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  Image

  An image or picture is a visual representation. An image can be two-dimensional, such as a drawing, painting, or photograph, or three-dimensional, such as a carving or sculpture. Images may be displayed through other media, including a projection on a surface, activation of electronic signals, or digital displays; they can also be reproduced through mechanical means, such as photography, printmaking, or photocopying. Images can also be animated through digital or physical processes. In the context of signal processing, an image is a distributed amplitude of color(s). In optics, the term image (or optical image) refers specifically to the reproduction of an object formed by light waves coming from the object. A volatile image exists or is perceived only for a short period. This may be a reflection of an object by a mirror, a projection of a camera obscura, or a scene displayed on a cathode-ray tube. A fixed image, also called a hard copy, is one that has been recorded on a material object, such as paper or textile. A mental image exists in an individual's mind as something one remembers or imagines. The subject of an image does not need to be real; it may be an abstract concept such as a graph or function or an imaginary entity. For a mental image to be understood outside of an individual's mind, however, there must be a way of conveying that mental image through the words or visual productions of the subject. == Characteristics == === Two-dimensional images === The broader sense of the word 'image' also encompasses any two-dimensional figure, such as a map, graph, pie chart, painting, or banner. In this wider sense, images can also be rendered manually, such as by drawing, the art of painting, or the graphic arts (such as lithography or etching). Additionally, images can be rendered automatically through printing, computer graphics technology, or a combination of both methods. A two-dimensional image does not need to use the entire visual system to be a visual representation. An example of this is a grayscale ("black and white") image, which uses the visual system's sensitivity to brightness across all wavelengths without taking into account different colors. A black-and-white visual representation of something is still an image, even though it does not fully use the visual system's capabilities. On the other hand, some processes can be used to create visual representations of objects that are otherwise inaccessible to the human visual system. These include microscopy for the magnification of minute objects, telescopes that can observe objects at great distances, X-rays that can visually represent the interior structures of the human body (among other objects), magnetic resonance imaging (MRI), positron emission tomography (PET scans), and others. Such processes often rely on detecting electromagnetic radiation that occurs beyond the light spectrum visible to the human eye and converting such signals into recognizable images. === Three-dimensional images === Aside from sculpture and other physical activities that can create three-dimensional images from solid material, some modern techniques, such as holography, can create three-dimensional images that are reproducible but intangible to human touch. Some photographic processes can now render the illusion of depth in an otherwise "flat" image, but "3-D photography" (stereoscopy) or "3-D film" are optical illusions that require special devices such as eyeglasses to create the illusion of depth. === Moving images === "Moving" two-dimensional images are actually illusions of movement perceived when still images are displayed in sequence, each image lasting less, and sometimes much less, than a fraction of a second. The traditional standard for the display of individual frames by a motion picture projector has been 24 frames per second (FPS) since at least the commercial introduction of "talking pictures" in the late 1920s, which necessitated a standard for synchronizing images and sounds. Even in electronic formats such as television and digital image displays, the apparent "motion" is actually the result of many individual lines giving the impression of continuous movement. This phenomenon has often been described as "persistence of vision": a physiological effect of light impressions remaining on the retina of the eye for very brief periods. Even though the term is still sometimes used in popular discussions of movies, it is not a scientifically valid explanation. Other terms emphasize the complex cognitive operations of the brain and the human visual system. "Flicker fusion", the "phi phenomenon", and "beta movement" are among the terms that have replaced "persistence of vision", though no one term seems adequate to describe the process. == Cultural and other uses == Image-making seems to have been common to virtually all human cultures since at least the Paleolithic era. Prehistoric examples of rock art—including cave paintings, petroglyphs, rock reliefs, and geoglyphs—have been found on every inhabited continent. Many of these images seem to have served various purposes: as a form of record-keeping; as an element of spiritual, religious, or magical practice; or even as a form of communication. Early writing systems, including hieroglyphics, ideographic writing, and even the Roman alphabet, owe their origins in some respects to pictorial representations. === Meaning and signification === Images of any type may convey different meanings and sensations for individual viewers, regardless of whether the image's creator intended them. An image may be taken simply as a more or less "accurate" copy of a person, place, thing, or event. It may represent an abstract concept, such as the political power of a ruler or ruling class, a practical or moral lesson, an object for spiritual or religious veneration, or an object—human or otherwise—to be desired. It may also be regarded for its purely aesthetic qualities, rarity, or monetary value. Such reactions can depend on the viewer's context. A religious image in a church may be regarded differently than the same image mounted in a museum. Some might view it simply as an object to be bought or sold. Viewers' reactions will also be guided or shaped by their education, class, race, and other contexts. The study of emotional sensations and their relationship to any given image falls into the categories of aesthetics and the philosophy of art. While such studies inevitably deal with issues of meaning, another approach to signification was suggested by the American philosopher, logician, and semiotician Charles Sanders Peirce. "Images" are one type of the broad category of "signs" proposed by Peirce. Although his ideas are complex and have changed over time, the three categories of signs that he distinguished stand out: The "icon," which relates to an object by resemblance to some quality of the object. A painted or photographed portrait is an icon by virtue of its resemblance to the painting's or photograph's subject. A more abstract representation, such as a map or diagram, can also be an icon. The "index," which relates to an object by some real connection. For example, smoke may be an index of fire, or the temperature recorded on a thermometer may be an index of a patient's illness or health. The "symbol," which lacks direct resemblance or connection to an object but whose association is arbitrarily assigned by the creator or dictated by cultural and historical habit, convention, etc. The color red, for example, may connote rage, beauty, prosperity, political affiliation, or other meanings within a given culture or context; the Swedish film director Ingmar Bergman claimed that his use of the color in his 1972 film Cries and Whispers came from his personal visualization of the human soul. A single image may exist in all three categories at the same time. The Statue of Liberty provides an example. While there have been countless two-dimensional and three-dimensional "reproductions" of the statue (i.e., "icons" themselves), the statue itself exists as an "icon" by virtue of its resemblance to a human woman (or, more specifically, previous representations of the Roman goddess Libertas or the female model used by the artist Frederic-Auguste Bartholdi). an "index" representing New York City or the United States of America in general due to its placement in New York Harbor, or with "immigration" from its proximity to the immigration center at Ellis Island. a "symbol" as a visualization of the abstract concept of "liberty" or "freedom" or even "opportunity" or "diversity". === Critiques of imagery === The nature of images, whether three-dimensional or two-dimensional, created for a specific purpose or only for aesthetic pleasure, has continued to provoke questions and even condemnation at different times and places. In his dialogue, The Republic, the Greek philosopher Plato described our apparent reality as a copy of a higher order of universal forms.

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  Artificial general intelligence

  Artificial general intelligence (AGI) is a hypothetical type of artificial intelligence that matches or surpasses human capabilities across virtually all cognitive tasks. Beyond AGI, artificial superintelligence (ASI) would outperform the best human abilities across every domain by a wide margin. Unlike artificial narrow intelligence (ANI), whose competence is confined to well‑defined tasks, an AGI system can generalise knowledge, transfer skills between domains, and solve novel problems without task‑specific reprogramming. Creating AGI is a stated goal of technology companies such as OpenAI, Google, xAI, and Meta. A 2020 survey identified 72 active AGI research and development projects across 37 countries. AGI is a common topic in science fiction and futures studies. Contention exists over whether AGI represents an existential risk. Some AI experts and industry figures have stated that mitigating the risk of human extinction posed by AGI should be a global priority. Others find the development of AGI to be in too remote a stage to present such a risk. == Terminology == AGI is also known as strong AI, full AI, human-level AI, human-level intelligent AI, or general intelligent action. The term "artificial general intelligence" was used in 1997 by Mark Gubrud in a discussion of the implications of fully automated military production and operations. A mathematical formalism of AGI named AIXI was proposed in 2000 by Marcus Hutter, who defines intelligence as "an agent’s ability to achieve goals or succeed in a wide range of environments". This type of AGI has also been called "universal artificial intelligence". The term AGI was re-introduced and popularized by Shane Legg and Ben Goertzel around 2002. Some academic sources reserve the term "strong AI" for computer programs that will experience sentience or consciousness. In contrast, weak AI (or narrow AI) can solve a specific problem but lacks general cognitive abilities. Some academic sources use "weak AI" to refer more broadly to any programs that neither experience consciousness nor have a mind in the same sense as humans. Related concepts include artificial superintelligence and transformative AI. An artificial superintelligence (ASI) is a hypothetical type of AGI that is much more generally intelligent than humans, while the notion of transformative AI relates to AI having a large impact on society, for example, similar to the agricultural or industrial revolution. A framework for classifying AGI was proposed in 2023 by Google DeepMind researchers. They define five performance levels of AGI: emerging, competent, expert, virtuoso, and superhuman. For example, a competent AGI is defined as an AI that outperforms 50% of skilled adults in a wide range of non-physical tasks, and a superhuman AGI (i.e., an artificial superintelligence) is similarly defined but with a threshold of 100%. They consider large language models like ChatGPT or LLaMA 2 to be instances of emerging AGI (comparable to unskilled humans). Regarding the autonomy of AGI and associated risks, they define five levels: tool (fully in human control), consultant, collaborator, expert, and agent (fully autonomous). == Characteristics == There is no single agreed-upon definition of intelligence as applied to computers. Computer scientist John McCarthy wrote in 2007: "We cannot yet characterize in general what kinds of computational procedures we want to call intelligent." === Intelligence traits === Researchers generally hold that a system is required to do all of the following to be regarded as an AGI: reason, use strategy, solve puzzles, and make judgments under uncertainty, represent knowledge, including common sense knowledge, plan, learn, communicate in natural language, if necessary, integrate these skills in completion of any given goal. Many interdisciplinary approaches (e.g. cognitive science, computational intelligence, and decision making) consider additional traits such as imagination (the ability to form novel mental images and concepts) and autonomy. Computer-based systems exhibiting these capabilities are now widespread, with modern large language models demonstrating computational creativity, automated reasoning, and decision support simultaneously across domains. === Physical traits === Other capabilities are considered desirable in intelligent systems, as they may affect intelligence or aid in its expression. These include: the ability to sense (e.g. see, hear, etc.), and the ability to act (e.g. move and manipulate objects, change location to explore, etc.) This includes the ability to detect and respond to hazard. === Tests for human-level AGI === Several tests meant to confirm human-level AGI have been considered. ==== Turing test ==== The Turing test was proposed by Alan Turing in his 1950 paper "Computing Machinery and Intelligence". This test involves a human judge engaging in natural language conversations with both a human and a machine designed to generate human-like responses. The machine passes the test if it can convince the judge that it is human a significant fraction of the time. Turing proposed this as a practical measure of machine intelligence, focusing on the ability to produce human-like responses rather than on the internal workings of the machine. The idea of the test is that the machine has to try and pretend to be a man, by answering questions put to it, and it will only pass if the pretence is reasonably convincing. A considerable portion of a jury, who should not be experts about machines, must be taken in by the pretence. In 2014, a chatbot named Eugene Goostman, designed to imitate a 13-year-old Ukrainian boy, reportedly passed a Turing Test event by convincing 33% of judges that it was human. However, this claim was met with significant skepticism from the AI research community, who questioned the test's implementation and its relevance to AGI. A 2025 pre‑registered, three‑party Turing‑test study by Cameron R. Jones and Benjamin K. Bergen showed that GPT-4.5 was judged to be the human in 73% of five‑minute text conversations—surpassing the 67% humanness rate of real confederates and meeting the researchers' criterion for having passed the test. ==== Ikea test ==== The "Ikea test", also known as the Flat Pack Furniture Test, involves an AI controlling a robot which attempts to assemble an Ikea flat-pack furniture product after having been shown the parts and instructions. As early as 2013, MIT's IkeaBot demonstrated fully autonomous multi-robot assembly of an IKEA Lack table in ten minutes, with no human intervention and no pre-programmed assembly instructions. The robots inferred the assembly sequence from the geometry of the parts alone. ==== Coffee test ==== Steve Wozniak proposed a test where a machine is required to enter an average American home and figure out how to make coffee. It must find the coffee machine, find the coffee, add water, find a mug, and brew the coffee by pushing the proper buttons. This test has been substantially approached across multiple systems. In January 2024, Figure AI's Figure 01 humanoid learned to operate a Keurig coffee machine autonomously after watching video demonstrations, using end-to-end neural networks to translate visual input into motor actions. In 2025, researchers at the University of Edinburgh published the ELLMER framework in Nature Machine Intelligence, demonstrating a robotic arm that interprets verbal instructions, analyses its surroundings, and autonomously makes coffee in dynamic kitchen environments — adapting to unforeseen obstacles in real time rather than following pre-programmed sequences. ==== Suleyman's test ==== Mustafa Suleyman's test proposes giving an AI model US$100,000 and asking it to obtain US$1 million. ==== Use of video-games ==== Adams, et al. propose that the ability to learn and succeed in a wide range of video games can be used to test AI intelligence. This range would include games unknown to the AGI developers before the test is administered. === AI-complete problems === A problem is informally called "AI-complete" or "AI-hard" if it is believed that AGI would be needed to solve it, because the solution is beyond the capabilities of a purpose-specific algorithm. == History == === Classical AI === Modern AI research began in the mid-1950s. The first generation of AI researchers were convinced that artificial general intelligence was possible and that it would exist in just a few decades. AI pioneer Herbert A. Simon wrote in 1965: "machines will be capable, within twenty years, of doing any work a man can do". Their predictions were the inspiration for Stanley Kubrick and Arthur C. Clarke's fictional character HAL 9000, who embodied what AI researchers believed they could create by the year 2001. AI pioneer Marvin Minsky was a consultant on the project of making HAL 9000 as realistic as possible according to the consensus predictions of the time. He said in 1967, "Within a generation... the problem of

