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.
Digital video effect
Digital video effects (DVEs) are visual effects that provide comprehensive live video image manipulation, in the same form as optical printer effects in film. DVEs differ from standard video switcher effects (often referred to as analog effects) such as wipes or dissolves, in that they deal primarily with resizing, distortion or movement of the image. Modern video switchers often contain internal DVE functionality. Modern DVE devices are incorporated in high-end broadcast video switchers. Early examples of DVE devices found in the broadcast post-production industry include the Ampex Digital Optics (ADO), Quantel DPE-5000, Vital Squeezoom, NEC E-Flex and the Abekas A5x series of DVEs. By 1988, Grass Valley Group caught up with the competition with their Kaleidoscope, which integrated ADO-type effects with their widely used line of broadcast switching gear. DVEs are used by the broadcast television industry in live television production environments like television studios and outside broadcasts. They are commonly used in video post-production.
Project Maven
Project Maven (officially Algorithmic Warfare Cross Functional Team) is a United States Department of Defense initiative launched in 2017 to accelerate the adoption of machine learning and data integration across U.S. military intelligence workflows, specifically in intelligence, surveillance, target acquisition, and reconnaissance as well as in geospatial intelligence. It initially focused on applying computer vision for processing images and videos for intelligence purposes. Currently, the program operates under the National Geospatial-Intelligence Agency (NGA) and encompasses multiple applications across the Department of Defense spanning military operation targeting support, data integration and visualization for analysts, and training machine learning models on labeled datasets of military assets and infrastructure. It integrates data from drones, satellites, and other sensors to flag potential targets, present findings to human analysts, and relay their decisions to operational systems. The program originated under Deputy Secretary Robert O. Work after he raised concerns about China's advances in defense applications of artificial intelligence. Project leaders, Colonel Drew Cukor, USMC, and Lt. Gen. Jack Shanahan, framed the program as human-in-the-loop decision support inside the Department of Defense rather than as an autonomous weapons platform. Contractors supporting Maven have included Google, which withdrew in 2018 after internal protests, and follow-on integrators such as Palantir, Anduril, Amazon Web Services, and Anthropic (withdrew in 2026). The Pentagon credits Maven with providing 2024 targeting support for U.S. airstrikes in Iraq, Syria, and Yemen, along with locating hostile maritime assets in the Red Sea. == Administrative history == Initially, the effort was led by Robert O. Work who was concerned about China's military use of the emerging technology. Reportedly, Pentagon development stops short of acting as an AI weapons system capable of firing on self-designated targets. The project was established in a memo by the U.S. Deputy Secretary of Defense on 26 April 2017 proposing an "Algorithmic Warfare Cross-Functional Team". With the help of Defense Innovation Unit, the project obtained the support of top talents in AI outside of the traditional defense contracting base. It was initially funded for $70 million. Jack Shanahan was the director of the project during April 2017 to December 2018. At the second Defense One Tech Summit in July 2017, Cukor said that the investment in a "deliberate workflow process" was funded by the Department [of Defense] through its "rapid acquisition authorities" for about "the next 36 months". In the defense industry, the standard procedure for the military to acquire hardware is by way of research, development, test, and evaluation (RDT&E), followed by production and sustainment. In 2017, acquiring software was done in the same way as hardware. This created a problem, since software is constantly updated. Project Maven procured software using Broad Agency Announcements, a flexible contracting vehicle that categorized software as consistently RDT&E, allowing constant updating. Another issue was that the government usually acquired the intellectual property (IP) for procured software, and with the project, only parts of the IP of the software was acquired. Cukor used the principle of "platform IP belongs to the vendor, configurations on top are the customer's". For example, Palantir retained IP to their core platform, while the government obtained the IP to Maven-specific logic configured on top of it. According to US Air Force Lt. Gen. Jack Shanahan in November 2017, it is "designed to be that pilot project, that pathfinder, that spark that kindles the flame front of artificial intelligence across the rest of the [Defense] Department". Its