AI Email Builder

AI Email Builder — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

• 

  Automated dispensing cabinet

  An automated dispensing cabinet (ADC), also called a unit-based cabinet (UBC), automated dispensing device (ADD), or automated dispensing machine (ADM)[1], is a computerized medicine cabinet for hospitals and healthcare settings. ADCs allow medications to be stored and dispensed near the point of care while controlling and tracking drug distribution. == Overview == Hospital pharmacies have provided medications for patients by filling patient-specific cassettes of unit-dose medications that were then delivered to the nursing unit and stored in medication cabinets or carts. ADCs, originally designed for hospital use, were introduced in hospitals in the 1980s and have facilitated the transition to alternative delivery models and more decentralized medication distribution systems.[2] Implementing automated dispensing cabinets as part of a decentralized or hybrid medication distribution system can improve patient safety and the accountability of the inventory, streamline certain billing processes. However, in the 2000s, the technology began to be deployed into other care settings where medication doses were stored onsite, and higher security methods were needed to control inventory, access, and dispensing of each patient dose. Settings that now deploy ADCs include long-term care facilities, hospice, critical access hospitals, surgery centers, group homes, residential care facilities, rehab and psych environments, animal health, dental clinics, and nursing education simulation. These diverse care settings share a common need to safely store, account for, and dispense individual doses of medications, especially narcotics and high-value medications, at the point of care.[3] ADCs track user access and dispensed medications, and their use can improve control over medication inventory. The real-time inventory reports generated by many cabinets can simplify the filling process and help the pharmacy track expired drugs. Furthermore, by restricting individual drugs – such as high-risk medications and controlled substances – to unique drawers within the cabinet, overall inventory management, patient safety, and medication security can be improved. Automated dispensing cabinets allow the pharmacy department to profile physician orders before they are dispensed.[4] ADCs can also enable providers to record medication charges upon dispensing, reducing the billing paperwork the pharmacy is responsible for. In addition, nurses can note returned medications using the cabinets' computers, enabling direct credits to patients' accounts. Since automated cabinets can be located on the nursing unit floor, nursing have speedier access to a patient's medications. Also, shorter waiting time ensures improved patient comfort and care.[5] == Role of automated dispensing in healthcare == Automated dispensing is a pharmacy practice in which a device dispenses medications and fills prescriptions. ADCs, which can handle many different medications, are available from a number of manufacturers such as BD, ARxIUM, and Omnicell. Though members of the pharmacy community have been utilizing automation technology since the 1980s, companies are constantly improving ADCs to meet changing needs and health standards in the industry. Several goals can be met by implementing an automated product in a healthcare facility. Patient safety can be ensured with the use of ADC technology such as barcoding. Anesthesia ADCs in operating rooms and perioperative areas may include label printing to prevent mix-ups such as errors between morphine and hydromorphone, two different opioid analgesics that frequently get confused. These systems also communicate with the pharmacy and its information management system to track medications removed and support inventory replenishment. == Key features == ADCs are like automated teller machines whose specific technologies such as barcode scanning and clinical decision support can improve medication safety. Some have metal locking drawers for added security and some have automated single-dose dispensing to prevent the need for a blind count each time a controlled substance is accessed. Over the years, ADCs have been adapted to facilitate compliance with emerging regulatory requirements such as pharmacy review of medication orders and safe practice recommendations. ADCs incorporate advanced software and electronic interfaces to synthesize high-risk steps in the medication use process. These unit-based medication repositories provide computer-controlled storage, dispensation, tracking, and documentation of medication distribution in the resident care unit. Since automated dispensing cabinets are not located in the pharmacy, they are considered "decentralized" medication distribution systems. Instead, they can be found at the point of care on the resident care unit. Tracking of the stocking and distribution process can occur by interfacing the unit with a central pharmacy computer. These cabinets can also be interfaced with other external databases such as resident profiles, the facility's admission/discharge/transfer system, and billing systems. Most ADC providers offer scalable systems since several important factors vary widely by facility such as budget, physical room size, patient population/demographics, type of healthcare facility, etc.

  Read more →
• 

  Cartesian genetic programming

  Cartesian genetic programming is a form of genetic programming that uses a graph representation to encode computer programs. It grew from a method of evolving digital circuits developed by Julian F. Miller and Peter Thomson in 1997. The term ‘Cartesian genetic programming’ first appeared in 1999 and was proposed as a general form of genetic programming in 2000. It is called ‘Cartesian’ because it represents a program using a two-dimensional grid of nodes. Miller's keynote explains how CGP works. He edited a book entitled Cartesian Genetic Programming, published in 2011 by Springer. The open source project dCGP implements a differentiable version of CGP developed at the European Space Agency by Dario Izzo, Francesco Biscani and Alessio Mereta able to approach symbolic regression tasks, to find solution to differential equations, find prime integrals of dynamical systems, represent variable topology artificial neural networks and more.

  Read more →
• 

  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.