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  Learning rate

  In machine learning and statistics, the learning rate is a tuning parameter in an optimization algorithm that determines the step size at each iteration while moving toward a minimum of a loss function. Since it influences to what extent newly acquired information overrides old information, it metaphorically represents the speed at which a machine learning model "learns". In the adaptive control literature, the learning rate is commonly referred to as gain. In setting a learning rate, there is a trade-off between the rate of convergence and overshooting. While the descent direction is usually determined from the gradient of the loss function, the learning rate determines how big a step is taken in that direction. Too high a learning rate will make the learning jump over minima, but too low a learning rate will either take too long to converge or get stuck in an undesirable local minimum. In order to achieve faster convergence, prevent oscillations and getting stuck in undesirable local minima the learning rate is often varied during training either in accordance to a learning rate schedule or by using an adaptive learning rate. The learning rate and its adjustments may also differ per parameter, in which case it is a diagonal matrix that can be interpreted as an approximation to the inverse of the Hessian matrix in Newton's method. The learning rate is related to the step length determined by inexact line search in quasi-Newton methods and related optimization algorithms. == Learning rate schedule == Initial rate can be left as system default or can be selected using a range of techniques. A learning rate schedule changes the learning rate during learning and is most often changed between epochs/iterations. This is mainly done with two parameters: decay and momentum. There are many different learning rate schedules but the most common are time-based, step-based and exponential. Decay serves to settle the learning in a nice place and avoid oscillations, a situation that may arise when too high a constant learning rate makes the learning jump back and forth over a minimum, and is controlled by a hyperparameter. Momentum is analogous to a ball rolling down a hill; we want the ball to settle at the lowest point of the hill (corresponding to the lowest error). Momentum both speeds up the learning (increasing the learning rate) when the error cost gradient is heading in the same direction for a long time and also avoids local minima by 'rolling over' small bumps. Momentum is controlled by a hyperparameter analogous to a ball's mass which must be chosen manually—too high and the ball will roll over minima which we wish to find, too low and it will not fulfil its purpose. The formula for factoring in the momentum is more complex than for decay but is most often built in with deep learning libraries such as Keras. Time-based learning schedules alter the learning rate depending on the learning rate of the previous time iteration. Factoring in the decay the mathematical formula for the learning rate is: η n + 1 = η 0 1 + d n {\displaystyle \eta _{n+1}={\frac {\eta _{0}}{1+dn}}} where η {\displaystyle \eta } is the learning rate, η 0 {\displaystyle \eta _{0}} is the original learning rate, d {\displaystyle d} is a decay parameter and n {\displaystyle n} is the iteration step. Step-based learning schedules changes the learning rate according to some predefined steps. The decay application formula is here defined as: η n = η 0 d ⌊ 1 + n r ⌋ {\displaystyle \eta _{n}=\eta _{0}d^{\left\lfloor {\frac {1+n}{r}}\right\rfloor }} where η n {\displaystyle \eta _{n}} is the learning rate at iteration n {\displaystyle n} , η 0 {\displaystyle \eta _{0}} is the initial learning rate, d {\displaystyle d} is how much the learning rate should change at each drop (0.5 corresponds to a halving) and r {\displaystyle r} corresponds to the drop rate, or how often the rate should be dropped (10 corresponds to a drop every 10 iterations). The floor function ( ⌊ … ⌋ {\displaystyle \lfloor \dots \rfloor } ) here drops the value of its input to 0 for all values smaller than 1. Exponential learning schedules are similar to step-based, but instead of steps, a decreasing exponential function is used. The mathematical formula for factoring in the decay is: η n = η 0 e − d n {\displaystyle \eta _{n}=\eta _{0}e^{-dn}} where d {\displaystyle d} is a decay parameter. == Adaptive learning rate == The issue with learning rate schedules is that they all depend on hyperparameters that must be manually chosen for each given learning session and may vary greatly depending on the problem at hand or the model used. To combat this, there are many different types of adaptive gradient descent algorithms such as Adagrad, Adadelta, RMSprop, and Adam which are generally built into deep learning libraries such as Keras.

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  Artificial wisdom

  Artificial wisdom (AW) is an artificial intelligence (AI) system which is able to display the human traits of wisdom and morals while being able to contemplate its own “endpoint”. Artificial wisdom can be described as artificial intelligence reaching the top-level of decision-making when confronted with the most complex challenging situations. The term artificial wisdom is used when the "intelligence" is based on more than by chance collecting and interpreting data, but by design enriched with smart and conscience strategies that wise people would use. == Overview == The goal of artificial wisdom is to create artificial intelligence that can successfully replicate the “uniquely human trait[s]” of having wisdom and morals as closely as possible. Thus, artificial wisdom, must “incorporate [the] ethical and moral considerations” of the data it uses. There are also many significant ethical and legal implications of AW which are compounded by the rapid advances in AI and related technologies alongside the lack of the development of ethics, guidelines, and regulations without the oversight of any kind of overarching advisory board. Additionally, there are challenges in how to develop, test, and implement AW in real world scenarios. Existing tests do not test the internal thought process by which a computer system reaches its conclusion, only the result of said process. When examining computer-aided wisdom; the partnership of artificial intelligence and contemplative neuroscience, concerns regarding the future of artificial intelligence shift to a more optimistic viewpoint. This artificial wisdom forms the basis of Louis Molnar's monographic article on artificial philosophy, where he coined the term and proposes how artificial intelligence might view its place in the grand scheme of things. == Definitions == There are no universal or standardized definitions for human intelligence, artificial intelligence, human wisdom, or artificial wisdom. However, the DIKW pyramid, describes the continuum of relationship between data, information, knowledge, and wisdom, puts wisdom at the highest level in its hierarchy. Gottfredson defines intelligence as “the ability to reason, plan, solve problems, think abstractly, comprehend complex ideas, learn quickly, and learn from experience”. Definitions for wisdom typically include requiring: The ability for emotional regulation, Pro-social behaviors (e.g., empathy, compassion, and altruism), Self-reflection, “A balance between decisiveness and acceptance of uncertainty and diversity of perspectives, and social advising.” As previously defined, Artificial Wisdom would then be an AI system which is able to solve problems via “an understanding of…context, ethics and moral principles,” rather than simple pre-defined inputs or “learned patterns.” Some scientists have also considered the field of artificial consciousness. However, Jeste states that “…it is generally agreed that only humans can have consciousness, autonomy, will, and theory of mind.” An artificially wise system must also be able to contemplate its end goal and recognize its own ignorance. Additionally, to contemplate its end goal, a wise system must have a “correct conception of worthwhile goals (broadly speaking) or well-being (narrowly speaking)”. "Stephen Grimm further suggests that the following three types of knowledge are individually necessary for wisdom: first, "knowledge of what is good or important for well-being", second, "knowledge of one’s standing, relative to what is good or important for well-being", and third, "knowledge of a strategy for obtaining what is good or important for wellbeing."" == Problems == There are notable problems with attempting to create an artificially wise system. Consciousness, autonomy, and will are considered strictly human features. === Values === There are significant ethical and philosophical issues when attempting to create an intelligent or a wise system. Notably, whose moral values will be used to train the system to be wise. Differing moral values and prejudice can already be seen from various organizations and governments in artificial intelligence. Deployment strategies and values of Artificial Wisdom will conflict between leaders, companies, and countries. Nusbaum states, “When values are in conflict, leaders often make choices that are clever or smart about their own needs, but are often not wise.” === Ethics === Science fiction author Isaac Asimov realized the need to control the technology in the 1940s when he wrote the three laws of robotics as follows: A robot may not injure a human directly or indirectly. A robot must obey human’s orders. A robot should seek to protect its own existence. Additionally, the pace at which technology is rapidly advancing artificial intelligence and thus the need for artificial wisdom may “have outpaced the development of societal guidelines have raised serious questions about the ethics and morality of AI, and called for international oversight and regulations to ensure safety.” === Principal impossibility === One argument, coined by Tsai as the “argument against AW,” or AAAW, postulates the principal impossibility of Artificial Wisdom. The argument is based on the philosophical differences between practical wisdom, also called phronesis, and practical intelligence. Said difference isn’t in “selecting the correct means, but reasoning correctly about what ends to follow”. Tsai puts the argument into a logical proposition as follows: “(P1) An agent is genuinely wise only if the agent can deliberate about the final goal of the domain in which the agent is situated.” “(P2) An intelligent agent cannot deliberate about the final goal of the domain in which the agent is situated.” “(C1) An intelligent agent cannot be genuinely wise.” “(P3) An AW is, at its core, intelligent.” “(C2) An AW cannot be genuinely wise.”