chief, U.S. Marine Corps Col. Drew Cukor, said: "People and computers will work symbiotically to increase the ability of weapon systems to detect objects." Project Maven has been noted by allies, such as Australia's Ian Langford, for the ability to identify adversaries by harvesting data from sensors on UAVs and satellites. As of 2017 December, 150,000 images had been manually labelled to establish the first training data sets, and it was projected to reach one million by January 2018. Project Maven was funded for $221 million in fiscal 2020. In 2020, the House and Senate conferees on the National Defense Authorization Act for Fiscal Year 2021, agreed to the Senate's recommendation to fund the Pentagon's $250 million request for Project Maven. At the GEOINT Symposium of 2022, it was announced that Project Maven was transferred from the Office of the Under Secretary of Defense for Intelligence and Security to the NGA, under President Biden’s proposed budget for Fiscal Year 2023. It became a Program of Record on 2023 November 7. Frank "Trey" Whitworth, vice admiral, was the director of NGA from June 2022 to November 2025. Whitworth was initially skeptical of the program, suspecting it was incautious about the targeting principles, but later regarded it as "important work". As of 2024, the project is jointly administered by the NGA and the CDAO, and its director is Rachel Martin. Before 2025, Biden appointees within CDAO had held back AI development for safety and reliability concerns, though as of 2025, this has stopped. As of 2024, Maven provided the cloud infrastructure, software capabilities, and AI for CDAO's Combined Joint All-Domain Command and Control initiatives. As of summer 2025, there were eight Maven initiatives. Of these, five were in the NGA, including analyzing drone feeds and satellite imagery. On 18 September 2025, the UK government announced a new partnership with Palantir to develop AI-powered military capabilities for decision-making and targeting, identifying opportunities worth up to £750 million over five years. On 25 March 2025, the NATO Communications and Information Agency and Palantir finalized the acquisition of the Palantir Maven Smart System NATO (MSS NATO) for employment within NATO's Allied Command Operations. It was planned to be used within 30 days of acquisition. In a letter to Pentagon on 9 March 2026, Steve Feinberg stated that Project Maven will become an official program of record by September 2026, the close of the current fiscal year. The project would transfer from the NGA to the CDAO within 30 days. Future contracting with Palantir would be handled by the US Army. In 2026-03, it was announced that the US Army Combined Arms Command would integrate Maven into its training. == Technology == Project Maven uses machine learning algorithms to analyze and fuse vast amounts of surveillance data from multiple sources made possible through data integration using Palantir Technologies. The data sources include photographs, satellite imagery, geolocation data (IP address, geotag, metadata, etc) from communications intercepts, infrared sensors, synthetic-aperture radar, and more. The system is mainly used for assisting analysts in intelligence, surveillance, target acquisition, and reconnaissance. Machine learning systems, including object recognition systems, process the data and identify potential targets, such as enemy tanks or location of new military facility. The training dataset included at least 4 million images of military objects such as warships, labelled by humans. The user interface is called Maven Smart System. It could display information such as aircraft movements, logistics, locations of key personnel, locations on the no-strike list, ships, etc. Yellow-outlined boxes show potential targets. Blue-outlined boxes show friendly forces or no-strike zones. It could also transmit, directly to weapons, a human decision to fire weapons. Internal documentation referred to "Maven ATR: automatic target recognition". Initially the project focused on applications of computer vision. The project's leaders were particularly impressed by model performance on ImageNet. As of 2018, the purpose of the system was AI-enabled analysis of full-motion video. In 2022 it expanded to combatant commands under the AI and Data Acceleration Initiative. In 2022, it was reported that the project expanded to non-image data, including captured enemy material, maritime intelligence, and publicly available information. In 2024, it was stated that Maven's key technical contribution was data management: Maven standardizes heterogeneous data through an ontology layer so data can be fused, exchanged across cloud and edge systems, and used by multiple applications. The system was presented as a broader data-centric warfighting system that feeds apps for planning, preparing, and executing operations. In 2024, the Broad Area Surveillance-Targeting (BAS-T) is a part of Maven. The system detects objects in images and uses data fusion to produce a common operational picture containing "priority based, in-depth assessment of the enemy systems pre
