Digital data or digital information, in information theory and information systems, is data or information represented as a string of discrete symbols, each of which can take on one of only a finite number of values from some alphabet, such as letters or digits. An example is a text document, which consists of a string of alphanumeric characters. The most common form of digital data in modern information systems is binary data, which is represented by a string of binary digits (bits) each of which can have one of two values, either 0 or 1. Digital data can be contrasted with analog data, which is represented by a value from a continuous range of real numbers. Analog data is transmitted by an analog signal, which not only takes on continuous values but can vary continuously with time, a continuous real-valued function of time. An example is the air pressure variation in a sound wave. Data requires interpretation to become information. In modern (post-1960) computer systems, all data is digital. The word digital comes from the same source as the words digit and digitus (the Latin word for finger), as fingers are often used for counting. Mathematician George Stibitz of Bell Telephone Laboratories used the word digital in reference to the fast electric pulses emitted by a device designed to aim and fire anti-aircraft guns in 1942. The term is most commonly used in computing and electronics, especially where real-world information is converted to binary numeric form as in digital audio and digital photography. == Symbol to digital conversion == Since symbols (for example, alphanumeric characters) are not continuous, representing symbols digitally is rather simpler than conversion of continuous or analog information to digital. Instead of sampling and quantization as in analog-to-digital conversion, such techniques as polling and encoding are used. A symbol input device usually consists of a group of switches that are polled at regular intervals to see which switches are switched. Data will be lost if, within a single polling interval, two switches are pressed, or a switch is pressed, released, and pressed again. This polling can be done by a specialized processor in the device to prevent burdening the main CPU. When a new symbol has been entered, the device typically sends an interrupt, in a specialized format, so that the CPU can read it. For devices with only a few switches (such as the buttons on a joystick), the status of each can be encoded as bits (usually 0 for released and 1 for pressed) in a single word. This is useful when combinations of key presses are meaningful, and is sometimes used for passing the status of modifier keys on a keyboard (such as shift and control). But it does not scale to support more keys than the number of bits in a single byte or word. Devices with many switches (such as a computer keyboard) usually arrange these switches in a scan matrix, with the individual switches on the intersections of x and y lines. When a switch is pressed, it connects the corresponding x and y lines together. Polling (often called scanning in this case) is done by activating each x line in sequence and detecting which y lines then have a signal, thus which keys are pressed. When the keyboard processor detects that a key has changed state, it sends a signal to the CPU indicating the scan code of the key and its new state. The symbol is then encoded or converted into a number based on the status of modifier keys and the desired character encoding. A custom encoding can be used for a specific application with no loss of data. However, using a standard encoding such as ASCII is problematic if a symbol such as 'ß' needs to be converted but is not in the standard. It is estimated that in the year 1986, less than 1% of the world's technological capacity to store information was digital and in 2007 it was already 94%. The year 2002 is assumed to be the year when humankind was able to store more information in digital than in analog format (the "beginning of the digital age"). == States == Digital data come in these three states: data at rest, data in transit, and data in use. The confidentiality, integrity, and availability have to be managed during the entire lifecycle from 'birth' to the destruction of the data. === Data at rest === Data at rest in information technology means data that is housed physically on computer data storage in any digital form (e.g. cloud storage, file hosting services, databases, data warehouses, spreadsheets, archives, tapes, off-site or cloud backups, mobile devices etc.). Data at rest includes both structured and unstructured data. This type of data is subject to threats from hackers and other malicious threats to gain access to the data digitally or physical theft of the data storage media. To prevent this data from being accessed, modified or stolen, organizations will often employ security protection measures such as password protection, data encryption, or a combination of both. The security options used for this type of data are broadly referred to as data-at-rest protection (DARP). Definitions include: "...all data in computer storage while excluding data that is traversing a network or temporarily residing in computer memory to be read or updated." "...all data in storage but excludes any data that frequently traverses the network or that which resides in temporary memory. Data at rest includes but is not limited to archived data, data which is not accessed or changed frequently, files stored on hard drives, USB thumb drives, files stored on backup tape and disks, and also files stored off-site or on a storage area network (SAN)." While it is generally accepted that archive data (i.e. which never changes), regardless of its storage medium, is data at rest and active data subject to constant or frequent change is data in use. “Inactive data” could be taken to mean data which may change, but infrequently. The imprecise nature