  Read more →
• 

  Mathematics of neural networks in machine learning

  An artificial neural network (ANN) or neural network combines biological principles with advanced statistics to solve problems in domains such as pattern recognition and game-play. ANNs adopt the basic model of neuron analogues connected to each other in a variety of ways. == Structure == === Neuron === A neuron with label j {\displaystyle j} receiving an input p j ( t ) {\displaystyle p_{j}(t)} from predecessor neurons consists of the following components: an activation a j ( t ) {\displaystyle a_{j}(t)} , the neuron's state, depending on a discrete time parameter, an optional threshold θ j {\displaystyle \theta _{j}} , which stays fixed unless changed by learning, an activation function f {\displaystyle f} that computes the new activation at a given time t + 1 {\displaystyle t+1} from a j ( t ) {\displaystyle a_{j}(t)} , θ j {\displaystyle \theta _{j}} and the net input p j ( t ) {\displaystyle p_{j}(t)} giving rise to the relation a j ( t + 1 ) = f ( a j ( t ) , p j ( t ) , θ j ) , {\displaystyle a_{j}(t+1)=f(a_{j}(t),p_{j}(t),\theta _{j}),} and an output function f out {\displaystyle f_{\text{out}}} computing the output from the activation o j ( t ) = f out ( a j ( t ) ) . {\displaystyle o_{j}(t)=f_{\text{out}}(a_{j}(t)).} Often the output function is simply the identity function. An input neuron has no predecessor but serves as input interface for the whole network. Similarly an output neuron has no successor and thus serves as output interface of the whole network. === Propagation function === The propagation function computes the input p j ( t ) {\displaystyle p_{j}(t)} to the neuron j {\displaystyle j} from the outputs o i ( t ) {\displaystyle o_{i}(t)} and typically has the form p j ( t ) = ∑ i o i ( t ) w i j . {\displaystyle p_{j}(t)=\sum _{i}o_{i}(t)w_{ij}.} === Bias === A bias term can be added, changing the form to the following: p j ( t ) = ∑ i o i ( t ) w i j + w 0 j , {\displaystyle p_{j}(t)=\sum _{i}o_{i}(t)w_{ij}+w_{0j},} where w 0 j {\displaystyle w_{0j}} is a bias. == Neural networks as functions == Neural network models can be viewed as defining a function that takes an input (observation) and produces an output (decision) f : X → Y {\displaystyle \textstyle f:X\rightarrow Y} or a distribution over X {\displaystyle \textstyle X} or both X {\displaystyle \textstyle X} and Y {\displaystyle \textstyle Y} . Sometimes models are intimately associated with a particular learning rule. A common use of the phrase "ANN model" is really the definition of a class of such functions (where members of the class are obtained by varying parameters, connection weights, or specifics of the architecture such as the number of neurons, number of layers or their connectivity). Mathematically, a neuron's network function f ( x ) {\displaystyle \textstyle f(x)} is defined as a composition of other functions g i ( x ) {\displaystyle \textstyle g_{i}(x)} , that can further be decomposed into other functions. This can be conveniently represented as a network structure, with arrows depicting the dependencies between functions. A widely used type of composition is the nonlinear weighted sum, where f ( x ) = K ( ∑ i w i g i ( x ) ) {\displaystyle \textstyle f(x)=K\left(\sum _{i}w_{i}g_{i}(x)\right)} , where K {\displaystyle \textstyle K} (commonly referred to as the activation function) is some predefined function, such as the hyperbolic tangent, sigmoid function, softmax function, or rectifier function. The important characteristic of the activation function is that it provides a smooth transition as input values change, i.e. a small change in input produces a small change in output. The following refers to a collection of functions g i {\displaystyle \textstyle g_{i}} as a vector g = ( g 1 , g 2 , … , g n ) {\displaystyle \textstyle g=(g_{1},g_{2},\ldots ,g_{n})} . This figure depicts such a decomposition of f {\displaystyle \textstyle f} , with dependencies between variables indicated by arrows. These can be interpreted in two ways. The first view is the functional view: the input x {\displaystyle \textstyle x} is transformed into a 3-dimensional vector h {\displaystyle \textstyle h} , which is then transformed into a 2-dimensional vector g {\displaystyle \textstyle g} , which is finally transformed into f {\displaystyle \textstyle f} . This view is most commonly encountered in the context of optimization. The second view is the probabilistic view: the random variable F = f ( G ) {\displaystyle \textstyle F=f(G)} depends upon the random variable G = g ( H ) {\displaystyle \textstyle G=g(H)} , which depends upon H = h ( X ) {\displaystyle \textstyle H=h(X)} , which depends upon the random variable X {\displaystyle \textstyle X} . This view is most commonly encountered in the context of graphical models. The two views are largely equivalent. In either case, for this particular architecture, the components of individual layers are independent of each other (e.g., the components of g {\displaystyle \textstyle g} are independent of each other given their input h {\displaystyle \textstyle h} ). This naturally enables a degree of parallelism in the implementation. Networks such as the previous one are commonly called feedforward, because their graph is a directed acyclic graph. Networks with cycles are commonly called recurrent. Such networks are commonly depicted in the manner shown at the top of the figure, where f {\displaystyle \textstyle f} is shown as dependent upon itself. However, an implied temporal dependence is not shown. == Backpropagation == Backpropagation training algorithms fall into three categories: steepest descent (with variable learning rate and momentum, resilient backpropagation); quasi-Newton (Broyden–Fletcher–Goldfarb–Shanno, one step secant); Levenberg–Marquardt and conjugate gradient (Fletcher–Reeves update, Polak–Ribiére update, Powell–Beale restart, scaled conjugate gradient). === Algorithm === Let N {\displaystyle N} be a network with e {\displaystyle e} connections, m {\displaystyle m} inputs and n {\displaystyle n} outputs. Below, x 1 , x 2 , … {\displaystyle x_{1},x_{2},\dots } denote vectors in R m {\displaystyle \mathbb {R} ^{m}} , y 1 , y 2 , … {\displaystyle y_{1},y_{2},\dots } vectors in R n {\displaystyle \mathbb {R} ^{n}} , and w 0 , w 1 , w 2 , … {\displaystyle w_{0},w_{1},w_{2},\ldots } vectors in R e {\displaystyle \mathbb {R} ^{e}} . These are called inputs, outputs and weights, respectively. The network corresponds to a function y = f N ( w , x ) {\displaystyle y=f_{N}(w,x)} which, given a weight w {\displaystyle w} , maps an input x {\displaystyle x} to an output y {\displaystyle y} . In supervised learning, a sequence of training examples ( x 1 , y 1 ) , … , ( x p , y p ) {\displaystyle (x_{1},y_{1}),\dots ,(x_{p},y_{p})} produces a sequence of weights w 0 , w 1 , … , w p {\displaystyle w_{0},w_{1},\dots ,w_{p}} starting from some initial weight w 0 {\displaystyle w_{0}} , usually chosen at random. These weights are computed in turn: first compute w i {\displaystyle w_{i}} using only ( x i , y i , w i − 1 ) {\displaystyle (x_{i},y_{i},w_{i-1})} for i = 1 , … , p {\displaystyle i=1,\dots ,p} . The output of the algorithm is then w p {\displaystyle w_{p}} , giving a new function x ↦ f N ( w p , x ) {\displaystyle x\mapsto f_{N}(w_{p},x)} . The computation is the same in each step, hence only the case i = 1 {\displaystyle i=1} is described. w 1 {\displaystyle w_{1}} is calculated from ( x 1 , y 1 , w 0 ) {\displaystyle (x_{1},y_{1},w_{0})} by considering a variable weight w {\displaystyle w} and applying gradient descent to the function w ↦ E ( f N ( w , x 1 ) , y 1 ) {\displaystyle w\mapsto E(f_{N}(w,x_{1}),y_{1})} to find a local minimum, starting at w = w 0 {\displaystyle w=w_{0}} . This makes w 1 {\displaystyle w_{1}} the minimizing weight found by gradient descent. == Learning pseudocode == To implement the algorithm above, explicit formulas are required for the gradient of the function w ↦ E ( f N ( w , x ) , y ) {\displaystyle w\mapsto E(f_{N}(w,x),y)} where the function is E ( y , y ′ ) = | y − y ′ | 2 {\displaystyle E(y,y')=|y-y'|^{2}} . The learning algorithm can be divided into two phases: propagation and weight update. === Propagation === Propagation involves the following steps: Propagation forward through the network to generate the output value(s) Calculation of the cost (error term) Propagation of the output activations back through the network using the training pattern target to generate the deltas (the difference between the targeted and actual output values) of all output and hidden neurons. === Weight update === For each weight: Multiply the weight's output delta and input activation to find the gradient of the weight. Subtract the ratio (percentage) of the weight's gradient from the weight. The learning rate is the ratio (percentage) that influences the speed and quality of learning. The greater the ratio, the faster the neuron trains, but the lower the ratio, the more accurat