AutoGPT
AutoGPT is an open-source autonomous software agent that uses OpenAI's large language models, such as GPT-4, to attempt to achieve a goal specified by a user in natural language. Unlike chatbots that require continuous user commands, AutoGPT works autonomously by breaking the main goal into smaller sub-tasks and using tools like web browsing and file management to complete them. Released in March 2023, the project quickly gained popularity on GitHub and social media, with users creating agents for tasks like software development, market research, and content creation. One notable experiment, ChaosGPT, was tasked with destroying humanity, which brought mainstream attention to the technology's potential. However, AutoGPT is known for significant limitations, including a tendency to get stuck in loops, hallucinate information, and incur high operational costs due to its reliance on paid APIs. == Background == AutoGPT was released on March 30, 2023, by Toran Bruce Richards, the founder of video game company Significant Gravitas Ltd. It was one of the first widely accessible applications to showcase the autonomous capabilities of GPT-4, which had been released weeks earlier. Richards's goal was to create a model that could respond to real-time feedback and pursue objectives with a long-term outlook without needing constant human intervention. The application operates by prompting a user to define an agent's name, role, and main objective, including up to five sub-goals to achieve it. AutoGPT then works independently to reach its objective. The project is publicly available on GitHub but requires users to install it in a development environment like Docker and have a paid OpenAI account to obtain the necessary API key. In October 2023, the project's parent company, Significant Gravitas Ltd., raised $12 million in venture funding to support further development. == Capabilities == The overarching capability of AutoGPT is the breaking down of a large task into various sub-tasks without the need for user input. These sub-tasks are then chained together and performed sequentially to yield a larger result as originally laid out by the user input. One of the distinguishing features of AutoGPT is its ability to connect to the internet. This allows for up-to-date information retrieval to help complete tasks. In addition, AutoGPT maintains short-term memory for the current task, which allows it to provide context to subsequent sub-tasks needed to achieve the larger goal. Another feature is its ability to store and organize files so users can better structure their data for future analysis and extension. AutoGPT is also multimodal, which means that it can take in both text and images as input. With these features, AutoGPT is claimed to be capable of automating workflows, analyzing data, and coming up with new suggestions. == Applications == === Software === AutoGPT can be used to develop software applications from scratch. AutoGPT can also debug code and generate test cases. Observers suggest that AutoGPT's ability to write, debug, test, and edit code may extend to AutoGPT's own source code, enabling self-improvement. === Business === AutoGPT can be used to do market research, analyze investments, research products and write product reviews, create a business plan or improve operations, and create content such as a blog or podcast. One user has used AutoGPT to conduct product research and write a summary on the best headphones. Another user has used AutoGPT to summarize recent news events and prepare an outline for a podcast. === Other === AutoGPT was used to create ChefGPT, an AI agent able to independently explore the internet to generate and save unique recipes. AutoGPT was also used to create ChaosGPT, an AI agent tasked to “destroy humanity, establish global dominance, cause chaos and destruction, control humanity through manipulation, and attain immortality”. ChaosGPT reportedly researched nuclear weapons and tweeted disparagingly about humankind. == Limitations == AutoGPT is susceptible to frequent mistakes, primarily because it relies on its own feedback, which can compound errors. In contrast, non-autonomous models can be corrected by users overseeing their outputs. Furthermore, AutoGPT has a tendency to hallucinate or to present false or misleading information as fact when responding. AutoGPT can be constrained by the cost associated with running it as its recursive nature requires it to continually call the OpenAI API on which it is built. Every step required in one of AutoGPT's tasks requires a corresponding call to GPT-4 at a cost of at least about $0.03 for every 1000 tokens used for inputs and $0.06 for every 1000 tokens for output when choosing the cheapest option. For reference, 1000 tokens roughly result in 750 words. Another limitation is AutoGPT's tendency to get stuck