of terms such as “constant” and “frequent” means that some stored data cannot be comprehensively defined as either data at rest or in use. These definitions could be taken to assume that Data at Rest is a superset of data in use; however, data in use, subject to frequent change, has distinct processing requirements from data at rest, whether completely static or subject to occasional change. ==== Security ==== Because of its nature data at rest is of increasing concern to businesses, government agencies and other institutions. Mobile devices are often subject to specific security protocols to protect data at rest from unauthorized access when lost or stolen and there is an increasing recognition that database management systems and file servers should also be considered as at risk; the longer data is left unused in storage, the more likely it might be retrieved by unauthorized individuals outside the network. Data encryption, which prevents data visibility in the event of its unauthorized access or theft, is commonly used to protect data in motion and increasingly promoted for protecting data at rest. The encryption of data at rest should only include strong encryption methods such as AES or RSA. Encrypted data should remain encrypted when access controls such as usernames and password fail. Increasing encryption on multiple levels is recommended. Cryptography can be implemented on the database housing the data and on the physical storage where the databases are stored. Data encryption keys should be updated on a regular basis. Encryption keys should be stored separately from the data. Encryption also enables crypto-shredding at the end of the data or hardware lifecycle. Periodic auditing of sensitive data should be part of policy and should occur on scheduled occurrences. Finally, only store the minimum possible amount of sensitive data. Tokenization is a non-mathematical approach to protecting data at rest that replaces sensitive data with non-sensitive substitutes, referred to as tokens, which have no extrinsic or exploitable meaning or value. This process does not alter the type or length of data, which means it can be processed by legacy systems such as databases that may be sensitive to data length and type. Tokens require significantly less computational resources to process and less storage space in databases than traditionally encrypted data. This is achieved by keeping specific data fully or partially visible for processing and analytics while sensitive information is kept hidden. Lower processing and storage requirements makes tokenization an ideal method of securing data at rest in systems that manage large volumes of data. A further method of preventing unwanted access to data at rest is the use of data federation especially when data is distributed globally (e.g. in off-shore archives). An example of this would be a European organisation which stores its archived data off-site in the US. Under the terms of the USA PATRIOT Act the American authorities can demand
Opponent process
The opponent process is a hypothesis of color vision that states that the human visual system interprets information about color by processing signals from the three types of photoreceptor cells in an antagonistic manner. The three types of cones are called L, M, and S. The names stand for "Long wavelength sensitive,” "middle wavelength sensitive," and "short wavelength sensitive." The opponent-process theory implicates three opponent channels: L versus M, S versus (L+M), and a luminance channel (+ versus -). These cone-opponent mechanisms were at one time thought to be the neural substrate for a psychological theory called Hering's Opponent Colors Theory, which calls for three psychologically important opponent color processes: red versus green, blue versus yellow, and black versus white (luminance). The Opponent Colors Theory is named for the German physiologist Ewald Hering who proposed the idea in the late 19th century. However, it has been argued that Hering’s Opponent Colors Theory lacks adequate phenomenological and empirical support, and may not be a necessary feature of normal human color experience. Correspondingly, considerable physiological and behavioral evidence proves that the physiological cone opponent mechanisms do not constitute the neurobiological basis for Hering's Opponent Colors Theory. == Color theory == === Complementary colors === When staring at a bright color for a while (e.g. red), then looking away at a white field, an afterimage is perceived, such that the original color will evoke its complementary color (cyan, in the case of red input). When complementary colors are combined or mixed, they "cancel each other out" and become neutral (white or gray). That is, complementary colors are never perceived as a mixture; there is no "greenish red" or "yellowish blue", despite claims to the contrary. The strongest color contrast that a color can have is its complementary color. Complementary colors may also be called "opposite colors" and they were originally considered the primary evidence in support of Hering's Opponent Colors Theory. There are two fatal problems with this evidence. First, the complement of red is not green, as called for by Hering's theory; it is bluish-green. And second, there exists a complementary color for every color, so there is nothing special about the set of complementary pairs picked out by Hering's theory. === Unique hues === The colors that define the extremes for each opponent channel are called unique hues, as opposed to composite (mixed) hues. Ewald Hering first defined the unique hues as red, green, blue, and yellow, and based them on the concept that these colors could not be simultaneously perceived. For example, a color cannot appear both red and green. These definitions have been experimentally refined and are represented today by average hue angles of 353° (carmine red), 128° (cobalt green), 228° (cobalt blue), 