  Read more →
• 

  Headway (app)

  Headway, also known as the Headway App, is an educational technology (EdTech) product that provides short text and audio summaries of nonfiction books. The product was launched in 2019 by Anton Pavlovsky and is developed by Headway Inc, a global consumer tech company that operates in the lifelong learning space. == History == The Headway app was launched in January 2019, with the first version of the application released the same year. In 2021, Headway ranked first globally in downloads within the book summary application niche. In 2022, the application received the Golden Novum Design Award for product design. In 2023 and 2024, Headway appeared in several App Store editorial selections, including App of the Day in multiple countries, and received an Editors’ Choice label in the United States. In April 2025, the application was listed as a Webby Honoree in the Learning & Education category. The company has also launched the Headway Scholarship for Book Lovers. As of 2025, publicly available reporting notes that the Headway app has surpassed 50 million downloads and is among the Top 10 iOS applications by revenue in the Education category worldwide. == Products and features == The Headway app provides short-form summaries of nonfiction books in both text and audio formats. Content is produced by an in-house team of writers, editors, and voice actors. Features include highlighting and saving key insights, spaced repetition for knowledge retention, and offline access to downloaded summaries. The app is available on iOS, iPadOS, watchOS, Android, CarPlay, and Android Auto, and supports multiple languages. == Pricing == Headway operates on a subscription business model, with optional paid plans alongside free access. The company publicly provides its terms of use, privacy policy, subscription details, and AI usage policy on its official website. == Technology and integrations == Headway reports that its book summaries are written and edited manually, while artificial intelligence tools are used in limited supporting functions, such as experimental conversational features and selected marketing processes. == Adoption == According to figures released by the company, the app has exceeded 50 million downloads worldwide. Sensor Tower data indicates that Headway has been the most downloaded application in its niche since October 2020. In January 2025, the app claimed the #1 position in the Education category in both the United States and United Kingdom App Stores and remained among the Top 10 iOS applications globally by revenue within the Education category. == Awards == The Headway app has received several product-level distinctions. In 2023 and 2024, it appeared in multiple App Store editorial selections, including App of the Day features and an Editors’ Choice label in the United States. In 2025, the app was recognized as a Webby Honoree in the Learning & Education category. The product has also been featured in independent media roundups of notable educational applications.

  Read more →
• 

  Soft independent modelling of class analogies

  Soft independent modelling by class analogy (SIMCA) is a statistical method for supervised classification of data. The method requires a training data set consisting of samples (or objects) with a set of attributes and their class membership. The term soft refers to the fact the classifier can identify samples as belonging to multiple classes and not necessarily producing a classification of samples into non-overlapping classes. == Method == In order to build the classification models, the samples belonging to each class need to be analysed using principal component analysis (PCA); only the significant components are retained. For a given class, the resulting model then describes either a line (for one Principal Component or PC), plane (for two PCs) or hyper-plane (for more than two PCs). For each modelled class, the mean orthogonal distance of training data samples from the line, plane, or hyper-plane (calculated as the residual standard deviation) is used to determine a critical distance for classification. This critical distance is based on the F-distribution and is usually calculated using 95% or 99% confidence intervals. New observations are projected into each PC model and the residual distances calculated. An observation is assigned to the model class when its residual distance from the model is below the statistical limit for the class. The observation may be found to belong to multiple classes and a measure of goodness of the model can be found from the number of cases where the observations are classified into multiple classes. The classification efficiency is usually indicated by Receiver operating characteristics. In the original SIMCA method, the ends of the hyper-plane of each class are closed off by setting statistical control limits along the retained principal components axes (i.e., score value between plus and minus 0.5 times score standard deviation). More recent adaptations of the SIMCA method close off the hyper-plane by construction of ellipsoids (e.g. Hotelling's T2 or Mahalanobis distance). With such modified SIMCA methods, classification of an object requires both that its orthogonal distance from the model and its projection within the model (i.e. score value within the region defined by the ellipsoid) are not significant. == Application == SIMCA as a method of classification has gained widespread use especially in applied statistical fields such as chemometrics and spectroscopic data analysis.

  Read more →
• 

  Inductive logic programming

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

  Read more →
• 

  Santa Fe Trail problem

  The Santa Fe Trail problem is a genetic programming exercise in which artificial ants search for food pellets according to a programmed set of instructions. The layout of food pellets in the Santa Fe Trail problem has become a standard for comparing different genetic programming algorithms and solutions. One method for programming and testing algorithms on the Santa Fe Trail problem is by using the NetLogo application. There is at least one case of a student creating a Lego robotic ant to solve the problem.

  Read more →
• 

  RIPAC (microprocessor)

  RIPAC was a VLSI single-chip microprocessor designed for automatic recognition of the connected speech, one of the first of this use. The project of the microprocessor RIPAC started in 1984. RIPAC was aimed to provide efficient real-time speech recognition services to the italian telephone system provided by SIP. The microprocessor was presented in September 1986 at The Hague (Netherlands) at EUSPICO conference. It was composed of 70.000 transistors and structured as Harvard architecture. The name RIPAC is the acronym for "Riconoscimento del PArlato Connesso", that means "Recognition of the connected speech" in Italian. The microprocessor was designed by the Italian companies CSELT and ELSAG and was produced by SGS: a combination of Hidden Markov Model and Dynamic Time Warping algorithms was used for processing speech signals. It was able to do real-time speech recognition of Italian and many languages with a good affordability. The chip, issued by U.S. Patent No. 4,907,278, worked at first run.