in infinite loops. Developers believe that this is a result of AutoGPT's inability to remember, as it is unaware of what it has already done and repeatedly attempts the same subtask without end. Andrej Karpathy, co-founder of OpenAI which creates GPT-4, further explains that it is AutoGPT's “finite context window” that can limit its performance and cause it to “go off the rails”. Like other autonomous agents, AutoGPT is prone to distraction and unable to focus on its objective due to its lack of long-term memory, leading to unpredictable and unintended behavior. == Reception == AutoGPT became the top trending repository on GitHub after its release and has since repeatedly trended on Twitter. In April 2023, Avram Piltch wrote for Tom's Hardware that AutoGPT 'might be too autonomous to be useful,' as it did not ask questions to clarify requirements or allow corrective interventions by users. Piltch nonetheless noted that such tools have "a ton of potential" and should improve with better language models and further development. Malcolm McMillan from Tom's Guide mentioned that AutoGPT may not be better than ChatGPT for tasks involving conversation, as ChatGPT is well-suited for situations in which advice, rather than task completion, is sought. Will Knight from Wired wrote that AutoGPT is not a foolproof task-completion tool. When given a test task of finding a public figure's email address, he noted that it was not able to accurately find the email address. Clara Shih, Salesforce Service Cloud CEO commented that "AutoGPT illustrates the power and unknown risks of generative AI," and that due to usage risks, enterprises should include a human in the loop when using such technologies. Performance is reportedly enhanced when using AutoGPT with GPT-4 compared to GPT-3.5. For example, one reviewer who tested it on a task of finding the best laptops on the market with pros and cons found that AutoGPT with GPT-4 created a more comprehensive report than one by GPT 3.5.
Ian Goodfellow
Ian J. Goodfellow (born 1987) is an American computer scientist, engineer, and executive, most noted for his work on artificial neural networks and deep learning. He is a research scientist at Google DeepMind, was previously employed as a research scientist at Google Brain and director of machine learning at Apple as well as one of the first employees at OpenAI, and has made several important contributions to the field of deep learning, including the invention of the generative adversarial network (GAN). Goodfellow co-wrote, as the first author, the textbook Deep Learning (2016) and wrote the chapter on deep learning in the authoritative textbook of the field of artificial intelligence, Artificial Intelligence: A Modern Approach (used in more than 1,500 universities in 135 countries). == Education == Goodfellow obtained his BSc and MSc in computer science from Stanford University under the supervision of Andrew Ng, and his PhD in machine learning from the Université de Montréal in February 2015, under the supervision of Yoshua Bengio and Aaron Courville. Goodfellow's thesis is titled Deep learning of representations and its application to computer vision. == Career == After graduation, Goodfellow joined Google as part of the Google Brain research team. In March 2016, he left Google to join the newly founded OpenAI research laboratory. 11 months later, in March 2017, Goodfellow returned to Google Research, but left again in 2019. In 2019, Goodfellow joined Apple as director of machine learning in the Special Projects Group. He resigned from Apple in April 2022 to protest Apple's plan to require in-person work for its employees. Shortly after, Goodfellow then joined Google DeepMind as a research scientist. In 2025, Goodfellow left Google. As of July 2026, based on information on Goodfellow's LinkedIn profile, he is co-founding a startup company. == Research == Goodfellow is best known for inventing generative adversarial networks (GANs), using deep learning to generate images. This approach uses two neural networks to competitively improve an image's quality. A “generator” network creates a synthetic image based on an initial set of images such as a collection of faces. A “discriminator” network tries to determine whether images are authentic or created by the generator. The generate-detect cycle is repeated. For each iteration, the generator and the discriminator use the other's feedback to improve or detect the generated images, until the discriminator can no longer distinguish between generated and authentic images. However, GANs have also been used to create deepfakes. At Google, Goodfellow developed a system enabling Google Maps to automatically transcribe addresses from photos taken by Street View cars and demonstrated security vulnerabilities of machine learning systems. == Recognition == In 2017, Goodfellow was cited in MIT Technology Review's 35 Innovators Under 35. In 2019, he was included in Foreign Policy's list of 100 Global Thinkers.
