Multifactor dimensionality reduction (MDR) is a statistical approach, also used in machine learning automatic approaches, for detecting and characterizing combinations of attributes or independent variables that interact to influence a dependent or class variable. MDR was designed specifically to identify nonadditive interactions among discrete variables that influence a binary outcome and is considered a nonparametric and model-free alternative to traditional statistical methods such as logistic regression. The basis of the MDR method is a constructive induction or feature engineering algorithm that converts two or more variables or attributes to a single attribute. This process of constructing a new attribute changes the representation space of the data. The end goal is to create or discover a representation that facilitates the detection of nonlinear or nonadditive interactions among the attributes such that prediction of the class variable is improved over that of the original representation of the data. == Illustrative example == Consider the following simple example using the exclusive OR (XOR) function. XOR is a logical operator that is commonly used in data mining and machine learning as an example of a function that is not linearly separable. The table below represents a simple dataset where the relationship between the attributes (X1 and X2) and the class variable (Y) is defined by the XOR function such that Y = X1 XOR X2. Table 1 A machine learning algorithm would need to discover or approximate the XOR function in order to accurately predict Y using information about X1 and X2. An alternative strategy would be to first change the representation of the data using constructive induction to facilitate predictive modeling. The MDR algorithm would change the representation of the data (X1 and X2) in the following manner. MDR starts by selecting two attributes. In this simple example, X1 and X2 are selected. Each combination of values for X1 and X2 are examined and the number of times Y=1 and/or Y=0 is counted. In this simple example, Y=1 occurs zero times and Y=0 occurs once for the combination of X1=0 and X2=0. With MDR, the ratio of these counts is computed and compared to a fixed threshold. Here, the ratio of counts is 0/1 which is less than our fixed threshold of 1. Since 0/1 < 1 we encode a new attribute (Z) as a 0. When the ratio is greater than one we encode Z as a 1. This process is repeated for all unique combinations of values for X1 and X2. Table 2 illustrates our new transformation of the data. Table 2 The machine learning algorithm now has much less work to do to find a good predictive function. In fact, in this very simple example, the function Y = Z has a classification accuracy of 1. A nice feature of constructive induction methods such as MDR is the ability to use any data mining or machine learning method to analyze the new representation of the data. Decision trees, neural networks, or a naive Bayes classifier could be used in combination with measures of model quality such as balanced accuracy and mutual information. == Machine learning with MDR == As illustrated above, the basic constructive induction algorithm in MDR is very simple. However, its implementation for mining patterns from real data can be computationally complex. As with any machine learning algorithm there is always concern about overfitting. That is, machine learning algorithms are good at finding patterns in completely random data. It is often difficult to determine whether a reported pattern is an important signal or just chance. One approach is to estimate the generalizability of a model to independent datasets using methods such as cross-validation. Models that describe random data typically don't generalize. Another approach is to generate many random permutations of the data to see what the data mining algorithm finds when given the chance to overfit. Permutation testing makes it possible to generate an empirical p-value for the result. Replication in independent data may also provide evidence for an MDR model but can be sensitive to difference in the data sets. These approaches have all been shown to be useful for choosing and evaluating MDR models. An important step in a machine learning exercise is interpretation. Several approaches have been used with MDR including entropy analysis and pathway analysis. Tips and approaches for using MDR to model gene-gene interactions have been reviewed. == Extensions to MDR == Numerous extensions to MDR have been introduced. These include family-based methods, fuzzy methods, covariate adjustment, odds ratios, risk scores, survival methods, robust methods, methods for quantitative traits, and many others. == Applications of MDR == MDR has mostly been applied to detecting gene-gene interactions or epistasis in genetic studies of common human diseases such as atrial fibrillation, autism, bladder cancer, breast cancer, cardiovascular disease, hypertension, obesity, pancreatic cancer, prostate cancer and tuberculosis. It has also been applied to other biomedical problems such as the genetic analysis of pharmacology outcomes. A central challenge is the scaling of MDR to big data such as that from genome-wide association studies (GWAS). Several approaches have been used. One approach is to filter the features prior to MDR analysis. This can be done using biological knowledge through tools such as BioFilter. It can also be done using computational tools such as ReliefF. Another approach is to use stochastic search algorithms such as genetic programming to explore the search space of feature combinations. Yet another approach is a brute-force search using high-performance computing. == Implementations == www.epistasis.org provides an open-source and freely-available MDR software package. An R package for MDR. An sklearn-compatible Python implementation. An R package for Model-Based MDR. MDR in Weka. Generalized MDR.