58° (yellow). The unique hues are a defining feature of many psychological color spaces, but there is substantial evidence showing that the unique hues are not hard wired in the nervous system, contrary to the stipulations of Hering's Opponent Colors Theory. Unique hues can differ between individuals and are often used in psychophysical research to measure variations in color perception due to color-vision deficiencies or color adaptation. While there is considerable inter-subject variability when defining unique hues experimentally, an individual's unique hues are very consistent, to within a few nanometers of wavelength. == Physiological basis == === Relation to LMS color space === The trichromatic theory is in conflict with Hering's Opponent Colors Theory, although it is compatible with a physiological opponent process that compares the outputs of the different classes of cone types. The poles of these cone opponent mechanisms do not correspond to the unique hues of Hering's Opponent Colors Theory and unlike the unique hues, have no privilege in color perception. Most humans have three different cone cells in their retinas that facilitate trichromatic color vision. Colors are determined by the proportional excitation of these three cone types, i.e. their quantum catch. The levels of excitation of each cone type are the parameters that define LMS color space. To calculate the opponent process tristimulus values from the LMS color space, the cone excitations must be compared: The luminous (achromatic) opponent channel is a weighted sum of all three cone cells (plus the rod cells in some conditions). The red–green opponent channel is equal to the difference of the L- and M-cones. The blue–yellow opponent channel is equal to the difference of the S-cone and the average/weighted sum of the L- and M-cones. Most mammals have no L cone (the primate L cone arose from a gene duplication of the M cone opsin gene). These mammals still show two kinds of opponent channels in their retinal ganglion cells: the achromatic channel and the blue-yellow opponency channel. === Cone opponent mechanisms are encoded in the retina === The output of different types of cones are compared by cells in the retina including retina bipolar cells (which compare signals from L and M cones) and bistratified retinal ganglion cells (which compare S cone signals with L and M cone signals). The output of bipolar cells is relayed to the visual cortex by the retinal ganglion cells (RGCs) by way of a thalamic relay station called the lateral geniculate nucleus (LGN) of the thalamus. Much of the scientific knowledge of retinal ganglion cell physiology was obtained by neural recordings of cells in the LGN. The cone-opponent mechanisms in the retina and LGN represent a fundamental physiological opponent process but do not represent the unique hues (or Hering's Opponent Colors Theory). For example, the colors that best elicit responses of the bistratified S-(L+M)-opponent neurons are best described as purplish (or lavender) and lime-green, not "blue" and "yellow". The neurons are sometimes referred to as "blue–yellow" neurons, but this is a historical artifact dating to the time when it was thought that Hering's Opponent Colors Theory was hardwired by the retina and the mismatch between the colors to which they are optimally tuned and Hering's Opponent Colors was overlooked. Cone opponent mechanisms exist in the retinas of many mammals, including monkeys, mice, and cats. In primates, the LGN contains three major classes of layers: Magnocellular layers (M, large-cell) – responsible largely for the luminance channel Parvocellular layers (P, small-cell) – responsible largely for red–green opponency Koniocellular layers (K) – responsible largely for blue–yellow opponency, poor spatial resolution, long latency Other mammals such as cats also have three cell types denoted as X (magno), Y (parvo), and W (konio). The W type is beyond most doubt homologous to the primate K type. There are some subtle differences between the M and X types as well as the Y and P types to make the correspondence unclear. === Advantage === Transmitting information in opponent-channel color space could be advantageous over transmitting it in LMS color space ("raw" signals from each cone type). There is some overlap in the wavelengths of light to which the three types of cones (L for long-wave, M for medium-wave, and S for short-wave light) respond, so it is more efficient for the visual system (from a perspective of dynamic range) to record differences between the responses of cones, rather than each type of cone's individual response. Hurvich and Jameson argued that the use of opponent-channel color space would increase color contrast, making the information easier to process by later stages of vision. === Color blindness === Color blindness can be classified by the cone cell that is affected (protan, deutan, tritan) or by the opponent channel that is affected (red–green or blue–yellow). In either case, the channel can either be inactive (in the case of dichromacy) or have a lower dynamic range (in the case of anomalous trichromacy). For example, individuals with deuteranopia see little difference between the red and green unique hues. == History == Johann Wolfgang von Goethe first studied the physiological effect of opposed colors in his Theory of Colours in 1810. Goethe arranged his color wheel symmetrically "for the colours diametrically opposed to each other in this diagram are those which reciprocally evoke each other in the eye. Thus, yellow demands purple; orange, blue; red, green; and vice versa: Thus again all intermediate gradations reciprocally evoke each other." Ewald Hering proposed opponent color theory in 1892. He thought that the colors red, yellow, green, and blue are special in that any other color can be described as a mix of them, and that they exist in opposite pairs. That is, either red or green is perceived and never greenish-red: Even though yellow is a mixture of red and green in the RGB color theory, humans