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  Learnable function class

  In statistical learning theory, a learnable function class is a set of functions for which an algorithm can be devised to asymptotically minimize the expected risk, uniformly over all probability distributions. The concept of learnable classes are closely related to regularization in machine learning, and provides large sample justifications for certain learning algorithms. == Definition == === Background === Let Ω = X × Y = { ( x , y ) } {\displaystyle \Omega ={\mathcal {X}}\times {\mathcal {Y}}=\{(x,y)\}} be the sample space, where y {\displaystyle y} are the labels and x {\displaystyle x} are the covariates (predictors). F = { f : X ↦ Y } {\displaystyle {\mathcal {F}}=\{f:{\mathcal {X}}\mapsto {\mathcal {Y}}\}} is a collection of mappings (functions) under consideration to link x {\displaystyle x} to y {\displaystyle y} . L : Y × Y ↦ R {\displaystyle L:{\mathcal {Y}}\times {\mathcal {Y}}\mapsto \mathbb {R} } is a pre-given loss function (usually non-negative). Given a probability distribution P ( x , y ) {\displaystyle P(x,y)} on Ω {\displaystyle \Omega } , define the expected risk I P ( f ) {\displaystyle I_{P}(f)} to be: I P ( f ) = ∫ L ( f ( x ) , y ) d P ( x , y ) {\displaystyle I_{P}(f)=\int L(f(x),y)dP(x,y)} The general goal in statistical learning is to find the function in F {\displaystyle {\mathcal {F}}} that minimizes the expected risk. That is, to find solutions to the following problem: f ^ = arg ⁡ min f ∈ F I P ( f ) {\displaystyle {\hat {f}}=\arg \min _{f\in {\mathcal {F}}}I_{P}(f)} But in practice the distribution P {\displaystyle P} is unknown, and any learning task can only be based on finite samples. Thus we seek instead to find an algorithm that asymptotically minimizes the empirical risk, i.e., to find a sequence of functions { f ^ n } n = 1 ∞ {\displaystyle \{{\hat {f}}_{n}\}_{n=1}^{\infty }} that satisfies lim n → ∞ P ( I P ( f ^ n ) − inf f ∈ F I P ( f ) > ϵ ) = 0 {\displaystyle \lim _{n\rightarrow \infty }\mathbb {P} (I_{P}({\hat {f}}_{n})-\inf _{f\in {\mathcal {F}}}I_{P}(f)>\epsilon )=0} One usual algorithm to find such a sequence is through empirical risk minimization. === Learnable function class === We can make the condition given in the above equation stronger by requiring that the convergence is uniform for all probability distributions. That is: The intuition behind the more strict requirement is as such: the rate at which sequence { f ^ n } {\displaystyle \{{\hat {f}}_{n}\}} converges to the minimizer of the expected risk can be very different for different P ( x , y ) {\displaystyle P(x,y)} . Because in real world the true distribution P {\displaystyle P} is always unknown, we would want to select a sequence that performs well under all cases. However, by the no free lunch theorem, such a sequence that satisfies (1) does not exist if F {\displaystyle {\mathcal {F}}} is too complex. This means we need to be careful and not allow too "many" functions in F {\displaystyle {\mathcal {F}}} if we want (1) to be a meaningful requirement. Specifically, function classes that ensure the existence of a sequence { f ^ n } {\displaystyle \{{\hat {f}}_{n}\}} that satisfies (1) are known as learnable classes. It is worth noting that at least for supervised classification and regression problems, if a function class is learnable, then the empirical risk minimization automatically satisfies (1). Thus in these settings not only do we know that the problem posed by (1) is solvable, we also immediately have an algorithm that gives the solution. == Interpretations == If the true relationship between y {\displaystyle y} and x {\displaystyle x} is y ∼ f ∗ ( x ) {\displaystyle y\sim f^{}(x)} , then by selecting the appropriate loss function, f ∗ {\displaystyle f^{}} can always be expressed as the minimizer of the expected loss across all possible functions. That is, f ∗ = arg ⁡ min f ∈ F ∗ I P ( f ) {\displaystyle f^{}=\arg \min _{f\in {\mathcal {F}}^{}}I_{P}(f)} Here we let F ∗ {\displaystyle {\mathcal {F}}^{}} be the collection of all possible functions mapping X {\displaystyle {\mathcal {X}}} onto Y {\displaystyle {\mathcal {Y}}} . f ∗ {\displaystyle f^{}} can be interpreted as the actual data generating mechanism. However, the no free lunch theorem tells us that in practice, with finite samples we cannot hope to search for the expected risk minimizer over F ∗ {\displaystyle {\mathcal {F}}^{}} . Thus we often consider a subset of F ∗ {\displaystyle {\mathcal {F}}^{}} , F {\displaystyle {\mathcal {F}}} , to carry out searches on. By doing so, we risk that f ∗ {\displaystyle f^{}} might not be an element of F {\displaystyle {\mathcal {F}}} . This tradeoff can be mathematically expressed as In the above decomposition, part ( b ) {\displaystyle (b)} does not depend on the data and is non-stochastic. It describes how far away our assumptions ( F {\displaystyle {\mathcal {F}}} ) are from the truth ( F ∗ {\displaystyle {\mathcal {F}}^{}} ). ( b ) {\displaystyle (b)} will be strictly greater than 0 if we make assumptions that are too strong ( F {\displaystyle {\mathcal {F}}} too small). On the other hand, failing to put enough restrictions on F {\displaystyle {\mathcal {F}}} will cause it to be not learnable, and part ( a ) {\displaystyle (a)} will not stochastically converge to 0. This is the well-known overfitting problem in statistics and machine learning literature. == Example: Tikhonov regularization == A good example where learnable classes are used is the so-called Tikhonov regularization in reproducing kernel Hilbert space (RKHS). Specifically, let F ∗ {\displaystyle {\mathcal {F^{}}}} be an RKHS, and | | ⋅ | | 2 {\displaystyle ||\cdot ||_{2}} be the norm on F ∗ {\displaystyle {\mathcal {F^{}}}} given by its inner product. It is shown in that F = { f : | | f | | 2 ≤ γ } {\displaystyle {\mathcal {F}}=\{f:||f||_{2}\leq \gamma \}} is a learnable class for any finite, positive γ {\displaystyle \gamma } . The empirical minimization algorithm to the dual form of this problem is arg ⁡ min f ∈ F ∗ { ∑ i = 1 n L ( f ( x i ) , y i ) + λ | | f | | 2 } {\displaystyle \arg \min _{f\in {\mathcal {F}}^{}}\left\{\sum _{i=1}^{n}L(f(x_{i}),y_{i})+\lambda ||f||_{2}\right\}} This was first introduced by Tikhonov to solve ill-posed problems. Many statistical learning algorithms can be expressed in such a form (for example, the well-known ridge regression). The tradeoff between ( a ) {\displaystyle (a)} and ( b ) {\displaystyle (b)} in (2) is geometrically more intuitive with Tikhonov regularization in RKHS. We can consider a sequence of { F γ } {\displaystyle \{{\mathcal {F}}_{\gamma }\}} , which are essentially balls in F ∗ {\displaystyle {\mathcal {F^{}}}} with centers at 0. As γ {\displaystyle \gamma } gets larger, F γ {\displaystyle {\mathcal {F}}_{\gamma }} gets closer to the entire space, and ( b ) {\displaystyle (b)} is likely to become smaller. However we will also suffer smaller convergence rates in ( a ) {\displaystyle (a)} . The way to choose an optimal γ {\displaystyle \gamma } in finite sample settings is usually through cross-validation. == Relationship to empirical process theory == Part ( a ) {\displaystyle (a)} in (2) is closely linked to empirical process theory in statistics, where the empirical risk { ∑ i = 1 n L ( y i , f ( x i ) ) , f ∈ F } {\displaystyle \{\sum _{i=1}^{n}L(y_{i},f(x_{i})),f\in {\mathcal {F}}\}} are known as empirical processes. In this field, the function class F {\displaystyle {\mathcal {F}}} that satisfies the stochastic convergence are known as uniform Glivenko–Cantelli classes. It has been shown that under certain regularity conditions, learnable classes and uniformly Glivenko-Cantelli classes are equivalent. Interplay between ( a ) {\displaystyle (a)} and ( b ) {\displaystyle (b)} in statistics literature is often known as the bias-variance tradeoff. However, note that in the authors gave an example of stochastic convex optimization for General Setting of Learning where learnability is not equivalent with uniform convergence.