  Read more →
• 

  Error-driven learning

  In reinforcement learning, error-driven learning is a method for adjusting a model's (intelligent agent's) parameters based on the difference between its output results and the ground truth. These models stand out as they depend on environmental feedback, rather than explicit labels or categories. They are based on the idea that language acquisition involves the minimization of the prediction error (MPSE). By leveraging these prediction errors, the models consistently refine expectations and decrease computational complexity. Typically, these algorithms are operated by the GeneRec algorithm. Error-driven learning has widespread applications in cognitive sciences and computer vision. These methods have also found successful application in natural language processing (NLP), including areas like part-of-speech tagging, parsing, named entity recognition (NER), machine translation (MT), speech recognition (SR), and dialogue systems. == Formal Definition == Error-driven learning models are ones that rely on the feedback of prediction errors to adjust the expectations or parameters of a model. The key components of error-driven learning include the following: A set S {\displaystyle S} of states representing the different situations that the learner can encounter. A set A {\displaystyle A} of actions that the learner can take in each state. A prediction function P ( s , a ) {\displaystyle P(s,a)} that gives the learner's current prediction of the outcome of taking action a {\displaystyle a} in state s {\displaystyle s} . An error function E ( o , p ) {\displaystyle E(o,p)} that compares the actual outcome o {\displaystyle o} with the prediction p {\displaystyle p} and produces an error value. An update rule U ( p , e ) {\displaystyle U(p,e)} that adjusts the prediction p {\displaystyle p} in light of the error e {\displaystyle e} . == Algorithms == Error-driven learning algorithms refer to a category of reinforcement learning algorithms that leverage the disparity between the real output and the expected output of a system to regulate the system's parameters. Typically applied in supervised learning, these algorithms are provided with a collection of input-output pairs to facilitate the process of generalization. The widely utilized error backpropagation learning algorithm is known as GeneRec, a generalized recirculation algorithm primarily employed for gene prediction in DNA sequences. Many other error-driven learning algorithms are derived from alternative versions of GeneRec. == Applications == === Cognitive science === Simpler error-driven learning models effectively capture complex human cognitive phenomena and anticipate elusive behaviors. They provide a flexible mechanism for modeling the brain's learning process, encompassing perception, attention, memory, and decision-making. By using errors as guiding signals, these algorithms adeptly adapt to changing environmental demands and objectives, capturing statistical regularities and structure. Furthermore, cognitive science has led to the creation of new error-driven learning algorithms that are both biologically acceptable and computationally efficient. These algorithms, including deep belief networks, spiking neural networks, and reservoir computing, follow the principles and constraints of the brain and nervous system. Their primary aim is to capture the emergent properties and dynamics of neural circuits and systems. === Computer vision === Computer vision is a complex task that involves understanding and interpreting visual data, such as images or videos. In the context of error-driven learning, the computer vision model learns from the mistakes it makes during the interpretation process. When an error is encountered, the model updates its internal parameters to avoid making the same mistake in the future. This repeated process of learning from errors helps improve the model's performance over time. For NLP to do well at computer vision, it employs deep learning techniques. This form of computer vision is sometimes called neural computer vision (NCV), since it makes use of neural networks. NCV therefore interprets visual data based on a statistical, trial and error approach and can deal with context and other subtleties of visual data. === Natural Language Processing === ==== Part-of-speech tagging ==== Part-of-speech (POS) tagging is a crucial component in Natural Language Processing (NLP). It helps resolve human language ambiguity at different analysis levels. In addition, its output (tagged data) can be used in various applications of NLP such as information extraction, information retrieval, question Answering, speech eecognition, text-to-speech conversion, partial parsing, and grammar correction. ==== Parsing ==== Parsing in NLP involves breaking down a text into smaller pieces (phrases) based on grammar rules. If a sentence cannot be parsed, it may contain grammatical errors. In the context of error-driven learning, the parser learns from the mistakes it makes during the parsing process. When an error is encountered, the parser updates its internal model to avoid making the same mistake in the future. This iterative process of learning from errors helps improve the parser's performance over time. In conclusion, error-driven learning plays a crucial role in improving the accuracy and efficiency of NLP parsers by allowing them to learn from their mistakes and adapt their internal models accordingly. ==== Named entity recognition (NER) ==== NER is the task of identifying and classifying entities (such as persons, locations, organizations, etc.) in a text. Error-driven learning can help the model learn from its false positives and false negatives and improve its recall and precision on (NER). In the context of error-driven learning, the significance of NER is quite profound. Traditional sequence labeling methods identify nested entities layer by layer. If an error occurs in the recognition of an inner entity, it can lead to incorrect identification of the outer entity, leading to a problem known as error propagation of nested entities. This is where the role of NER becomes crucial in error-driven learning. By accurately recognizing and classifying entities, it can help minimize these errors and improve the overall accuracy of the learning process. Furthermore, deep learning-based NER methods have shown to be more accurate as they are capable of assembling words, enabling them to understand the semantic and syntactic relationship between various words better. ==== Machine translation ==== Machine translation is a complex task that involves converting text from one language to another. In the context of error-driven learning, the machine translation model learns from the mistakes it makes during the translation process. When an error is encountered, the model updates its internal parameters to avoid making the same mistake in the future. This iterative process of learning from errors helps improve the model's performance over time. ==== Speech recognition ==== Speech recognition is a complex task that involves converting spoken language into written text. In the context of error-driven learning, the speech recognition model learns from the mistakes it makes during the recognition process. When an error is encountered, the model updates its internal parameters to avoid making the same mistake in the future. This iterative process of learning from errors helps improve the model's performance over time. ==== Dialogue systems ==== Dialogue systems are a popular NLP task as they have promising real-life applications. They are also complicated tasks since many NLP tasks deserving study are involved. In the context of error-driven learning, the dialogue system learns from the mistakes it makes during the dialogue process. When an error is encountered, the model updates its internal parameters to avoid making the same mistake in the future. This iterative process of learning from errors helps improve the model's performance over time. == Advantages == Error-driven learning has several advantages over other types of machine learning algorithms: They can learn from feedback and correct their mistakes, which makes them adaptive and robust to noise and changes in the data. They can handle large and high-dimensional data sets, as they do not require explicit feature engineering or prior knowledge of the data distribution. They can achieve high accuracy and performance, as they can learn complex and nonlinear relationships between the input and the output. == Limitations == Although error driven learning has its advantages, their algorithms also have the following limitations: They can suffer from overfitting, which means that they memorize the training data and fail to generalize to new and unseen data. This can be mitigated by using regularization techniques, such as adding a penalty term to the loss function, or reducing the complexity of the model. They can be sensitive to the choice of