Drools
Drools is a business rule management system (BRMS) with a forward and backward chaining inference-based rules engine, more correctly known as a production rule system, using an enhanced implementation of the Rete algorithm. Drools supports the Java Rules Engine API (Java Specification Request 94) standard for its business rule engine and enterprise framework for the construction, maintenance, and enforcement of business policies in an organization, application, or service. == Drools in Apache Kie == Drools, as part of the Kie Community has entered Apache Incubator in January, 2023. == Red Hat Decision Manager == Red Hat Decision Manager (formerly Red Hat JBoss BRMS) is a business rule management system and reasoning engine for business policy and rules development, access, and change management. JBoss Enterprise BRMS is a productized version of Drools with enterprise-level support available. JBoss Rules is also a productized version of Drools, but JBoss Enterprise BRMS is the flagship product. Components of the enterprise version: JBoss Enterprise Web Platform – the software infrastructure, supported to run the BRMS components only JBoss Enterprise Application Platform or JBoss Enterprise SOA Platform – the software infrastructure, supported to run the BRMS components only Business Rules Engine – Drools Expert using the Rete algorithm and the Drools Rule Language (DRL) Business Rules Manager – Drools Guvnor - Guvnor is a centralized repository for Drools Knowledge Bases, with rich web-based GUIs, editors, and tools to aid in the management of large numbers of rules. Business Rules Repository – Drools Guvnor Drools and Guvnor are JBoss Community open source projects. As they are mature, they are brought into the enterprise-ready product JBoss Enterprise BRMS. Components of the JBoss Community version: Drools Guvnor (Business Rules Manager) – a centralized repository for Drools Knowledge Bases Drools Expert (rule engine) – uses the rules to perform reasoning Drools Flow (process/workflow), or jBPM 5 – provides for workflow and business processes Drools Fusion (event processing/temporal reasoning) – provides for complex event processing Drools Planner/OptaPlanner (automated planning) – optimizes automated planning, including NP-hard planning problems == Example == This example illustrates a simple rule to print out information about a holiday in July. It checks a condition on an instance of the Holiday class, and executes Java code if that condition is true. The purpose of dialect "mvel" is to point the getter and setters of the variables of your Plain Old Java Object (POJO) classes. Consider the above example, in which a Holiday class is used and inside the circular brackets (parentheses) "month" is used. So with the help of dialect "mvel" the getter and setters of the variable "month" can be accessed. Dialect "java" is used to help us write our Java code in our rules. There is one restriction or characteristic on this. We cannot use Java code inside the "when" part of the rule but we can use Java code in the "then" part. We can also declare a Reference variable $h1 without the $ symbol. There is no restriction on this. The main purpose of putting the $ symbol before the variable is to mark the difference between variables of POJO classes and Rules.
Liveness test
A liveness test, liveness check or liveness detection is an automated method for determining whether a subject is a real person or part of a spoofing attack. The technique is used as part of know your customer checks in financial services and during facial age estimation. Liveness detection is a cornerstone of digital safety. == Test process == The threat in face spoofing attacks is that "the attacker only needs to find a good face swap library on Github and understand how to inject the model into the camera feed during the KYC process". Fraudsters usually buy stolen IDs on the dark web to start a deepfake attack. An AI-powered generative adversarial network (GAN) can then generate the face swapping model that many online verification services fail to detect. Low level hackers may use face swapping apps such as SwapFace, DeepFaceLive, and Swapstream (increasing interest for those apps in 2023 according to Google Trends). In a video liveness test, users are typically asked to look into a camera and to move, smile or blink, and features of their moving face may then be compared to that of a still image. Artificial intelligence is used to counter presentation attacks such as deepfakes or users wearing hyperrealistic masks, or video injection attacks. Other forms of liveness test include checking for a pulse when using a fingerprint scanner or checking that a person's voice is not a recording or artificially generated during speaker recognition. == Adoption and certification == In a 2022 report published by the security firm Sensity, it was demonstrated that the liveness test of most US banks was easily cheated with new and publicly-available AI-powered techniques. Many of these banks disregarded the results of the report. In the first half of 2023, the security firm iProov detected a 704% increase in face-swap attacks. In 2023, in the UK, many customers of Ryanair were upset to have to go through many ID verification checks, including liveness tests, before boarding, as the airline was using it as a mean to deter customers to buy tickets through third-party websites. In the first half of 2024 iBeta Quality Assurance issued 18 new ISO/IEC 30107-3 Presentation Attack Detection certificates, raising the cumulative total to 85 since 2018. In January 2024, the Department of Homeland Security (DHS) opened applications from vendors to test their Liveness test. Identity frauds peaked during the COVID-19 lockdown, leading government agencies to take reinforced measures to secure their digital applications.