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  Symbolic regression

  Symbolic regression (SR) is a type of regression analysis that searches the space of mathematical expressions to find the model that best fits a given dataset, both in terms of accuracy and simplicity. No particular model is provided as a starting point for symbolic regression. Instead, initial expressions are formed by randomly combining mathematical building blocks such as mathematical operators, analytic functions, constants, and state variables. Usually, a subset of these primitives will be specified by the person operating it, but that's not a requirement of the technique. The symbolic regression problem for mathematical functions has been tackled with a variety of methods, including recombining equations most commonly using genetic programming, as well as more recent methods utilizing Bayesian methods and neural networks. Another non-classical alternative method to SR is called Universal Functions Originator (UFO), which has a different mechanism, search-space, and building strategy. Further methods such as Exact Learning attempt to transform the fitting problem into a moments problem in a natural function space, usually built around generalizations of the Meijer-G function. By not requiring a priori specification of a model, symbolic regression isn't affected by human bias, or unknown gaps in domain knowledge. It attempts to uncover the intrinsic relationships of the dataset, by letting the patterns in the data itself reveal the appropriate models, rather than imposing a model structure that is deemed mathematically tractable from a human perspective. The fitness function that drives the evolution of the models takes into account not only error metrics (to ensure the models accurately predict the data), but also special complexity measures, thus ensuring that the resulting models reveal the data's underlying structure in a way that's understandable from a human perspective. This facilitates reasoning and favors the odds of getting insights about the data-generating system, as well as improving generalisability and extrapolation behaviour by preventing overfitting. Accuracy and simplicity may be left as two separate objectives of the regression—in which case the optimum solutions form a Pareto front—or they may be combined into a single objective by means of a model selection principle such as minimum description length. It has been proven that symbolic regression is an NP-hard problem. Nevertheless, if the sought-for equation is not too complex it is possible to solve the symbolic regression problem exactly by generating every possible function (built from some predefined set of operators) and evaluating them on the dataset in question. == Difference from classical regression == While conventional regression techniques seek to optimize the parameters for a pre-specified model structure, symbolic regression avoids imposing prior assumptions, and instead infers the model from the data. In other words, it attempts to discover both model structures and model parameters. This approach has the disadvantage of having a much larger space to search, because not only the search space in symbolic regression is infinite, but there are an infinite number of models which will perfectly fit a finite data set (provided that the model complexity isn't artificially limited). This means that it will possibly take a symbolic regression algorithm longer to find an appropriate model and parametrization, than traditional regression techniques. This can be attenuated by limiting the set of building blocks provided to the algorithm, based on existing knowledge of the system that produced the data; but in the end, using symbolic regression is a decision that has to be balanced with how much is known about the underlying system. Nevertheless, this characteristic of symbolic regression also has advantages: because the evolutionary algorithm requires diversity in order to effectively explore the search space, the result is likely to be a selection of high-scoring models (and their corresponding set of parameters). Examining this collection could provide better insight into the underlying process, and allows the user to identify an approximation that better fits their needs in terms of accuracy and simplicity. == Benchmarking == === SRBench === In 2021, SRBench was proposed as a large benchmark for symbolic regression. In its inception, SRBench featured 14 symbolic regression methods, 7 other ML methods, and 252 datasets from PMLB. The benchmark intends to be a living project: it encourages the submission of improvements, new datasets, and new methods, to keep track of the state of the art in SR. === SRBench Competition 2022 === In 2022, SRBench announced the competition Interpretable Symbolic Regression for Data Science, which was held at the GECCO conference in Boston, MA. The competition pitted nine leading symbolic regression algorithms against each other on a novel set of data problems and considered different evaluation criteria. The competition was organized in two tracks, a synthetic track and a real-world data track. ==== Synthetic Track ==== In the synthetic track, methods were compared according to five properties: re-discovery of exact expressions; feature selection; resistance to local optima; extrapolation; and sensitivity to noise. Rankings of the methods were: QLattice PySR (Python Symbolic Regression) uDSR (Deep Symbolic Optimization) ==== Real-world Track ==== In the real-world track, methods were trained to build interpretable predictive models for 14-day forecast counts of COVID-19 cases, hospitalizations, and deaths in New York State. These models were reviewed by a subject expert and assigned trust ratings and evaluated for accuracy and simplicity. The ranking of the methods was: uDSR (Deep Symbolic Optimization) QLattice geneticengine (Genetic Engine) == Non-standard methods == Most symbolic regression algorithms prevent combinatorial explosion by implementing evolutionary algorithms that iteratively improve the best-fit expression over many generations. Recently, researchers have proposed algorithms utilizing other tactics in AI. Silviu-Marian Udrescu and Max Tegmark developed the "AI Feynman" algorithm, which attempts symbolic regression by training a neural network to represent the mystery function, then runs tests against the neural network to attempt to break up the problem into smaller parts. For example, if f ( x 1 , . . . , x i , x i + 1 , . . . , x n ) = g ( x 1 , . . . , x i ) + h ( x i + 1 , . . . , x n ) {\displaystyle f(x_{1},...,x_{i},x_{i+1},...,x_{n})=g(x_{1},...,x_{i})+h(x_{i+1},...,x_{n})} , tests against the neural network can recognize the separation and proceed to solve for g {\displaystyle g} and h {\displaystyle h} separately and with different variables as inputs. This is an example of divide and conquer, which reduces the size of the problem to be more manageable. AI Feynman also transforms the inputs and outputs of the mystery function in order to produce a new function which can be solved with other techniques, and performs dimensional analysis to reduce the number of independent variables involved. The algorithm was able to "discover" 100 equations from The Feynman Lectures on Physics, while a leading software using evolutionary algorithms, Eureqa, solved only 71. AI Feynman, in contrast to classic symbolic regression methods, requires a very large dataset in order to first train the neural network and is naturally biased towards equations that are common in elementary physics.

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  Wetware computer

  A wetware computer is an organic computer (which can also be known as an artificial organic brain or a neurocomputer) composed of organic material "wetware" such as "living" neurons. Wetware computers composed of neurons are different than conventional computers because they use biological materials, and offer the possibility of substantially more energy-efficient computing. While a wetware computer is still largely conceptual, there has been limited success with construction and prototyping, which has acted as a proof of the concept's realistic application to computing in the future. The most notable prototypes have stemmed from the research completed by biological engineer William Ditto during his time at the Georgia Institute of Technology. His work constructing a simple neurocomputer capable of basic addition from leech neurons in 1999 was a significant discovery for the concept. This research was a primary example driving interest in creating these artificially constructed, but still organic brains. == Origins and theoretical foundations == The term wetware came from cyberpunk fiction, notably through Gibson's Neuromancer, but was quickly taken up in scientific literature to explain computation by biological material. Theories of early biological computation borrowed from Alan Turing's morphogenesis model, which showed that chemical interactions could produce complex patterns without centralized control. Hopfield's associative memory networks also provided a foundation for biological information systems with fault tolerance and self-organization. == Major characteristics and processes == Biological wetware systems demonstrate dynamic reconfigurability underpinned by neuroplasticity and enable continuous learning and adaptation. Reaction-diffusion-based computing and molecular logic gates allow spatially parallel information processing unachievable in conventional systems. These systems also show fault tolerance and self-repair at the cellular and network level. The development of cerebral organoids—miniature lab-grown brains—demonstrates spontaneous learning behavior and suggests biological tissue as a viable computational substrate. == Overview == The concept of wetware is an application of specific interest to the field of computer manufacturing. Moore's law, which states that the number of transistors which can be placed on a silicon chip is doubled roughly every two years, has acted as a goal for the industry for decades, but as the size of computers continues to decrease, the ability to meet this goal has become more difficult, threatening to reach a plateau. Due to the difficulty in reducing the size of computers because of size limitations of transistors and integrated circuits, wetware provides an unconventional alternative. A wetware computer composed of neurons is an ideal concept because, unlike conventional materials which operate in binary (on/off), a neuron can shift between thousands of states, constantly altering its chemical conformation, and redirecting electrical pulses through over 200,000 channels in any of its many synaptic connections. Because of this large difference in the possible settings for any one neuron, compared to the binary limitations of conventional computers, the space limitations are far fewer. == Background == The concept of wetware is distinct and unconventional and draws slight resonance with both hardware and software from conventional computers. While hardware is understood as the physical architecture of traditional computational devices, comprising integrated circuits and supporting infrastructure, software represents the encoded architecture of storage and instructions. Wetware is a separate concept that uses the formation of organic molecules, mostly complex cellular structures (such as neurons), to create a computational device such as a computer. In wetware, the ideas of hardware and software are intertwined and interdependent. The molecular and chemical composition of the organic or biological structure would represent not only the physical structure of the wetware but also the software, being continually reprogrammed by the discrete shifts in electrical pulses and chemical concentration gradients as the molecules change their structures to communicate signals. The responsiveness of a cell, proteins, and molecules to changing conformations, both within their structures and around them, ties the idea of internal programming and external structure together in a way that is alien to the current model of conventional computer architecture. The structure of wetware represents a model where the external structure and internal programming are interdependent and unified; meaning that changes to the programming or internal communication between molecules of the device would represent a physical change in the structure. The dynamic nature of wetware borrows from the function of complex cellular structures in biological organisms. The combination of "hardware" and "software" into one dynamic, and interdependent system which uses organic molecules and complexes to create an unconventional model for computational devices is a specific example of applied biorobotics. === The cell as a model of wetware === Cells in many ways can be seen as their form of naturally occurring wetware, similar to the concept that the human brain is the preexisting model system for complex wetware. In his book Wetware: A Computer in Every Living Cell (2009) Dennis Bray explains his theory that cells, which are the most basic form of life, are just a highly complex computational structure, like a computer. To simplify one of his arguments a cell can be seen as a type of computer, using its structured architecture. In this architecture, much like a traditional computer, many smaller components operate in tandem to receive input, process the information, and compute an output. In an overly simplified, non-technical analysis, cellular function can be broken into the following components: Information and instructions for execution are stored as DNA in the cell, RNA acts as a source for distinctly encoded input, processed by ribosomes and other transcription factors to access and process the DNA and to output a protein. Bray's argument in favor of viewing cells and cellular structures as models of natural computational devices is important when considering the more applied theories of wetware to biorobotics. === Biorobotics === Wetware and biorobotics are closely related concepts, which both borrow from similar overall principles. A biorobotic structure can be defined as a system modeled from a preexisting organic complex or model such as cells (neurons) or more complex structures like organs (brain) or whole organisms. Unlike wetware, the concept of biorobotics is not always a system composed of organic molecules, but instead could be composed of conventional material which is designed and assembled in a structure similar or derived from a biological model. Biorobotics have many applications and are used to address the challenges of conventional computer architecture. Conceptually, designing a program, robot, or computational device after a preexisting biological model such as a cell, or even a whole organism, provides the engineer or programmer the benefits of incorporating into the structure the evolutionary advantages of the model. == Effects on users == Wetware technologies such as BCIs and neuromorphic chips offer new possibilities for user autonomy. For those with disabilities, such systems could restore motor or sensory functions and enhance quality of life. However, these technologies raise ethical questions: cognitive privacy, consent over biological data, and risk of exploitation. Without proper oversight, wetware technologies may also widen inequality, favoring those with access to cognitive enhancements. Open governance frameworks and ethical AI design grounded in neuro ethics will be essential. With the development of wetware devices, disparities in access could exacerbate social inequalities, benefiting those who have resources to enhance cognitive or physical abilities. It is necessary to create strong ethical frameworks, inclusive development practices, and open systems of governance to reduce risks and make sure that wetware advances are beneficial to all segments of society. == Applications and goals == === Basic neurocomputer composed of leech neurons === In 1999 William Ditto and his team of researchers at Georgia Institute of Technology and Emory University created a basic form of a wetware computer capable of simple addition by harnessing leech neurons. Leeches were used as a model organism due to the large size of their neuron, and the ease associated with their collection and manipulation. However, these results have never been published in a peer-reviewed journal, prompting questions about the validity of the claims. The computer was able to complete basic addition through electrical probes