  Read more →
• 

  Vapnik–Chervonenkis dimension

  In Vapnik–Chervonenkis theory, the Vapnik–Chervonenkis (VC) dimension is a measure of the size (capacity, complexity, expressive power, richness, or flexibility) of a class of sets. The notion can be extended to classes of binary functions. It is defined as the cardinality of the largest set of points that the function class can shatter—that is, for which all possible binary labelings can be realized by some function in the class. It was originally defined by Vladimir Vapnik and Alexey Chervonenkis. Informally, the capacity of a classification model is related to how complicated it can be. For example, consider the thresholding of a high-degree polynomial: if the polynomial evaluates above zero, that point is classified as positive, otherwise as negative. A high-degree polynomial can be wiggly, so that it can fit a given set of training points well. Such a polynomial has a high capacity. A much simpler alternative is to threshold a linear function. This function may not fit the training set well, because it has a low capacity. This notion of capacity is made rigorous below. == Definitions == === VC dimension of a set-family === Let C = { C } C ∈ C {\displaystyle {\mathcal {C}}=\{C\}_{C\in {\mathcal {C}}}} be a family of sets (also called set family, collection of sets or set of sets) and X {\displaystyle X} a set. Their intersection is defined as the following set family: C ∩ X := { C ∩ X ∣ C ∈ C } . {\displaystyle {\mathcal {C}}\cap X:=\{C\cap X\mid C\in {\mathcal {C}}\}.} Here typically X {\displaystyle X} and each C ∈ C {\displaystyle C\in {\mathcal {C}}} are subsets of a big "universe" of possibilities U {\displaystyle U} where intersection takes place. We say that a set X {\displaystyle X} is shattered by C {\displaystyle {\mathcal {C}}} if P ( X ) = C ∩ X {\displaystyle {\mathcal {P}}(X)={\mathcal {C}}\cap X} i.e. the set of intersections contains (hence is equal to) all the subsets of X {\displaystyle X} . For finite sets X {\displaystyle X} this is equivalent to | C ∩ X | = 2 | X | . {\displaystyle |{\mathcal {C}}\cap X|=2^{|X|}.} The VC dimension D {\displaystyle D} of C {\displaystyle {\mathcal {C}}} is the cardinality of the largest set that is shattered by C {\displaystyle {\mathcal {C}}} . If arbitrarily large sets can be shattered, the VC dimension of C {\displaystyle {\mathcal {C}}} is ∞ {\displaystyle \infty } . === VC dimension of a classification model === A binary classification model f {\displaystyle f} with some parameter vector θ {\displaystyle \theta } is said to shatter a set of generally positioned data points ( x 1 , x 2 , … , x n ) {\displaystyle (x_{1},x_{2},\ldots ,x_{n})} if, for every assignment of labels to those points, there exists a θ {\displaystyle \theta } such that the model f {\displaystyle f} makes no errors when evaluating that set of data points. The VC dimension of a model f {\displaystyle f} is the maximum number of points that can be arranged so that f {\displaystyle f} shatters them. More formally, it is the maximum cardinal D {\displaystyle D} such that there exists a generally positioned data point set of cardinality D {\displaystyle D} that can be shattered by f {\displaystyle f} . == Examples == f {\displaystyle f} is a constant classifier (with no parameters); Its VC dimension is 0 since it cannot shatter even a single point. In general, the VC dimension of a finite classification model, which can return at most 2 d {\displaystyle 2^{d}} different classifiers, is at most d {\displaystyle d} (this is an upper bound on the VC dimension; the Sauer–Shelah lemma gives a lower bound on the dimension). f {\displaystyle f} is a single-parametric threshold classifier on real numbers; i.e., for a certain threshold θ {\displaystyle \theta } , the classifier f θ {\displaystyle f_{\theta }} returns 1 if the input number is larger than θ {\displaystyle \theta } and 0 otherwise. The VC dimension of f {\displaystyle f} is 1 because: (a) It can shatter a single point. For every point x {\displaystyle x} , a classifier f θ {\displaystyle f_{\theta }} labels it as 0 if θ > x {\displaystyle \theta >x} and labels it as 1 if θ < x {\displaystyle \theta x + 2 {\displaystyle \theta >x+2} , as (1,0) if θ ∈ [ x − 4 , x − 2 ) {\displaystyle \theta \in [x-4,x-2)} , as (1,1) if θ ∈ [ x − 2 , x ] {\displaystyle \theta \in [x-2,x]} , and as (0,1) if θ ∈ ( x , x + 2 ] {\displaystyle \theta \in (x,x+2]} . (b) It cannot shatter any set of three points. For every set of three numbers, if the smallest and the largest are labeled 1, then the middle one must also be labeled 1, so not all labelings are possible. f {\displaystyle f} is a straight line as a classification model on points in a two-dimensional plane (this is the model used by a perceptron). The line should separate positive data points from negative data points. There exist sets of 3 points that can indeed be shattered using this model (any 3 points that are not collinear can be shattered). However, no set of 4 points can be shattered: by Radon's theorem, any four points can be partitioned into two subsets with intersecting convex hulls, so it is not possible to separate one of these two subsets from the other. Thus, the VC dimension of this particular classifier is 3. It is important to remember that while one can choose any arrangement of points, the arrangement of those points cannot change when attempting to shatter for some label assignment. Note, only 3 of the 23 = 8 possible label assignments are shown for the three points. f {\displaystyle f} is a single-parametric sine classifier, i.e., for a certain parameter θ {\displaystyle \theta } , the classifier f θ {\displaystyle f_{\theta }} returns 1 if the input number x {\displaystyle x} has sin ⁡ ( θ x ) > 0 {\displaystyle \sin(\theta x)>0} and 0 otherwise. The VC dimension of f {\displaystyle f} is infinite, since it can shatter any finite subset of the set { 2 − m ∣ m ∈ N } {\displaystyle \{2^{-m}\mid m\in \mathbb {N} \}} . == Uses == === In statistical learning theory === The VC dimension can predict a probabilistic upper bound on the test error of a classification model. Vapnik proved that the probability of the test error (i.e., risk with 0–1 loss function) distancing from an upper bound (on data that is drawn i.i.d. from the same distribution as the training set) is given by: Pr ( test error ⩽ training error + 1 N [ D ( log ⁡ ( 2 N D ) + 1 ) − log ⁡ ( η 4 ) ] ) = 1 − η , {\displaystyle \Pr \left({\text{test error}}\leqslant {\text{training error}}+{\sqrt {{\frac {1}{N}}\left[D\left(\log \left({\tfrac {2N}{D}}\right)+1\right)-\log \left({\tfrac {\eta }{4}}\right)\right]}}\,\right)=1-\eta ,} where D {\displaystyle D} is the VC dimension of the classification model, 0 < η ⩽ 1 {\displaystyle 0<\eta \leqslant 1} , and N {\displaystyle N} is the size of the training set (restriction: this formula is valid when D ≪ N {\displaystyle D\ll N} . When D {\displaystyle D} is larger, the test-error may be much higher than the training-error. This is due to overfitting). The VC dimension also appears in sample-complexity bounds. A space of binary functions with VC dimension D {\displaystyle D} can be learned with: N = Θ ( D + ln ⁡ 1 δ ε 2 ) {\displaystyle N=\Theta \left({\frac {D+\ln {1 \over \delta }}{\varepsilon ^{2}}}\right)} samples, where ε {\displaystyle \varepsilon } is the learning error and δ {\displaystyle \delta } is the failure probability. Thus, the sample-complexity is a linear function of the VC dimension of the hypothesis space. === In computational geometry === The VC dimension is one of the critical parameters in the size of ε-nets, which determines the complexity of approximation algorithms based on them; range sets without finite VC dimension may not have finite ε-nets at all. == Bounds == The VC dimension of the dual set-family of C {\displaystyle {\mathcal {C}}} is strictly less than 2 vc ⁡ ( C ) + 1 {\displaystyle 2^{\operatorname {vc} ({\mathcal {C}})+1}} , and this is best possible. The VC dimension of a finite set-family C {\displaystyle {\mathcal {C}}} is at most log 2 ⁡ | C | {\displaystyle \log _{2}|{\mathcal {C}}|} . This is because | C ∩ X | ≤ | X | {\displaystyle |{\mathcal {C}}\cap X|\leq |X|} by definition. Given a set-fa