Karen Hao
Karen Hao (born in the United States c. 1993) is an American journalist and author. Currently a freelancer for publications like The Atlantic and previously a foreign correspondent based in Hong Kong for The Wall Street Journal and senior artificial intelligence editor at the MIT Technology Review, she is best known for her coverage on AI research, technology ethics and the social impact of AI. Hao also co-produced the podcast In Machines We Trust and wrote the newsletter The Algorithm. Previously, she worked at Quartz as a tech reporter and data scientist and was an application engineer at the first startup to spin out of X Development. Hao's writing has also appeared in Mother Jones, Sierra Magazine, The New Republic, and other publications. == Early life and education == Hao is the daughter of Chinese immigrant parents, and grew up in New Jersey. She is a native speaker of both English and Mandarin Chinese. She graduated from The Lawrenceville School in 2011. She then studied at the Massachusetts Institute of Technology (MIT), graduating with a B.S. in mechanical engineering and a minor in energy studies in 2015. == Career == Hao is known in the technology world for her coverage of new AI research findings and their societal and ethical impacts. Her writing has spanned research and issues regarding big tech data privacy, misinformation, deepfakes, facial recognition, and AI healthcare tools. In March 2021, Hao published a piece that uncovered previously unknown information about how attempts to combat misinformation by different teams at Facebook using machine learning were impeded and constantly at odds with Facebook's drive to grow user engagement. Upon its release, leaders at Facebook including Mike Schroepfer and Yann LeCun immediately criticized the piece through Twitter responses. AI researchers and AI ethics experts Timnit Gebru and Margaret Mitchell responded in support of Hao's writing and advocated for more change and improvement for all. Hao also co-produced the podcast In Machines We Trust, which discusses the rise of AI with people developing, researching, and using AI technologies. The podcast won the 2020 Front Page Award in investigative reporting. Hao has occasionally created data visualizations that have been featured in her work at the MIT Technology Review and elsewhere. In 2018, her "What is AI?" flowchart visualization was exhibited as an installation at the Museum of Applied Arts in Vienna. She has been an invited speaker at TEDxGateway, the United Nations Foundation, EmTech, WNPR, and many other conferences and podcasts. Her TEDx talk discussed the importance of democratizing how AI is built. In March 2022, she was hired by The Wall Street Journal to cover China technology and society, while being based in Hong Kong. She left the WSJ in 2023. In May 2025, Hao released the book Empire of AI: Dreams and Nightmares in Sam Altman's OpenAI. The book became a New York Times Bestseller and was named a Book of the Year by the Financial Times. In December 2025, after criticism from readers, Hao issued a correction to her book where she had previously overestimated the water consumption of a data center in Chile compared to the community's water consumption by factor of 1,000, due to an error in a government document. In April 2026 the book won the New York Public Library's Helen Bernstein Book Award for Excellence in Journalism. === Selected awards and honors === 2019 Webby Award nominee for best newsletter, as a writer of The Algorithm 2021 Front Page Award in investigative reporting, as a co-producer for In Machines We Trust 2021 Ambies Award nominee for best knowledge and science podcast, as a co-producer for In Machines We Trust 2021 Webby Award nominee for best technology podcast, as a co-producer for In Machines We Trust 2024 American Humanist Media Award 2025 TIME100 AI, named by TIME magazine as one of the 100 most influential people in artificial intelligence 2026 New York Public Library's Helen Bernstein Book Award for Excellence in Journalism 2026 Whiting Award in Non-fiction
Histogram of oriented displacements
Histogram of oriented displacements (HOD) is a 2D trajectory descriptor. The trajectory is described using a histogram of the directions between each two consecutive points. Given a trajectory T = {P1, P2, P3, ..., Pn}, where Pt is the 2D position at time t. For each pair of positions Pt and Pt+1, calculate the direction angle θ(t, t+1). Value of θ is between 0 and 360. A histogram of the quantized values of θ is created. If the histogram is of 8 bins, the first bin represents all θs between 0 and 45. The histogram accumulates the lengths of the consecutive moves. For each θ, a specific histogram bin is determined. The length of the line between Pt and Pt+1 is then added to the specific histogram bin. To show the intuition behind the descriptor, consider the action of waving hands. At the end of the action, the hand falls down. When describing this down movement, the descriptor does not care about the position from which the hand started to fall. This fall will affect the histogram with the appropriate angles and lengths, regardless of the position where the hand started to fall. HOD records for each moving point: how much it moves in each range of directions. HOD has a clear physical interpretation. It proposes that, a simple way to describe the motion of an object, is to indicate how much distance it moves in each direction. If the movement in all directions are saved accurately, the movement can be repeated from the initial position to the final destination regardless of the displacements order. However, the temporal information will be lost, as the order of movements is not stored-this is what we solve by applying the temporal pyramid, as shown in section \ref{sec:temp-pyramid}. If the angles quantization range is small, classifiers that use the descriptor will overfit. Generalization needs some slack in directions-which can be done by increasing the quantization range.