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  Matrix regularization

  In the field of statistical learning theory, matrix regularization generalizes notions of vector regularization to cases where the object to be learned is a matrix. The purpose of regularization is to enforce conditions, for example sparsity or smoothness, that can produce stable predictive functions. For example, in the more common vector framework, Tikhonov regularization optimizes over min x ‖ A x − y ‖ 2 + λ ‖ x ‖ 2 {\displaystyle \min _{x}\left\|Ax-y\right\|^{2}+\lambda \left\|x\right\|^{2}} to find a vector x {\displaystyle x} that is a stable solution to the regression problem. When the system is described by a matrix rather than a vector, this problem can be written as min X ‖ A X − Y ‖ 2 + λ ‖ X ‖ 2 , {\displaystyle \min _{X}\left\|AX-Y\right\|^{2}+\lambda \left\|X\right\|^{2},} where the vector norm enforcing a regularization penalty on x {\displaystyle x} has been extended to a matrix norm on X {\displaystyle X} . Matrix regularization has applications in matrix completion, multivariate regression, and multi-task learning. Ideas of feature and group selection can also be extended to matrices, and these can be generalized to the nonparametric case of multiple kernel learning. == Basic definition == Consider a matrix W {\displaystyle W} to be learned from a set of examples, S = ( X i t , y i t ) {\displaystyle S=(X_{i}^{t},y_{i}^{t})} , where i {\displaystyle i} goes from 1 {\displaystyle 1} to n {\displaystyle n} , and t {\displaystyle t} goes from 1 {\displaystyle 1} to T {\displaystyle T} . Let each input matrix X i {\displaystyle X_{i}} be ∈ R D T {\displaystyle \in \mathbb {R} ^{DT}} , and let W {\displaystyle W} be of size D × T {\displaystyle D\times T} . A general model for the output y {\displaystyle y} can be posed as y i t = ⟨ W , X i t ⟩ F , {\displaystyle y_{i}^{t}=\left\langle W,X_{i}^{t}\right\rangle _{F},} where the inner product is the Frobenius inner product. For different applications the matrices X i {\displaystyle X_{i}} will have different forms, but for each of these the optimization problem to infer W {\displaystyle W} can be written as min W ∈ H E ( W ) + R ( W ) , {\displaystyle \min _{W\in {\mathcal {H}}}E(W)+R(W),} where E {\displaystyle E} defines the empirical error for a given W {\displaystyle W} , and R ( W ) {\displaystyle R(W)} is a matrix regularization penalty. The function R ( W ) {\displaystyle R(W)} is typically chosen to be convex and is often selected to enforce sparsity (using ℓ 1 {\displaystyle \ell ^{1}} -norms) and/or smoothness (using ℓ 2 {\displaystyle \ell ^{2}} -norms). Finally, W {\displaystyle W} is in the space of matrices H {\displaystyle {\mathcal {H}}} with Frobenius inner product ⟨ … ⟩ F {\displaystyle \langle \dots \rangle _{F}} . == General applications == === Matrix completion === In the problem of matrix completion, the matrix X i t {\displaystyle X_{i}^{t}} takes the form X i t = e t ⊗ e i ′ , {\displaystyle X_{i}^{t}=e_{t}\otimes e_{i}',} where ( e t ) t {\displaystyle (e_{t})_{t}} and ( e i ′ ) i {\displaystyle (e_{i}')_{i}} are the canonical basis in R T {\displaystyle \mathbb {R} ^{T}} and R D {\displaystyle \mathbb {R} ^{D}} . In this case the role of the Frobenius inner product is to select individual elements w i t {\displaystyle w_{i}^{t}} from the matrix W {\displaystyle W} . Thus, the output y {\displaystyle y} is a sampling of entries from the matrix W {\displaystyle W} . The problem of reconstructing W {\displaystyle W} from a small set of sampled entries is possible only under certain restrictions on the matrix, and these restrictions can be enforced by a regularization function. For example, it might be assumed that W {\displaystyle W} is low-rank, in which case the regularization penalty can take the form of a nuclear norm. R ( W ) = λ ‖ W ‖ ∗ = λ ∑ i | σ i | , {\displaystyle R(W)=\lambda \left\|W\right\|_{}=\lambda \sum _{i}\left|\sigma _{i}\right|,} where σ i {\displaystyle \sigma _{i}} , with i {\displaystyle i} from 1 {\displaystyle 1} to min D , T {\displaystyle \min D,T} , are the singular values of W {\displaystyle W} . === Multivariate regression === Models used in multivariate regression are parameterized by a matrix of coefficients. In the Frobenius inner product above, each matrix X {\displaystyle X} is X i t = e t ⊗ x i {\displaystyle X_{i}^{t}=e_{t}\otimes x_{i}} such that the output of the inner product is the dot product of one row of the input with one column of the coefficient matrix. The familiar form of such models is Y = X W + b {\displaystyle Y=XW+b} Many of the vector norms used in single variable regression can be extended to the multivariate case. One example is the squared Frobenius norm, which can be viewed as an ℓ 2 {\displaystyle \ell ^{2}} -norm acting either entrywise, or on the singular values of the matrix: R ( W ) = λ ‖ W ‖ F 2 = λ ∑ i ∑ j | w i j | 2 = λ Tr ⁡ ( W ∗ W ) = λ ∑ i σ i 2 . {\displaystyle R(W)=\lambda \left\|W\right\|_{F}^{2}=\lambda \sum _{i}\sum _{j}\left|w_{ij}\right|^{2}=\lambda \operatorname {Tr} \left(W^{}W\right)=\lambda \sum _{i}\sigma _{i}^{2}.} In the multivariate case the effect of regularizing with the Frobenius norm is the same as the vector case; very complex models will have larger norms, and, thus, will be penalized more. === Multi-task learning === The setup for multi-task learning is almost the same as the setup for multivariate regression. The primary difference is that the input variables are also indexed by task (columns of Y {\displaystyle Y} ). The representation with the Frobenius inner product is then X i t = e t ⊗ x i t . {\displaystyle X_{i}^{t}=e_{t}\otimes x_{i}^{t}.} The role of matrix regularization in this setting can be the same as in multivariate regression, but matrix norms can also be used to couple learning problems across tasks. In particular, note that for the optimization problem min W ‖ X W − Y ‖ 2 2 + λ ‖ W ‖ 2 2 {\displaystyle \min _{W}\left\|XW-Y\right\|_{2}^{2}+\lambda \left\|W\right\|_{2}^{2}} the solutions corresponding to each column of Y {\displaystyle Y} are decoupled. That is, the same solution can be found by solving the joint problem, or by solving an isolated regression problem for each column. The problems can be coupled by adding an additional regularization penalty on the covariance of solutions min W , Ω ‖ X W − Y ‖ 2 2 + λ 1 ‖ W ‖ 2 2 + λ 2 Tr ⁡ ( W T Ω − 1 W ) {\displaystyle \min _{W,\Omega }\left\|XW-Y\right\|_{2}^{2}+\lambda _{1}\left\|W\right\|_{2}^{2}+\lambda _{2}\operatorname {Tr} \left(W^{T}\Omega ^{-1}W\right)} where Ω {\displaystyle \Omega } models the relationship between tasks. This scheme can be used to both enforce similarity of solutions across tasks, and to learn the specific structure of task similarity by alternating between optimizations of W {\displaystyle W} and Ω {\displaystyle \Omega } . When the relationship between tasks is known to lie on a graph, the Laplacian matrix of the graph can be used to couple the learning problems. == Spectral regularization == Regularization by spectral filtering has been used to find stable solutions to problems such as those discussed above by addressing ill-posed matrix inversions (see for example Filter function for Tikhonov regularization). In many cases the regularization function acts on the input (or kernel) to ensure a bounded inverse by eliminating small singular values, but it can also be useful to have spectral norms that act on the matrix that is to be learned. There are a number of matrix norms that act on the singular values of the matrix. Frequently used examples include the Schatten p-norms, with p = 1 or 2. For example, matrix regularization with a Schatten 1-norm, also called the nuclear norm, can be used to enforce sparsity in the spectrum of a matrix. This has been used in the context of matrix completion when the matrix in question is believed to have a restricted rank. In this case the optimization problem becomes: min ‖ W ‖ ∗ subject to W i , j = Y i j . {\displaystyle \min \left\|W\right\|_{}~~{\text{ subject to }}~~W_{i,j}=Y_{ij}.} Spectral Regularization is also used to enforce a reduced rank coefficient matrix in multivariate regression. In this setting, a reduced rank coefficient matrix can be found by keeping just the top n {\displaystyle n} singular values, but this can be extended to keep any reduced set of singular values and vectors. == Structured sparsity == Sparse optimization has become the focus of much research interest as a way to find solutions that depend on a small number of variables (see e.g. the Lasso method). In principle, entry-wise sparsity can be enforced by penalizing the entry-wise ℓ 0 {\displaystyle \ell ^{0}} -norm of the matrix, but the ℓ 0 {\displaystyle \ell ^{0}} -norm is not convex. In practice this can be implemented by convex relaxation to the ℓ 1 {\displaystyle \ell ^{1}} -norm. While entry-wise regularization with an ℓ 1 {\displaystyle \ell ^{1}} -norm will find solutions with a small number of nonzero elements, applying an ℓ 1 {