Scale-space axioms
In image processing and computer vision, a scale space framework can be used to represent an image as a family of gradually smoothed images. This framework is very general and a variety of scale space representations exist. A typical approach for choosing a particular type of scale space representation is to establish a set of scale-space axioms, describing basic properties of the desired scale-space representation and often chosen so as to make the representation useful in practical applications. Once established, the axioms narrow the possible scale-space representations to a smaller class, typically with only a few free parameters. A set of standard scale space axioms, discussed below, leads to the linear Gaussian scale-space, which is the most common type of scale space used in image processing and computer vision. == Scale space axioms for the linear scale-space representation == The linear scale space representation L ( x , y , t ) = ( T t f ) ( x , y ) = g ( x , y , t ) ∗ f ( x , y ) {\displaystyle L(x,y,t)=(T_{t}f)(x,y)=g(x,y,t)f(x,y)} of signal f ( x , y ) {\displaystyle f(x,y)} obtained by smoothing with the Gaussian kernel g ( x , y , t ) {\displaystyle g(x,y,t)} satisfies a number of properties 'scale-space axioms' that make it a special form of multi-scale representation: linearity T t ( a f + b h ) = a T t f + b T t h {\displaystyle T_{t}(af+bh)=aT_{t}f+bT_{t}h} where f {\displaystyle f} and h {\displaystyle h} are signals while a {\displaystyle a} and b {\displaystyle b} are constants, shift invariance T t S ( Δ x , Δ y ) f = S ( Δ x , Δ y ) T t f {\displaystyle T_{t}S_{(\Delta x,\Delta _{y})}f=S_{(\Delta x,\Delta _{y})}T_{t}f} where S ( Δ x , Δ y ) {\displaystyle S_{(\Delta x,\Delta _{y})}} denotes the shift (translation) operator ( S ( Δ x , Δ y ) f ) ( x , y ) = f ( x − Δ x , y − Δ y ) {\displaystyle (S_{(\Delta x,\Delta _{y})}f)(x,y)=f(x-\Delta x,y-\Delta y)} semi-group structure g ( x , y , t 1 ) ∗ g ( x , y , t 2 ) = g ( x , y , t 1 + t 2 ) {\displaystyle g(x,y,t_{1})g(x,y,t_{2})=g(x,y,t_{1}+t_{2})} with the associated cascade smoothing property L ( x , y , t 2 ) = g ( x , y , t 2 − t 1 ) ∗ L ( x , y , t 1 ) {\displaystyle L(x,y,t_{2})=g(x,y,t_{2}-t_{1})L(x,y,t_{1})} existence of an infinitesimal generator A {\displaystyle A} ∂ t L ( x , y , t ) = ( A L ) ( x , y , t ) {\displaystyle \partial _{t}L(x,y,t)=(AL)(x,y,t)} non-creation of local extrema (zero-crossings) in one dimension, non-enhancement of local extrema in any number of dimensions ∂ t L ( x , y , t ) ≤ 0 {\displaystyle \partial _{t}L(x,y,t)\leq 0} at spatial maxima and ∂ t L ( x , y , t ) ≥ 0 {\displaystyle \partial _{t}L(x,y,t)\geq 0} at spatial minima, rotational symmetry g ( x , y , t ) = h ( x 2 + y 2 , t ) {\displaystyle g(x,y,t)=h(x^{2}+y^{2},t)} for some function h {\displaystyle h} , scale invariance g ^ ( ω x , ω y , t ) = h ^ ( ω x φ ( t ) , ω x φ ( t ) ) {\displaystyle {\hat {g}}(\omega _{x},\omega _{y},t)={\hat {h}}({\frac {\omega _{x}}{\varphi (t)}},{\frac {\omega _{x}}{\varphi (t)}})} for some functions φ {\displaystyle \varphi } and h ^ {\displaystyle {\hat {h}}} where g ^ {\displaystyle {\hat {g}}} denotes the Fourier transform of g {\displaystyle g} , positivity g ( x , y , t ) ≥ 0 {\displaystyle g(x,y,t)\geq 0} , normalization ∫ x = − ∞ ∞ ∫ y = − ∞ ∞ g ( x , y , t ) d x d y = 1 {\displaystyle \int _{x=-\infty }^{\infty }\int _{y=-\infty }^{\infty }g(x,y,t)\,dx\,dy=1} . In fact, it can be shown that the Gaussian kernel is a unique choice given several different combinations of subsets of these scale-space axioms: most of the axioms (linearity, shift-invariance, semigroup) correspond to scaling being a semigroup of shift-invariant linear operator, which is satisfied by a number of families integral transforms, while "non-creation of local extrema" for one-dimensional signals or "non-enhancement of local extrema" for higher-dimensional signals are the crucial axioms which relate scale-spaces to smoothing (formally, parabolic partial differential equations), and hence select for the Gaussian. The Gaussian kernel is also separable in Cartesian coordinates, i.e. g ( x , y , t ) = g ( x , t ) g ( y , t ) {\displaystyle g(x,y,t)=g(x,t)\,g(y,t)} . Separability is, however, not counted as a scale-space axiom, since it is a coordinate dependent property related to issues of implementation. In