Hyper basis function network
In machine learning, a Hyper basis function network, or HyperBF network, is a generalization of radial basis function (RBF) networks concept, where the Mahalanobis-like distance is used instead of the Euclidean distance measure. Hyper basis function networks were first introduced by Poggio and Girosi in the 1990 paper “Networks for Approximation and Learning”. == Network Architecture == The typical HyperBF network structure consists of a real input vector x ∈ R n {\displaystyle x\in \mathbb {R} ^{n}} , a hidden layer of activation functions and a linear output layer. The output of the network is a scalar function of the input vector, ϕ : R n → R {\displaystyle \phi :\mathbb {R} ^{n}\to \mathbb {R} } , is given by where N {\displaystyle N} is a number of neurons in the hidden layer, μ j {\displaystyle \mu _{j}} and a j {\displaystyle a_{j}} are the center and weight of neuron j {\displaystyle j} . The activation function ρ j ( | | x − μ j | | ) {\displaystyle \rho _{j}(||x-\mu _{j}||)} at the HyperBF network takes the following form where R j {\displaystyle R_{j}} is a positive definite d × d {\displaystyle d\times d} matrix. Depending on the application, the following types of matrices R j {\displaystyle R_{j}} are usually considered R j = 1 2 σ 2 I d × d {\displaystyle R_{j}={\frac {1}{2\sigma ^{2}}}\mathbb {I} _{d\times d}} , where σ > 0 {\displaystyle \sigma >0} . This case corresponds to the regular RBF network. R j = 1 2 σ j 2 I d × d {\displaystyle R_{j}={\frac {1}{2\sigma _{j}^{2}}}\mathbb {I} _{d\times d}} , where σ j > 0 {\displaystyle \sigma _{j}>0} . In this case, the basis functions are radially symmetric, but are scaled with different width. R j = d i a g ( 1 2 σ j 1 2 , . . . , 1 2 σ j z 2 ) I d × d {\displaystyle R_{j}=diag\left({\frac {1}{2\sigma _{j1}^{2}}},...,{\frac {1}{2\sigma _{jz}^{2}}}\right)\mathbb {I} _{d\times d}} , where σ j i > 0 {\displaystyle \sigma _{ji}>0} . Every neuron has an elliptic shape with a varying size. Positive definite matrix, but not diagonal. == Training == Training HyperBF networks involves estimation of weights a j {\displaystyle a_{j}} , shape and centers of neurons R j {\displaystyle R_{j}} and μ j {\displaystyle \mu _{j}} . Poggio and Girosi (1990) describe the training method with moving centers and adaptable neuron shapes. The outline of the method is provided below. Consider the quadratic loss of the network H [ ϕ ∗ ] = ∑ i = 1 N ( y i − ϕ ∗ ( x i ) ) 2 {\displaystyle H[\phi ^{}]=\sum _{i=1}^{N}(y_{i}-\phi ^{}(x_{i}))^{2}} . The following conditions must be satisfied at the optimum: where R j = W T W {\displaystyle R_{j}=W^{T}W} . Then in the gradient descent method the values of a j , μ j , W {\displaystyle a_{j},\mu _{j},W} that minimize H [ ϕ ∗ ] {\displaystyle H[\phi ^{}]} can be found as a stable fixed point of the following dynamic system: where ω {\displaystyle \omega } determines the rate of convergence. Overall, training HyperBF networks can be computationally challenging. Moreover, the high degree of freedom of HyperBF leads to overfitting and poor generalization. However, HyperBF networks have an important advantage that a small number of neurons is enough for learning complex functions.