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  Secure coding

  Secure coding is the practice of developing computer software in such a way that guards against the accidental introduction of security vulnerabilities. Defects, bugs and logic flaws are consistently the primary cause of commonly exploited software vulnerabilities. Through the analysis of thousands of reported vulnerabilities, security professionals have discovered that most vulnerabilities stem from a relatively small number of common software programming errors. By identifying the insecure coding practices that lead to these errors and educating developers on secure alternatives, organizations can take proactive steps to help significantly reduce or eliminate vulnerabilities in software before deployment. Some scholars have suggested that in order to effectively confront threats related to cybersecurity, proper security should be coded or "baked in" to the systems. With security being designed into the software, this ensures that there will be protection against insider attacks and reduces the threat to application security. Implementing secure coding practices is part of the secure by design approach to security engineering. == Buffer-overflow prevention == Buffer overflows, a common software security vulnerability, happen when a process tries to store data beyond a fixed-length buffer. For example, if there are 8 slots to store items in, there will be a problem if there is an attempt to store 9 items. In computer memory the overflowed data may overwrite data in the next location which can result in a security vulnerability (stack smashing) or program termination (segmentation fault). An example of a C program prone to a buffer overflow is If the user input is larger than the destination buffer, a buffer overflow will occur. To fix this unsafe program, use strncpy to prevent a possible buffer overflow. Another secure alternative is to dynamically allocate memory on the heap using malloc. In the above code snippet, the program attempts to copy the contents of src into dst, while also checking the return value of malloc() to ensure that enough memory was able to be allocated for the destination buffer. == Format-string attack prevention == A Format String Attack is when a malicious user supplies specific inputs that will eventually be entered as an argument to a function that performs formatting, such as printf(). The attack involves the adversary reading from or writing to the stack. The C printf function writes output to stdout. If the parameter of the printf function is not properly formatted, several security bugs can be introduced. Below is a program that is vulnerable to a format string attack. A malicious argument passed to the program could be "%s%s%s%s%s%s%s", which can crash the program from improper memory reads. == Integer-overflow prevention == Integer overflow occurs when an arithmetic operation results in an integer too large to be represented within the available space. A program which does not properly check for integer overflow introduces potential software bugs and exploits. Below is a function in C++ which attempts to confirm that the sum of x and y is less than or equal to a defined value MAX: The problem with the code is it does not check for integer overflow on the addition operation. If the sum of x and y is greater than the maximum possible value of an unsigned int, the addition operation will overflow and perhaps result in a value less than or equal to MAX, even though the sum of x and y is greater than MAX. Below is a function which checks for overflow by confirming the sum is greater than or equal to both x and y. If the sum did overflow, the sum would be less than x or less than y. == Path traversal prevention == Path traversal is a vulnerability whereby paths provided from an untrusted source are interpreted in such a way that unauthorised file access is possible. For example, consider a script that fetches an article by taking a filename, which is then read by the script and parsed. Such a script might use the following hypothetical URL to retrieve an article about dog food: https://www.example.net/cgi-bin/article.sh?name=dogfood.html If the script has no input checking, instead trusting that the filename is always valid, a malicious user could forge a URL to retrieve configuration files from the web server: https://www.example.net/cgi-bin/article.sh?name=../../../../../etc/passwd Depending on the script, this may expose the /etc/passwd file, which on Unix-like systems contains (among others) user IDs, their login names, home directory paths and shells. (See SQL injection for a similar attack.) == Regulatory drivers == Secure coding practices are increasingly mandated by regulatory frameworks governing the development and maintenance of software systems that process sensitive data. The Health Insurance Portability and Accountability Act (HIPAA) Security Rule requires covered entities to protect the integrity of protected health information through technical safeguards under 45 CFR 164.312(c)(1) and to implement mechanisms to authenticate electronic protected health information under 45 CFR 164.312(c)(2). The Payment Card Industry Data Security Standard (PCI DSS) version 4.0 Requirement 6.2 mandates that custom software is developed securely, including training developers in secure coding techniques (6.2.2), reviewing custom code for vulnerabilities before release (6.2.3), and addressing common software attacks in development practices (6.2.4).

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  Kernel density estimation

  In statistics, kernel density estimation (KDE) is the application of kernel smoothing for probability density estimation, i.e., a non-parametric method to estimate the probability density function of a random variable based on kernels as weights. KDE answers a fundamental data smoothing problem where inferences about the population are made based on a finite data sample. In some fields such as signal processing and econometrics it is also termed the Parzen–Rosenblatt window method, after Emanuel Parzen and Murray Rosenblatt, who are usually credited with independently creating it in its current form. One of the famous applications of kernel density estimation is in estimating the class-conditional marginal densities of data when using a naive Bayes classifier, which can improve its prediction accuracy. == Definition == Let x = ( x 1 , x 2 , x 3 , . . . ) {\displaystyle \mathbf {x} =\left(x_{1},x_{2},x_{3},...\right)} be independent and identically distributed samples drawn from some univariate distribution with an unknown density f at any given point x. We are interested in estimating the shape of this function f. Its kernel density estimator is f ^ h ( x ) = 1 n ∑ i = 1 n K h ( x − x i ) = 1 n h ∑ i = 1 n K ( x − x i h ) , {\displaystyle {\hat {f}}_{h}(x)={\frac {1}{n}}\sum _{i=1}^{n}K_{h}(x-x_{i})={\frac {1}{nh}}\sum _{i=1}^{n}K{\left({\frac {x-x_{i}}{h}}\right)},} where K is the kernel — a non-negative function — and h > 0 is a smoothing parameter called the bandwidth or simply width. A kernel with subscript h is called the scaled kernel and defined as Kh(x) = ⁠1/h⁠ K(⁠x/h⁠). Intuitively one wants to choose h as small as the data will allow; however, there is always a trade-off between the bias of the estimator and its variance. The choice of bandwidth is discussed in more detail below. A range of kernel functions are commonly used: uniform, triangular, biweight, triweight, Epanechnikov (parabolic), normal, and others. The Epanechnikov kernel is optimal in a mean square error sense, though the loss of efficiency is small for the kernels listed previously. Due to its convenient mathematical properties, the normal kernel is often used, which means K(x) = ϕ(x), where ϕ is the standard normal density function. The kernel density estimator then becomes f ^ h ( x ) = 1 n ∑ i = 1 n 1 h 2 π exp ⁡ ( − ( x − x i ) 2 2 h 2 ) , {\displaystyle {\hat {f}}_{h}(x)={\frac {1}{n}}\sum _{i=1}^{n}{\frac {1}{h{\sqrt {2\pi }}}}\exp \left({\frac {-(x-x_{i})^{2}}{2h^{2}}}\right),} where h {\displaystyle h} is the standard deviation of the sample x {\displaystyle \mathbf {x} } . The construction of a kernel density estimate finds interpretations in fields outside of density estimation. For example, in thermodynamics, this is equivalent to the amount of heat generated when heat kernels (the fundamental solution to the heat equation) are placed at each data point locations xi. Similar methods are used to construct discrete Laplace operators on point clouds for manifold learning (e.g. diffusion map). == Example == Kernel density estimates are closely related to histograms, but can be endowed with properties such as smoothness or continuity by using a suitable kernel. The diagram below based on these 6 data points illustrates this relationship: For the histogram, first, the horizontal axis is divided into sub-intervals or bins which cover the range of the data: In this case, six bins each of width 2. Whenever a data point falls inside this interval, a box of height 1/12 is placed there. If more than one data point falls inside the same bin, the boxes are stacked on top of each other. For the kernel density estimate, normal kernels with a standard deviation of 1.5 (indicated by the red dashed lines) are placed on each of the data points xi. The kernels are summed to make the kernel density estimate (solid blue curve). The smoothness of the kernel density estimate (compared to the discreteness of the histogram) illustrates how kernel density estimates converge faster to the true underlying density for continuous random variables. == Bandwidth selection == The bandwidth of the kernel is a free parameter which exhibits a strong influence on the resulting estimate. To illustrate its effect, we take a simulated random sample from the standard normal distribution (plotted at the blue spikes in the rug plot on the horizontal axis). The grey curve is the true density (a normal density with mean 0 and variance 1). In comparison, the red curve is undersmoothed since it contains too many spurious data artifacts arising from using a bandwidth h = 0.05, which is too small. The green curve is oversmoothed since using the bandwidth h = 2 obscures much of the underlying structure. The black curve with a bandwidth of h = 0.337 is considered to be optimally smoothed since its density estimate is close to the true density. An extreme situation is encountered in the limit h → 0 {\displaystyle h\to 0} (no smoothing), where the estimate is a sum of n delta functions centered at the coordinates of analyzed samples. In the other extreme limit h → ∞ {\displaystyle h\to \infty } the estimate retains the shape of the used kernel, centered on the mean of the samples (completely smooth). The most common optimality criterion used to select this parameter is the expected L2 risk function, also termed the mean integrated squared error: MISE ⁡ ( h ) = E [ ∫ ( f ^ h ( x ) − f ( x ) ) 2 d x ] {\displaystyle \operatorname {MISE} (h)=\operatorname {E} \!\left[\int \!{\left({\hat {f}}\!_{h}(x)-f(x)\right)}^{2}dx\right]} Under weak assumptions on f and K, (f is the, generally unknown, real density function), MISE ⁡ ( h ) = AMISE ⁡ ( h ) + o ( ( n h ) − 1 + h 4 ) {\displaystyle \operatorname {MISE} (h)=\operatorname {AMISE} (h)+{\mathcal {o}}{\left((nh)^{-1}+h^{4}\right)}} where o is the little o notation, and n the sample size (as above). The AMISE is the asymptotic MISE, i. e. the two leading terms, AMISE ⁡ ( h ) = R ( K ) n h + 1 4 m 2 ( K ) 2 h 4 R ( f ″ ) {\displaystyle \operatorname {AMISE} (h)={\frac {R(K)}{nh}}+{\frac {1}{4}}m_{2}(K)^{2}h^{4}R(f'')} where R ( g ) = ∫ g ( x ) 2 d x {\textstyle R(g)=\int g(x)^{2}\,dx} for a function g, m 2 ( K ) = ∫ x 2 K ( x ) d x {\textstyle m_{2}(K)=\int x^{2}K(x)\,dx} and f ″ {\displaystyle f''} is the second derivative of f {\displaystyle f} and K {\displaystyle K} is the kernel. The minimum of this AMISE is the solution to this differential equation ∂ ∂ h AMISE ⁡ ( h ) = − R ( K ) n h 2 + m 2 ( K ) 2 h 3 R ( f ″ ) = 0 {\displaystyle {\frac {\partial }{\partial h}}\operatorname {AMISE} (h)=-{\frac {R(K)}{nh^{2}}}+m_{2}(K)^{2}h^{3}R(f'')=0} or h AMISE = R ( K ) 1 / 5 m 2 ( K ) 2 / 5 R ( f ″ ) 1 / 5 n − 1 / 5 = C n − 1 / 5 {\displaystyle h_{\operatorname {AMISE} }={\frac {R(K)^{1/5}}{m_{2}(K)^{2/5}R(f'')^{1/5}}}n^{-1/5}=Cn^{-1/5}} Neither the AMISE nor the hAMISE formulas can be used directly since they involve the unknown density function f {\displaystyle f} or its second derivative f ″ {\displaystyle f''} . To overcome that difficulty, a variety of automatic, data-based methods have been developed to select the bandwidth. Several review studies have been undertaken to compare their efficacies, with the general consensus that the plug-in selectors and cross validation selectors are the most useful over a wide range of data sets. Substituting any bandwidth h which has the same asymptotic order n−1/5 as hAMISE into the AMISE gives that AMISE(h) = O(n−4/5), where O is the big O notation. It can be shown that, under weak assumptions, there cannot exist a non-parametric estimator that converges at a faster rate than the kernel estimator. Note that the n−4/5 rate is slower than the typical n−1 convergence rate of parametric methods. If the bandwidth is not held fixed, but is varied depending upon the location of either the estimate (balloon estimator) or the samples (pointwise estimator), this produces a particularly powerful method termed adaptive or variable bandwidth kernel density estimation. Bandwidth selection for kernel density estimation of heavy-tailed distributions is relatively difficult. === A rule-of-thumb bandwidth estimator === If Gaussian basis functions are used to approximate univariate data, and the underlying density being estimated is Gaussian, the optimal choice for h (that is, the bandwidth that minimises the mean integrated squared error) is: h = ( 4 σ ^ 5 3 n ) 1 / 5 ≈ 1.06 σ ^ n − 1 / 5 , {\displaystyle h={\left({\frac {4{\hat {\sigma }}^{5}}{3n}}\right)}^{1/5}\approx 1.06\,{\hat {\sigma }}\,n^{-1/5},} An h {\displaystyle h} value is considered more robust when it improves the fit for long-tailed and skewed distributions or for bimodal mixture distributions. This is often done empirically by replacing the standard deviation σ ^ {\displaystyle {\hat {\sigma }}} by the parameter A {\displaystyle A} below: A = min ( σ ^ , I Q R 1.34 ) {\displaystyle A=\min \left({\hat {\sigma }},{\frac {\mathrm {IQR} }{1.34}}\right)} where IQR is the