  Read more →
• 

  Charge based boundary element fast multipole method

  The charge-based formulation of the boundary element method (BEM) is a dimensionality reduction numerical technique that is used to model quasistatic electromagnetic phenomena in highly complex conducting media (targeting, e.g., the human brain) with a very large (up to approximately 1 billion) number of unknowns. The charge-based BEM solves an integral equation of the potential theory written in terms of the induced surface charge density. This formulation is naturally combined with fast multipole method (FMM) acceleration, and the entire method is known as charge-based BEM-FMM. The combination of BEM and FMM is a common technique in different areas of computational electromagnetics and, in the context of bioelectromagnetism, it provides improvements over the finite element method. == Historical development == Along with more common electric potential-based BEM, the quasistatic charge-based BEM, derived in terms of the single-layer (charge) density, for a single-compartment medium has been known in the potential theory since the beginning of the 20th century. For multi-compartment conducting media, the surface charge density formulation first appeared in discretized form (for faceted interfaces) in the 1964 paper by Gelernter and Swihart. A subsequent continuous form, including time-dependent and dielectric effects, appeared in the 1967 paper by Barnard, Duck, and Lynn. The charge-based BEM has also been formulated for conducting, dielectric, and magnetic media, and used in different applications. In 2009, Greengard et al. successfully applied the charge-based BEM with fast multipole acceleration to molecular electrostatics of dielectrics. A similar approach to realistic modeling of the human brain with multiple conducting compartments was first described by Makarov et al. in 2018. Along with this, the BEM-based multilevel fast multipole method has been widely used in radar and antenna studies at microwave frequencies as well as in acoustics. == Physical background - surface charges in biological media == The charge-based BEM is based on the concept of an impressed (or primary) electric field E i {\displaystyle \mathbf {E} ^{i}} and a secondary electric field E s {\displaystyle \mathbf {E} ^{s}} . The impressed field is usually known a priori or is trivial to find. For the human brain, the impressed electric field can be classified as one of the following: A conservative field E i {\displaystyle \mathbf {E} ^{i}} derived from an impressed density of EEG or MEG current sources in a homogeneous infinite medium with the conductivity σ {\displaystyle \sigma } at the source location; An instantaneous solenoidal field E i {\displaystyle \mathbf {E} ^{i}} of an induction coil obtained from Faraday's law of induction in a homogeneous infinite medium (air), when transcranial magnetic stimulation (TMS) problems are concerned; A surface field E i {\displaystyle \mathbf {E} ^{i}} derived from an impressed surface current density J i = σ E i {\displaystyle \mathbf {J} ^{i}=\sigma \mathbf {E} ^{i}} of current electrodes injecting electric current at a boundary of a compartment with conductivity σ {\displaystyle \sigma } when transcranial direct-current stimulation (tDCS) or deep brain stimulation (DBS) are concerned; A conservative field E i {\displaystyle \mathbf {E} ^{i}} of charges deposited on voltage electrodes for tDCS or DBS. This specific problem requires a coupled treatment since these charges will depend on the environment; In application to multiscale modeling, a field E i {\displaystyle \mathbf {E} ^{i}} obtained from any other macroscopic numerical solution in a small (mesoscale or microscale) spatial domain within the brain. For example, a constant field can be used. When the impressed field is "turned on", free charges located within a conducting volume D immediately begin to redistribute and accumulate at the boundaries (interfaces) of regions of different conductivity in D. A surface charge density ρ ( r ) {\displaystyle \rho (\mathbf {r} )} appears on the conductivity interfaces. This charge density induces a secondary conservative electric field E s {\displaystyle \mathbf {E} ^{s}} following Coulomb's law. One example is a human under a direct current powerline with the known field E i {\displaystyle \mathbf {E} ^{i}} directed down. The superior surface of the human's conducting body will be charged negatively while its inferior portion is charged positively. These surface charges create a secondary electric field that effectively cancels or blocks the primary field everywhere in the body so that no current will flow within the body under DC steady state conditions. Another example is a human head with electrodes attached. At any conductivity interface with a normal vector n {\displaystyle \mathbf {n} } pointing from an "inside" (-) compartment of conductivity σ − {\displaystyle \sigma ^{-}} to an "outside" (+) compartment of conductivity σ + {\displaystyle \sigma ^{+}} , Kirchhoff's current law requires continuity of the normal component of the electric current density. This leads to the interfacial boundary condition in the form for every facet at a triangulated interface. As long as σ ± {\displaystyle \sigma ^{\pm }} are different from each other, the two normal components of the electric field, E ± ⋅ n {\displaystyle \mathbf {E} ^{\pm }\cdot \mathbf {n} } , must also be different. Such a jump across the interface is only possible when a sheet of surface charge exists at that interface. Thus, if an electric current or voltage is applied, the surface charge density follows. The goal of the numerical analysis is to find the unknown surface charge distribution and thus the total electric field E = E i + E s {\displaystyle \mathbf {E} =\mathbf {E} ^{i}+\mathbf {E} ^{s}} (and the total electric potential if required) anywhere in space. == System of equations for surface charges == Below, a derivation is given based on Gauss's law and Coulomb's law. All conductivity interfaces, denoted by S, are discretized into planar triangular facets t m {\displaystyle t_{m}} with centers r m {\displaystyle \mathbf {r} _{m}} . Assume that an m-th facet with the normal vector n m {\displaystyle \mathbf {n} _{m}} and area A m {\displaystyle A_{m}} carries a uniform surface charge density ρ m {\displaystyle \rho _{m}} . If a volumetric tetrahedral mesh were present, the charged facets would belong to tetrahedra with different conductivity values. We first compute the electric field E m + {\displaystyle \mathbf {E} _{m}^{+}} at the point r m + δ n m {\displaystyle \mathbf {r} _{m}+\delta \mathbf {n} _{m}} , for δ → 0 + {\displaystyle \delta \rightarrow 0^{+}} i.e., just outside facet 𝑚 at its center. This field contains three contributions: The continuous impressed electric field E i {\displaystyle \mathbf {E} ^{i}} itself; An electric field of the m-th charged facet itself. Very close to the facet, it can be approximated as the electric field of an infinite sheet of uniform surface charge ρ m {\displaystyle \rho _{m}} . By Gauss's law, it is given by + ρ m / 2 ε 0 ⋅ n m {\displaystyle +\rho _{m}/2\varepsilon _{0}\cdot \mathbf {n} _{m}} where ε 0 {\displaystyle \varepsilon _{0}} is a background electrical permittivity; An electric field generated by all other facets t n {\displaystyle t_{n}} , which we approximate as point charges of charge A n ρ n {\displaystyle A_{n}\rho _{n}} at each center r n {\displaystyle \mathbf {r} _{n}} . A similar treatment holds for the electric field E m − {\displaystyle \mathbf {E} _{m}^{-}} just inside facet 𝑚, but the electric field of the flat sheet of charge changes its sign. Using Coulomb's law to calculate the contribution of facets different from t m {\displaystyle t_{m}} , we find From this equation, we see that the normal component of the electric field indeed undergoes a jump through the charged interface. This is equivalent to a jump relation of the potential theory. As a second step, the two expressions for E m ± {\displaystyle \mathbf {E} _{m}^{\pm }} are substituted into the interfacial boundary condition σ − E m − ⋅ n m = σ + E m + ⋅ n m {\displaystyle \sigma ^{-}\mathbf {E} _{m}^{-}\cdot \mathbf {n} _{m}=\sigma ^{+}\mathbf {E} _{m}^{+}\cdot \mathbf {n} _{m}} , applied to every facet 𝑚. This operation leads to a system of linear equations for unknown charge densities ρ m {\displaystyle \rho _{m}} which solves the problem: where K m = σ − − σ + σ − + σ + {\displaystyle K_{m}={\frac {\sigma ^{-}-\sigma ^{+}}{\sigma ^{-}+\sigma ^{+}}}} is the electric conductivity contrast at the m-th facet. The normalization constant ε 0 {\displaystyle \varepsilon _{0}} will cancel out after the solution is substituted in the expression for E s {\displaystyle \mathbf {E} ^{s}} and becomes redundant. == Application of fast multipole method == For modern characterizations of brain topologies with ever-increasing levels of complexity, the above system of equations for ρ m {\displaystyle \rho _{m}} is very large; it is t

