neuromodulators. == Description == Three-factor learning introduces the concept of eligibility traces, which flag synapses for potential modification pending the arrival of the third factor, and helps temporal credit assignement by bridging the gap between rapid neuronal firing and slower behavioral timescales, from which learning can be done. Biological basis for Three-factor learning rules have been supported by experimental evidence. This approach addresses the instability of classical Hebbian learning by minimizing autocorrelation and maximizing cross-correlation between inputs.

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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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  Online machine learning

  In computer science, online machine learning is a method of machine learning in which data becomes available in a sequential order and is used to update the best predictor for future data at each step, as opposed to batch learning techniques which generate the best predictor by learning on the entire training data set at once. Online learning is a common technique used in areas of machine learning where it is computationally infeasible to train over the entire dataset, requiring the need of out-of-core algorithms. It is also used in situations where it is necessary for the algorithm to dynamically adapt to new patterns in the data, or when the data itself is generated as a function of time, e.g., prediction of prices in the financial international markets. Online learning algorithms may be prone to catastrophic interference, a problem that can be addressed by incremental learning approaches. Online machine learning algorithms find applications in a wide variety of fields such as sponsored search to maximize ad revenue, portfolio optimization, shortest path prediction (with stochastic weights, e.g. traffic on roads for a maps application), spam filtering, real-time fraud detection, dynamic pricing for e-commerce, etc. There is also growing interest in usage of online learning paradigms for LLMs to enable continuous, real-time adaptation after the initial training. == Introduction == In the setting of supervised learning, a function of f : X → Y {\displaystyle f:X\to Y} is to be learned, where X {\displaystyle X} is thought of as a space of inputs and Y {\displaystyle Y} as a space of outputs, that predicts well on instances that are drawn from a joint probability distribution p ( x , y ) {\displaystyle p(x,y)} on X × Y {\displaystyle X\times Y} . In reality, the learner never knows the true distribution p ( x , y ) {\displaystyle p(x,y)} over instances. Instead, the learner usually has access to a training set of examples ( x 1 , y 1 ) , … , ( x n , y n ) {\displaystyle (x_{1},y_{1}),\ldots ,(x_{n},y_{n})} . In this setting, the loss function is given as V : Y × Y → R {\displaystyle V:Y\times Y\to \mathbb {R} } , such that V ( f ( x ) , y ) {\displaystyle V(f(x),y)} measures the difference between the predicted value f ( x ) {\displaystyle f(x)} and the true value y {\displaystyle y} . The ideal goal is to select a function f ∈ H {\displaystyle f\in {\mathcal {H}}} , where H {\displaystyle {\mathcal {H}}} is a space of functions called a hypothesis space, so that some notion of total loss is minimized. Depending on the type of model (statistical or adversarial), one can devise different notions of loss, which lead to different learning algorithms. == Statistical view of online learning == In statistical learning models, the training sample ( x i , y i ) {\displaystyle (x_{i},y_{i})} are assumed to have been drawn from the true distribution p ( x , y ) {\displaystyle p(x,y)} and the objective is to minimize the expected "risk" I [ f ] = E [ V ( f ( x ) , y ) ] = ∫ V ( f ( x ) , y ) d p ( x , y ) . {\displaystyle I[f]=\mathbb {E} [V(f(x),y)]=\int V(f(x),y)\,dp(x,y)\ .} A common paradigm in this situation is to estimate a function f ^ {\displaystyle {\hat {f}}} through empirical risk minimization or regularized empirical risk minimization (usually Tikhonov regularization). The choice of loss function here gives rise to several well-known learning algorithms such as regularized least squares and support vector machines. A purely online model in this category would learn based on just the new input ( x t + 1 , y t + 1 ) {\displaystyle (x_{t+1},y_{t+1})} , the current best predictor f t {\displaystyle f_{t}} and some extra stored information (which is usually expected to have storage requirements independent of training data size). For many formulations, for example nonlinear kernel methods, true online learning is not possible, though a form of hybrid online learning with recursive algorithms can be used where f t + 1 {\displaystyle f_{t+1}} is permitted to depend on f t {\displaystyle f_{t}} and all previous data points ( x 1 , y 1 ) , … , ( x t , y t ) {\displaystyle (x_{1},y_{1}),\ldots ,(x_{t},y_{t})} . In this case, the space requirements are no longer guaranteed to be constant since it requires storing all previous data points, but the solution may take less time to compute with the addition of a new data point, as compared to batch learning techniques. A common strategy to overcome the above issues is to learn using mini-batches, which process a small batch of b ≥ 1 {\displaystyle b\geq 1} data points at a time, this can be considered as pseudo-online learning for b {\displaystyle b} much smaller than the total number of training points. Mini-batch techniques are used with repeated passing over the training data to obtain optimized out-of-core versions of machine learning algorithms, for example, stochastic gradient descent. When combined with backpropagation, this is currently the de facto training method for training artificial neural networks. === Example: linear least squares === The simple example of linear least squares is used to explain a variety of ideas in online learning. The ideas are general enough to be applied to other settings, for example, with other convex loss functions. === Batch learning === Consider the setting of supervised learning with f {\displaystyle f} being a linear function to be learned: f ( x j ) = ⟨ w , x j ⟩ = w ⋅ x j {\displaystyle f(x_{j})=\langle w,x_{j}\rangle =w\cdot x_{j}} where x j ∈ R d {\displaystyle x_{j}\in \mathbb {R} ^{d}} is a vector of inputs (data points) and w ∈ R d {\displaystyle w\in \mathbb {R} ^{d}} is a linear filter vector. The goal is to compute the filter vector w {\displaystyle w} . To this end, a square loss function V ( f ( x j ) , y j ) = ( f ( x j ) − y j ) 2 = ( ⟨ w , x j ⟩ − y j ) 2 {\displaystyle V(f(x_{j}),y_{j})=(f(x_{j})-y_{j})^{2}=(\langle w,x_{j}\rangle -y_{j})^{2}} is used to compute the vector w {\displaystyle w} that minimizes the empirical loss I n [ w ] = ∑ j = 1 n V ( ⟨ w , x j ⟩ , y j ) = ∑ j = 1 n ( x j T w − y j ) 2 {\displaystyle I_{n}[w]=\sum _{j=1}^{n}V(\langle w,x_{j}\rangle ,y_{j})=\sum _{j=1}^{n}(x_{j}^{\mathsf {T}}w-y_{j})^{2}} where y j ∈ R . {\displaystyle y_{j}\in \mathbb {R} .} Let X {\displaystyle X} be the i × d {\displaystyle i\times d} data matrix and y ∈ R i {\displaystyle y\in \mathbb {R} ^{i}} is the column vector of target values after the arrival of the first i {\displaystyle i} data points. Assuming that the covariance matrix Σ i = X T X {\displaystyle \Sigma _{i}=X^{\mathsf {T}}X} is invertible (otherwise it is preferential to proceed in a similar fashion with Tikhonov regularization), the best solution f ∗ ( x ) = ⟨ w ∗ , x ⟩ {\displaystyle f^{}(x)=\langle w^{},x\rangle } to the linear least squares problem is given by w ∗ = ( X T X ) − 1 X T y = Σ i − 1 ∑ j = 1 i x j y j . {\displaystyle w^{}=(X^{\mathsf {T}}X)^{-1}X^{\mathsf {T}}y=\Sigma _{i}^{-1}\sum _{j=1}^{i}x_{j}y_{j}.} Now, calculating the covariance matrix Σ i = ∑ j = 1 i x j x j T {\displaystyle \Sigma _{i}=\sum _{j=1}^{i}x_{j}x_{j}^{\mathsf {T}}} takes time O ( i d 2 ) {\displaystyle O(id^{2})} , inverting the d × d {\displaystyle d\times d} matrix takes time O ( d 3 ) {\displaystyle O(d^{3})} , while the rest of the multiplication takes time O ( d 2 ) {\displaystyle O(d^{2})} , giving a total time of O ( i d 2 + d 3 ) {\displaystyle O(id^{2}+d^{3})} . When there are n {\displaystyle n} total points in the dataset, to recompute the solution after the arrival of every datapoint i = 1 , … , n {\displaystyle i=1,\ldots ,n} , the naive approach will have a total complexity O ( n 2 d 2 + n d 3 ) {\displaystyle O(n^{2}d^{2}+nd^{3})} . Note that when storing the matrix Σ i {\displaystyle \Sigma _{i}} , then updating it at each step needs only adding x i + 1 x i + 1 T {\displaystyle x_{i+1}x_{i+1}^{\mathsf {T}}} , which takes O ( d 2 ) {\displaystyle O(d^{2})} time, reducing the total time to O ( n d 2 + n d 3 ) = O ( n d 3 ) {\displaystyle O(nd^{2}+nd^{3})=O(nd^{3})} , but with an additional storage space of O ( d 2 ) {\displaystyle O(d^{2})} to store Σ i {\displaystyle \Sigma _{i}} . === Online learning: recursive least squares === The recursive least squares (RLS) algorithm considers an online approach to the least squares problem. It can be shown that by initialising w 0 = 0 ∈ R d {\displaystyle \textstyle w_{0}=0\in \mathbb {R} ^{d}} and Γ 0 = I ∈ R d × d {\displaystyle \textstyle \Gamma _{0}=I\in \mathbb {R} ^{d\times d}} , the solution of the linear least squares problem given in the previous section can be computed by the following iteration: Γ i = Γ i − 1 − Γ i − 1 x i x i T Γ i − 1 1 + x i T Γ i − 1 x i {\displaystyle \Gamma _{i}=\Gamma _{i-1}-{\frac {\Gamma _{i-1}x_{i}x_{i}^{\mathsf {T}}\Gamma _{i-1}}{1+x_{i}^{\mathsf {T}}\Gamma _{i-1}x_{i}}}} w i = w i − 1 − Γ i x i ( x i T w i − 1 − y i ) {\displaystyle w_{i}=w_{i-1}-\Gamma _{i}x_{i}\left(x_{i}^{\mathsf {T}}w_{

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