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  Artificial general intelligence

  Artificial general intelligence (AGI) is a hypothetical type of artificial intelligence that matches or surpasses human capabilities across virtually all cognitive tasks. Beyond AGI, artificial superintelligence (ASI) would outperform the best human abilities across every domain by a wide margin. Unlike artificial narrow intelligence (ANI), whose competence is confined to well‑defined tasks, an AGI system can generalise knowledge, transfer skills between domains, and solve novel problems without task‑specific reprogramming. Creating AGI is a stated goal of technology companies such as OpenAI, Google, xAI, and Meta. A 2020 survey identified 72 active AGI research and development projects across 37 countries. AGI is a common topic in science fiction and futures studies. Contention exists over whether AGI represents an existential risk. Some AI experts and industry figures have stated that mitigating the risk of human extinction posed by AGI should be a global priority. Others find the development of AGI to be in too remote a stage to present such a risk. == Terminology == AGI is also known as strong AI, full AI, human-level AI, human-level intelligent AI, or general intelligent action. The term "artificial general intelligence" was used in 1997 by Mark Gubrud in a discussion of the implications of fully automated military production and operations. A mathematical formalism of AGI named AIXI was proposed in 2000 by Marcus Hutter, who defines intelligence as "an agent’s ability to achieve goals or succeed in a wide range of environments". This type of AGI has also been called "universal artificial intelligence". The term AGI was re-introduced and popularized by Shane Legg and Ben Goertzel around 2002. Some academic sources reserve the term "strong AI" for computer programs that will experience sentience or consciousness. In contrast, weak AI (or narrow AI) can solve a specific problem but lacks general cognitive abilities. Some academic sources use "weak AI" to refer more broadly to any programs that neither experience consciousness nor have a mind in the same sense as humans. Related concepts include artificial superintelligence and transformative AI. An artificial superintelligence (ASI) is a hypothetical type of AGI that is much more generally intelligent than humans, while the notion of transformative AI relates to AI having a large impact on society, for example, similar to the agricultural or industrial revolution. A framework for classifying AGI was proposed in 2023 by Google DeepMind researchers. They define five performance levels of AGI: emerging, competent, expert, virtuoso, and superhuman. For example, a competent AGI is defined as an AI that outperforms 50% of skilled adults in a wide range of non-physical tasks, and a superhuman AGI (i.e., an artificial superintelligence) is similarly defined but with a threshold of 100%. They consider large language models like ChatGPT or LLaMA 2 to be instances of emerging AGI (comparable to unskilled humans). Regarding the autonomy of AGI and associated risks, they define five levels: tool (fully in human control), consultant, collaborator, expert, and agent (fully autonomous). == Characteristics == There is no single agreed-upon definition of intelligence as applied to computers. Computer scientist John McCarthy wrote in 2007: "We cannot yet characterize in general what kinds of computational procedures we want to call intelligent." === Intelligence traits === Researchers generally hold that a system is required to do all of the following to be regarded as an AGI: reason, use strategy, solve puzzles, and make judgments under uncertainty, represent knowledge, including common sense knowledge, plan, learn, communicate in natural language, if necessary, integrate these skills in completion of any given goal. Many interdisciplinary approaches (e.g. cognitive science, computational intelligence, and decision making) consider additional traits such as imagination (the ability to form novel mental images and concepts) and autonomy. Computer-based systems exhibiting these capabilities are now widespread, with modern large language models demonstrating computational creativity, automated reasoning, and decision support simultaneously across domains. === Physical traits === Other capabilities are considered desirable in intelligent systems, as they may affect intelligence or aid in its expression. These include: the ability to sense (e.g. see, hear, etc.), and the ability to act (e.g. move and manipulate objects, change location to explore, etc.) This includes the ability to detect and respond to hazard. === Tests for human-level AGI === Several tests meant to confirm human-level AGI have been considered. ==== Turing test ==== The Turing test was proposed by Alan Turing in his 1950 paper "Computing Machinery and Intelligence". This test involves a human judge engaging in natural language conversations with both a human and a machine designed to generate human-like responses. The machine passes the test if it can convince the judge that it is human a significant fraction of the time. Turing proposed this as a practical measure of machine intelligence, focusing on the ability to produce human-like responses rather than on the internal workings of the machine. The idea of the test is that the machine has to try and pretend to be a man, by answering questions put to it, and it will only pass if the pretence is reasonably convincing. A considerable portion of a jury, who should not be experts about machines, must be taken in by the pretence. In 2014, a chatbot named Eugene Goostman, designed to imitate a 13-year-old Ukrainian boy, reportedly passed a Turing Test event by convincing 33% of judges that it was human. However, this claim was met with significant skepticism from the AI research community, who questioned the test's implementation and its relevance to AGI. A 2025 pre‑registered, three‑party Turing‑test study by Cameron R. Jones and Benjamin K. Bergen showed that GPT-4.5 was judged to be the human in 73% of five‑minute text conversations—surpassing the 67% humanness rate of real confederates and meeting the researchers' criterion for having passed the test. ==== Ikea test ==== The "Ikea test", also known as the Flat Pack Furniture Test, involves an AI controlling a robot which attempts to assemble an Ikea flat-pack furniture product after having been shown the parts and instructions. As early as 2013, MIT's IkeaBot demonstrated fully autonomous multi-robot assembly of an IKEA Lack table in ten minutes, with no human intervention and no pre-programmed assembly instructions. The robots inferred the assembly sequence from the geometry of the parts alone. ==== Coffee test ==== Steve Wozniak proposed a test where a machine is required to enter an average American home and figure out how to make coffee. It must find the coffee machine, find the coffee, add water, find a mug, and brew the coffee by pushing the proper buttons. This test has been substantially approached across multiple systems. In January 2024, Figure AI's Figure 01 humanoid learned to operate a Keurig coffee machine autonomously after watching video demonstrations, using end-to-end neural networks to translate visual input into motor actions. In 2025, researchers at the University of Edinburgh published the ELLMER framework in Nature Machine Intelligence, demonstrating a robotic arm that interprets verbal instructions, analyses its surroundings, and autonomously makes coffee in dynamic kitchen environments — adapting to unforeseen obstacles in real time rather than following pre-programmed sequences. ==== Suleyman's test ==== Mustafa Suleyman's test proposes giving an AI model US$100,000 and asking it to obtain US$1 million. ==== Use of video-games ==== Adams, et al. propose that the ability to learn and succeed in a wide range of video games can be used to test AI intelligence. This range would include games unknown to the AGI developers before the test is administered. === AI-complete problems === A problem is informally called "AI-complete" or "AI-hard" if it is believed that AGI would be needed to solve it, because the solution is beyond the capabilities of a purpose-specific algorithm. == History == === Classical AI === Modern AI research began in the mid-1950s. The first generation of AI researchers were convinced that artificial general intelligence was possible and that it would exist in just a few decades. AI pioneer Herbert A. Simon wrote in 1965: "machines will be capable, within twenty years, of doing any work a man can do". Their predictions were the inspiration for Stanley Kubrick and Arthur C. Clarke's fictional character HAL 9000, who embodied what AI researchers believed they could create by the year 2001. AI pioneer Marvin Minsky was a consultant on the project of making HAL 9000 as realistic as possible according to the consensus predictions of the time. He said in 1967, "Within a generation... the problem of

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  H2O (software)

  H2O is an open-source, in-memory, distributed machine learning and predictive analytics platform developed by the company H2O.ai (previously 0xdata). The software uses a distributed architecture for parallel processing on standard hardware. It supports algorithms for large-scale data analysis and model deployment. H2O is primarily used by data scientists and developers for statistical modeling and data-driven decision-making. The platform is designed to handle in-memory computations across a distributed computing environment. It offers implementations for numerous statistical and machine learning algorithms, which are accessible through various programming interfaces. The software is released under the Apache License 2.0. == Functionality and features == H2O provides a suite of supervised and unsupervised machine learning algorithms. Its core functions include: Supervised learning: algorithms in the field of statistics, data mining and machine learning such as generalized linear models, random forests, gradient boosting and deep learning are implemented for classification and regression tasks. Unsupervised learning: including K-Means clustering and principal component analysis. Automated machine learning: a features designed to automate the processes of model selection, tuning, and ensemble creation. The software can ingest data from various sources, including the Hadoop Distributed File System, Amazon S3, SQL databases, as well as local file systems. It operates natively on Apache Spark clusters through Sparkling Water. Proponents claim that improved performance is achieved compared to other analysis tools. The software is distributed free of charge, under a business model based on the development of individual applications and support. == Architecture == H2O is primarily written in Java. It uses a distributed architecture that allows the platform to cluster nodes for parallel processing and in-memory storage of data and models. Users interact with the H2O platform through several primary interfaces: Programming language interfaces: APIs are provided for the R and Python programming languages, and various Apache offerings (Apache Hadoop and Spark, as well as Maven). H2O Flow: a graphical web-based interactive computational environment that functions as a notebook interface for data exploration, model building, and scripting. REST-API: allows for integration with other applications and frameworks such as Microsoft Excel or RStudio. With the H2O Machine Learning Integration Nodes, KNIME offers algorithmic workflows. While the algorithm executes, approximate results are displayed, so that users can track the progress and intervene if needed. == History, influences, and extensions == The software project was initiated by the company 0xdata, which later changed its name to H2O.ai. The three Stanford professors Stephen P. Boyd, Robert Tibshirani and Trevor Hastie form a panel that advises H2O on scientific issues. Since its inception, H2O provides open-source machine learning libraries for enterprise use. The core H2O platform is often complemented by offerings from H2O.ai, such as H2O Driverless AI. == Reception == H2O is referenced in peer-reviewed literature regarding automated machine learning (AutoML). The platform has been categorized as a "Leader" and a "Strong Performer" in industry reports by Forrester Research. H2O (the open-source platform) and the associated commercial platform Driverless AI have been recurring winners of InfoWorld's most prestigious awards, including both the Best of Open Source Software ("Bossies") and the Technology of the Year awards.