addition, the requirement of separability in combination with rotational symmetry per se fixates the smoothing kernel to be a Gaussian. There exists a generalization of the Gaussian scale-space theory to more general affine and spatio-temporal scale-spaces. In addition to variabilities over scale, which original scale-space theory was designed to handle, this generalized scale-space theory also comprises other types of variabilities, including image deformations caused by viewing variations, approximated by local affine transformations, and relative motions between objects in the world and the observer, approximated by local Galilean transformations. In this theory, rotational symmetry is not imposed as a necessary scale-space axiom and is instead replaced by requirements of affine and/or Galilean covariance. The generalized scale-space theory leads to predictions about receptive field profiles in good qualitative agreement with receptive field profiles measured by cell recordings in biological vision. In the computer vision, image processing and signal processing literature there are many other multi-scale approaches, using wavelets and a variety of other kernels, that do not exploit or require the same requirements as scale space descriptions do; please see the article on related multi-scale approaches. There has also been work on discrete scale-space concepts that carry the scale-space properties over to the discrete domain; see the article on scale space implementation for examples and references.
Norm (artificial intelligence)
Norms can be considered from different perspectives in artificial intelligence to create computers and computer software that are capable of intelligent behaviour. In artificial intelligence and law, legal norms are considered in computational tools to automatically reason upon them. In multi-agent systems (MAS), a branch of artificial intelligence (AI), a norm is a guide for the common conduct of agents, thereby easing their decision-making, coordination and organization. Since most problems concerning regulation of the interaction of autonomous agents are linked to issues traditionally addressed by legal studies, and since law is the most pervasive and developed normative system, efforts to account for norms in artificial intelligence and law and in normative multi-agent systems often overlap. == Artificial intelligence and law == With the arrival of computer applications into the legal domain, and especially artificial intelligence applied to it, logic has been used as the major tool to formalize legal reasoning and has been developed in many directions, ranging from deontic logics to formal systems of argumentation. The knowledge base of legal reasoning systems usually includes legal norms (such as governmental regulations and contracts), and as a consequence, legal rules are the focus of knowledge representation and reasoning approaches to automatize and solve complex legal tasks. Legal norms are typically represented into a logic-based formalism, such as deontic logic. Artificial intelligence and law applications using an explicit representation of norms range from checking the compliance of business processes and the automatic execution of smart contracts to legal expert systems advising people on legal matters. == Multi-agent systems == Norms in multi-agent systems may appear with different degrees of explicitness ranging from fully unambiguous written prescriptions to implicit unwritten norms or tacit emerging patterns. Computer scientists’ studies mirror this polarity. Explicit norms are typically investigated in formal logics (e.g. deontic logics and argumentation) to represent and reason upon them, leading eventually to architecture for cognitive agents, while implicit norms are accounted as patterns emerging from repeated interactions amongst agents (typically reinforced learning agents). Explicit and implicit norms can be used together to coordinate agents. Explicit norms are typically represented as a deontic statement that aims at regulating the life of software agents and the interactions among them. It can be an obligation, a permission or a prohibition, and is often represented with some dialect or extension of Deontic logic. At the opposite, implicit norms are social norms that are not written, and they usually emerge from the repetitive interactions of agents.