  Read more →
• 

  Depth peeling

  In computer graphics, depth peeling is an exact multipass method of order-independent transparency that extracts transparent fragments into depth layers and composites those layers in depth order. Depth peeling has the advantage of being able to generate correct results even for complex images containing intersecting transparent objects. == Method == Depth peeling works by rendering the image multiple times. Depth peeling uses two Z buffers, one that works conventionally, and one that is not modified, and sets the minimum distance at which a fragment can be drawn without being discarded. For each pass, the previous pass' conventional Z-buffer is used as the minimal Z-buffer, so each pass removes already-captured nearer fragments and draws the next depth layer behind them. The resulting images can then be composited in depth order to form a single image. A major drawback of classical depth peeling is performance: it requires one geometry pass per peeled layer, so scenes with high depth complexity require many passes that each re-rasterize the transparent geometry. Later variants reduce the number of passes by peeling multiple layers or both front and back layers in a pass. Dual depth peeling reduces the geometry-pass count from N to N/2+1 by peeling one layer from the front and one from the back in each pass, while multi-layer depth peeling peels several layers per pass and reported up to an 8x speed-up in RGBA8 settings.

  Read more →
• 

  Quadratic classifier

  In statistics, a quadratic classifier is a statistical classifier that uses a quadratic decision surface to separate measurements of two or more classes of objects or events. It is a more general version of the linear classifier. == The classification problem == Statistical classification considers a set of vectors of observations x of an object or event, each of which has a known type y. This set is referred to as the training set. The problem is then to determine, for a given new observation vector, what the best class should be. For a quadratic classifier, the correct solution is assumed to be quadratic in the measurements, so y will be decided based on x T A x + b T x + c {\displaystyle \mathbf {x^{T}Ax} +\mathbf {b^{T}x} +c} In the special case where each observation consists of two measurements, this means that the surfaces separating the classes will be conic sections (i.e., either a line, a circle or ellipse, a parabola or a hyperbola). In this sense, we can state that a quadratic model is a generalization of the linear model, and its use is justified by the desire to extend the classifier's ability to represent more complex separating surfaces. == Quadratic discriminant analysis == Quadratic discriminant analysis (QDA) is closely related to linear discriminant analysis (LDA), where it is assumed that the measurements from each class are normally distributed. Unlike LDA however, in QDA there is no assumption that the covariance of each of the classes is identical. When the normality assumption is true, the best possible test for the hypothesis that a given measurement is from a given class is the likelihood ratio test. Suppose there are only two groups, with means μ 0 , μ 1 {\displaystyle \mu _{0},\mu _{1}} and covariance matrices Σ 0 , Σ 1 {\displaystyle \Sigma _{0},\Sigma _{1}} corresponding to y = 0 {\displaystyle y=0} and y = 1 {\displaystyle y=1} respectively. Then the likelihood ratio is given by Likelihood ratio = | 2 π Σ 1 | − 1 exp ⁡ ( − 1 2 ( x − μ 1 ) T Σ 1 − 1 ( x − μ 1 ) ) | 2 π Σ 0 | − 1 exp ⁡ ( − 1 2 ( x − μ 0 ) T Σ 0 − 1 ( x − μ 0 ) ) < t {\displaystyle {\text{Likelihood ratio}}={\frac {{\sqrt {|2\pi \Sigma _{1}|}}^{-1}\exp \left(-{\frac {1}{2}}(\mathbf {x} -{\boldsymbol {\mu }}_{1})^{T}\Sigma _{1}^{-1}(\mathbf {x} -{\boldsymbol {\mu }}_{1})\right)}{{\sqrt {|2\pi \Sigma _{0}|}}^{-1}\exp \left(-{\frac {1}{2}}(\mathbf {x} -{\boldsymbol {\mu }}_{0})^{T}\Sigma _{0}^{-1}(\mathbf {x} -{\boldsymbol {\mu }}_{0})\right)}} Read more →