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  INDIAai

  INDIAai is a web portal launched by the Government of India on 07 March 2024 for artificial intelligence-related developments in India. It is known as the National AI Portal of India, which was jointly started by the Ministry of Electronics and Information Technology (MeitY), the National e-Governance Division (NeGD) and the National Association of Software and Service Companies (NASSCOM) with support from the Department of School Education and Literacy (DoSE&L) and Ministry of Human Resource Development. == History == The portal was launched on 30 May 2020, by Ravi Shankar Prasad, the Union Minister for Electronics and IT, Law and Justice and Communications, on the first anniversary of the second tenure of Prime Minister Narendra Modi-led government. A national program for the youth, 'Responsible AI for Youth', was also launched on the same day. As of 2022, the website was visited by more than 4.5 lakh users with 1.2 million page views. It has 1151 articles on artificial intelligence, 701 news stories, 98 reports, 95 case studies and 213 videos on its portal. It maintains a database on AI ecosystem of India featuring 121 government initiatives and 281 startups. In May 2022, INDIAai released a book titled 'AI for Everyone' that covers the basics of AI. Cabinet chaired by the Prime Minister Narendra Modi has approved the comprehensive national-level IndiaAI mission with a budget outlay of Rs.10,371.92 crore. The Mission will be implemented by ‘IndiaAI’ Independent Business Division (IBD) under Digital India Corporation (DIC). == Objective and features == It aims to function as a one-stop portal for all AI-related development in India. The platform publishes resources such as articles, news, interviews, and investment funding news and events for AI startups, AI companies, and educational firms related to artificial intelligence in India. It also distributes documents, case studies, and research reports. Additionally, the platform provides education and employment opportunities related to AI. It offers AI courses, both free and paid.

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  Manifold hypothesis

  The manifold hypothesis posits that many high-dimensional data sets that occur in the real world actually lie along low-dimensional latent manifolds inside that high-dimensional space. As a consequence of the manifold hypothesis, many data sets that appear to initially require many variables to describe, can actually be described by a comparatively small number of variables, linked to the local coordinate system of the underlying manifold. It is suggested that this principle underpins the effectiveness of machine learning algorithms in describing high-dimensional data sets by considering a few common features. The manifold hypothesis is related to the effectiveness of nonlinear dimensionality reduction techniques in machine learning. Many techniques of dimensional reduction make the assumption that data lies along a low-dimensional submanifold, such as manifold sculpting, manifold alignment, and manifold regularization. The major implications of this hypothesis is that Machine learning models only have to fit relatively simple, low-dimensional, highly structured subspaces within their potential input space (latent manifolds). Within one of these manifolds, it's always possible to interpolate between two inputs, that is to say, morph one into another via a continuous path along which all points fall on the manifold. The ability to interpolate between samples is the key to generalization in deep learning. == The information geometry of statistical manifolds == An empirically-motivated approach to the manifold hypothesis focuses on its correspondence with an effective theory for manifold learning under the assumption that robust machine learning requires encoding the dataset of interest using methods for data compression. This perspective gradually emerged using the tools of information geometry thanks to the coordinated effort of scientists working on the efficient coding hypothesis, predictive coding and variational Bayesian methods. The argument for reasoning about the information geometry on the latent space of distributions rests upon the existence and uniqueness of the Fisher information metric. In this general setting, we are trying to find a stochastic embedding of a statistical manifold. From the perspective of dynamical systems, in the big data regime this manifold generally exhibits certain properties such as homeostasis: We can sample large amounts of data from the underlying generative process. Machine Learning experiments are reproducible, so the statistics of the generating process exhibit stationarity. In a sense made precise by theoretical neuroscientists working on the free energy principle, the statistical manifold in question possesses a Markov blanket.

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  Pedagogical agent

  A pedagogical agent is a concept borrowed from computer science and artificial intelligence and applied to education, usually as part of an intelligent tutoring system (ITS). It is a simulated human-like interface between the learner and the content, in an educational environment. A pedagogical agent is designed to model the type of interactions between a student and another person. Mabanza and de Wet define it as "a character enacted by a computer that interacts with the user in a socially engaging manner". A pedagogical agent can be assigned different roles in the learning environment, such as tutor or co-learner, depending on the desired purpose of the agent. "A tutor agent plays the role of a teacher, while a co-learner agent plays the role of a learning companion". == History == The history of Pedagogical Agents is closely aligned with the history of computer animation. As computer animation progressed, it was adopted by educators to enhance computerized learning by including a lifelike interface between the program and the learner. The first versions of a pedagogical agent were more cartoon than person, like Microsoft's Clippy which helped users of Microsoft Office load and use the program's features in 1997. However, with developments in computer animation, pedagogical agents can now look lifelike. By 2006 there was a call to develop modular, reusable agents to decrease the time and expertise required to create a pedagogical agent. There was also a call in 2009 to enact agent standards. The standardization and re-usability of pedagogical agents is less of an issue since the decrease in cost and widespread availability of animation tools. Individualized pedagogical agents can be found across disciplines including medicine, math, law, language learning, automotive, and armed forces. They are used in applications directed to every age, from preschool to adult. == Learning theories related to pedagogical agent design == === Distributed cognition theory === Distributed cognition theory is the method in which cognition progresses in the context of collaboration with others. Pedagogical agents can be designed to assist the cognitive transfer to the learner, operating as artifacts or partners with collaborative role in learning. To support the performance of an action by the user, the pedagogical agent can act as a cognitive tool as long as the agent is equipped with the knowledge that the user lacks. The interactions between the user and the pedagogical agent can facilitate a social relationship. The pedagogical agent may fulfill the role of a working partner. === Socio-cultural learning theory === Socio-cultural learning theory is how the user develops when they are involved in learning activities in which there is interaction with other agents. A pedagogical agent can: intervene when the user requests, provide support for tasks that the user cannot address, and potentially extend the learners cognitive reach. Interaction with the pedagogical agent may elicit a variety of emotions from the learner. The learner may become excited, confused, frustrated, and/or discouraged. These emotions affect the learners' motivation. === Extraneous Cognitive Load === Extraneous cognitive load is the extra effort being exerted by an individual's working memory due to the way information is being presented. A pedagogical agent can increase the user's cognitive load by distracting them and becoming the focus of their attention, causing split attention between the instructional material and the agent. Agents can reduce the perceived cognitive load by providing narration and personalization that can also promote a user's interest and motivation. While research on the reduction of cognitive load from pedagogical agents is minimal, more studies have shown that agents do not increase it. == Effectiveness == It has been suggested by researchers that pedagogical agents may take on different roles in the learning environment. Examples of these roles are: supplanting, scaffolding, coaching, testing, or demonstrating or modelling a procedure. A pedagogical agent as a tutor has not been demonstrated to add any benefit to an educational strategy in equivalent lessons with and without a pedagogical agent. According to Richard Mayer, there is some support in research for pedagogical agent increasing learning, but only as a presenter of social cues. A co-learner pedagogical agent is believed to increase the student's self-efficacy. By pointing out important features of instructional content, a pedagogical agent can fulfill the signaling function, which research on multimedia learning has shown to enhance learning. Research has demonstrated that human-human interaction may not be completely replaced by pedagogical agents, but learners may prefer the agents to non-agent multimedia systems. This finding is supported by social agency theory. Much like the varying effectiveness of the pedagogical agent roles in the learning environment, agents that take into account the user's affect have had mixed results. Research has shown pedagogical agents that make use of the users’ affect have been found to increase user knowledge retention, motivation, and perceived self-efficacy. However, with such a broad range of modalities in affective expressions, it is often difficult to utilize them. Additionally, having agents detect a user's affective state with precision remains challenging, as displays of affect are different across individuals. == Design == === Attractiveness === The appearance of a pedagogical agent can be manipulated to meet the learning requirements. The attractiveness of a pedagogical agent can enhance student's learning when the users were the opposite gender of the pedagogical agent. Male students prefer a sexy appearance of a female pedagogical agents and dislike the sexy appearance of male agents. Female students were not attracted by the sexy appearance of either male or female pedagogical agents. === Affective Response === Pedagogical agents have reached a point where they can convey and elicit emotion, but also reason about and respond to it. These agents are often designed to elicit and respond to affective actions from users through various modalities such as speech, facial expressions, and body gestures. They respond to the affective state of the given user, and make use of these modalities using a wide array of sensors incorporated into the design of the agent. Specifically in education and training applications, pedagogical agents are often designed to increasingly recognize when users or learners exhibit frustration, boredom, confusion, and states of flow. The added recognition in these agents is a step toward making them more emotionally intelligent, comforting and motivating the users as they interact. === Digital Representation === The design of a pedagogical agent often begins with its digital representation, whether it will be 2D or 3D and static or animated. Several studies have developed pedagogical agents that were both static and animated, then evaluated the relative benefits. Similar to other design considerations, the improved learning from static or animated agents remains questionable. One study showed that the appearance of an agent portrayed using a static image can impact a user's recall, based on the visual appearance. Other research found results that suggest static agent images improve learning outcomes. However, several other studies found user's learned more when the pedagogical agent was animated rather than static. Recently a meta-analysis of such research found a negligible improvement in learning via pedagogical agents, suggesting more work needs to be done in the area to support any claims.

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