• 

  Shattered set

  A class of sets is said to shatter another set if it is possible to "pick out" any element of that set using intersection. The concept of shattered sets plays an important role in Vapnik–Chervonenkis theory, also known as VC-theory. Shattering and VC-theory are used in the study of empirical processes as well as in statistical computational learning theory. == Definition == Suppose A is a set and C is a class of sets. The class C shatters the set A if for each subset a of A, there is some element c of C such that a = c ∩ A . {\displaystyle a=c\cap A.} Equivalently, C shatters A when their intersection is equal to A's power set: P(A) = { c ∩ A | c ∈ C }. We employ the letter C to refer to a "class" or "collection" of sets, as in a Vapnik–Chervonenkis class (VC-class). The set A is often assumed to be finite because, in empirical processes, we are interested in the shattering of finite sets of data points. == Example == We will show that the class of all discs in the plane (two-dimensional space) does not shatter every set of four points on the unit circle, yet the class of all convex sets in the plane does shatter every finite set of points on the unit circle. Let A be a set of four points on the unit circle and let C be the class of all discs. To test where C shatters A, we attempt to draw a disc around every subset of points in A. First, we draw a disc around the subsets of each isolated point. Next, we try to draw a disc around every subset of point pairs. This turns out to be doable for adjacent points, but impossible for points on opposite sides of the circle. Any attempt to include those points on the opposite side will necessarily include other points not in that pair. Hence, any pair of opposite points cannot be isolated out of A using intersections with class C and so C does not shatter A. As visualized below: Because there is some subset which can not be isolated by any disc in C, we conclude then that A is not shattered by C. And, with a bit of thought, we can prove that no set of four points is shattered by this C. However, if we redefine C to be the class of all elliptical discs, we find that we can still isolate all the subsets from above, as well as the points that were formerly problematic. Thus, this specific set of 4 points is shattered by the class of elliptical discs. Visualized below: With a bit of thought, we could generalize that any set of finite points on a unit circle could be shattered by the class of all convex sets (visualize connecting the dots). == Shatter coefficient == To quantify the richness of a collection C of sets, we use the concept of shattering coefficients (also known as the growth function). For a collection C of sets s ⊂ Ω {\displaystyle s\subset \Omega } , Ω {\displaystyle \Omega } being any space, often a sample space, we define the nth shattering coefficient of C as S C ( n ) = max ∀ x 1 , x 2 , … , x n ∈ Ω card ⁡ { { x 1 , x 2 , … , x n } ∩ s , s ∈ C } {\displaystyle S_{C}(n)=\max _{\forall x_{1},x_{2},\dots ,x_{n}\in \Omega }\operatorname {card} \{\,\{\,x_{1},x_{2},\dots ,x_{n}\}\cap s,s\in C\}} where card {\displaystyle \operatorname {card} } denotes the cardinality of the set and x 1 , x 2 , … , x n ∈ Ω {\displaystyle x_{1},x_{2},\dots ,x_{n}\in \Omega } is any set of n points,. S C ( n ) {\displaystyle S_{C}(n)} is the largest number of subsets of any set A of n points that can be formed by intersecting A with the sets in collection C. For example, if set A contains 3 points, its power set, P ( A ) {\displaystyle P(A)} , contains 2 3 = 8 {\displaystyle 2^{3}=8} elements. If C shatters A, its shattering coefficient(3) would be 8 and S C ( 2 ) {\displaystyle S_{C}(2)} would be 2 2 = 4 {\displaystyle 2^{2}=4} . However, if one of those sets in P ( A ) {\displaystyle P(A)} cannot be obtained through intersections in c, then S C ( 3 ) {\displaystyle S_{C}(3)} would only be 7. If none of those sets can be obtained, S C ( 3 ) {\displaystyle S_{C}(3)} would be 0. Additionally, if S C ( 2 ) = 3 {\displaystyle S_{C}(2)=3} , for example, then there is an element in the set of all 2-point sets from A that cannot be obtained from intersections with C. It follows from this that S C ( 3 ) {\displaystyle S_{C}(3)} would also be less than 8 (i.e. C would not shatter A) because we have already located a "missing" set in the smaller power set of 2-point sets. This example illustrates some properties of S C ( n ) {\displaystyle S_{C}(n)} : S C ( n ) ≤ 2 n {\displaystyle S_{C}(n)\leq 2^{n}} for all n because { s ∩ A | s ∈ C } ⊆ P ( A ) {\displaystyle \{s\cap A|s\in C\}\subseteq P(A)} for any A ⊆ Ω {\displaystyle A\subseteq \Omega } . If S C ( n ) = 2 n {\displaystyle S_{C}(n)=2^{n}} , that means there is a set of cardinality n, which can be shattered by C. If S C ( N ) < 2 N {\displaystyle S_{C}(N)<2^{N}} for some N > 1 {\displaystyle N>1} then S C ( n ) < 2 n {\displaystyle S_{C}(n)<2^{n}} for all n ≥ N {\displaystyle n\geq N} . The third property means that if C cannot shatter any set of cardinality N then it can not shatter sets of larger cardinalities. == Vapnik–Chervonenkis class == If A cannot be shattered by C, there will be a smallest value of n that makes the shatter coefficient(n) less than 2 n {\displaystyle 2^{n}} because as n gets larger, there are more sets that could be missed. Alternatively, there is also a largest value of n for which the S C ( n ) {\displaystyle S_{C}(n)} is still 2 n {\displaystyle 2^{n}} , because as n gets smaller, there are fewer sets that could be omitted. The extreme of this is S C ( 0 ) {\displaystyle S_{C}(0)} (the shattering coefficient of the empty set), which must always be 2 0 = 1 {\displaystyle 2^{0}=1} . These statements lends themselves to defining the VC dimension of a class C as: V C ( C ) = min n { n : S C ( n ) < 2 n } {\displaystyle VC(C)={\underset {n}{\min }}\{n:S_{C}(n)<2^{n}\}\,} or, alternatively, as V C 0 ( C ) = max n { n : S C ( n ) = 2 n } . {\displaystyle VC_{0}(C)={\underset {n}{\max }}\{n:S_{C}(n)=2^{n}\}.\,} Note that V C ( C ) = V C 0 ( C ) + 1. {\displaystyle VC(C)=VC_{0}(C)+1.} . The VC dimension is usually defined as V C 0 {\displaystyle VC_{0}} , the largest cardinality of points chosen that will still shatter A (i.e. n such that S C ( n ) = 2 n {\displaystyle S_{C}(n)=2^{n}} ). Altneratively, if for any n there is a set of cardinality n which can be shattered by C, then S C ( n ) = 2 n {\displaystyle S_{C}(n)=2^{n}} for all n and the VC dimension of this class C is infinite. A class with finite VC dimension is called a Vapnik–Chervonenkis class or VC class. A class C is uniformly Glivenko–Cantelli if and only if it is a VC class.

  Read more →