Database application
A database application is a computer program whose primary purpose is retrieving information from a computerized database. From here, information can be inserted, modified or deleted which is subsequently conveyed back into the database. Early examples of database applications were accounting systems and airline reservations systems, such as SABRE, developed starting in 1957. A characteristic of modern database applications is that they facilitate simultaneous updates and queries from multiple users. Systems in the 1970s might have accomplished this by having each user in front of a 3270 terminal to a mainframe computer. By the mid-1980s it was becoming more common to give each user a personal computer and have a program running on that PC that is connected to a database server. Information would be pulled from the database, transmitted over a network, and then arranged, graphed, or otherwise formatted by the program running on the PC. Starting in the mid-1990s it became more common to build database applications with a Web interface. Rather than develop custom software to run on a user's PC, the user would use the same Web browser program for every application. A database application with a Web interface had the advantage that it could be used on devices of different sizes, with different hardware, and with different operating systems. Examples of early database applications with Web interfaces include amazon.com, which used the Oracle relational database management system, the photo.net online community, whose implementation on top of Oracle was described in the book Database-Backed Web Sites (Ziff-Davis Press; May 1997), and eBay, also running Oracle. Electronic medical records are referred to on emrexperts.com, in December 2010, as "a software database application". A 2005 O'Reilly book uses the term in its title: Database Applications and the Web. Some of the most complex database applications remain accounting systems, such as SAP, which may contain thousands of tables in only a single module. Many of today's most widely used computer systems are database applications, for example, Facebook, which was built on top of MySQL. The etymology of the phrase "database application" comes from the practice of dividing computer software into systems programs, such as the operating system, compilers, the file system, and tools such as the database management system, and application programs, such as a payroll check processor. On a standard PC running Microsoft Windows, for example, the Windows operating system contains all of the systems programs while games, word processors, spreadsheet programs, photo editing programs, etc. would be application programs. As "application" is short for "application program", "database application" is short for "database application program". Not every program that uses a database would typically be considered a "database application". For example, many physics experiments, e.g., the Large Hadron Collider, generate massive data sets that programs subsequently analyze. The data sets constitute a "database", though they are not typically managed with a standard relational database management system. The computer programs that analyze the data are primarily developed to answer hypotheses, not to put information back into the database and therefore the overall program would not be called a "database application". == Examples of database applications == Amazon Student Data CNN eBay Facebook Fandango Filemaker (Mac OS) LibreOffice Base Microsoft Access Oracle relational database SAP (Systems, Applications & Products in Data Processing) Ticketmaster Wikipedia Yelp YouTube Google MySQL
Smartglasses
Smartglasses or smart glasses are eye or head-worn wearable computers. Many smartglasses include displays that add information alongside or to what the wearer sees. Alternatively, smartglasses are sometimes defined as glasses that are able to change their optical properties, such as smart sunglasses that are programmed to change tint by electronic means. Alternatively, smartglasses are sometimes defined as glasses that include headphone functionality. A pair of smartglasses can be considered an augmented reality device if it performs pose tracking. Superimposing information onto a field of view is achieved through an optical head-mounted display (OHMD) or embedded wireless glasses with transparent heads-up display (HUD) or augmented reality (AR) overlay. These systems have the capability to reflect projected digital images as well as allowing the user to see through it or see better with it. While early models can perform basic tasks, such as serving as a front end display for a remote system, as in the case of smartglasses utilizing cellular technology or Wi-Fi, modern smart glasses are effectively wearable computers which can run self-contained mobile apps. Some are handsfree and can communicate with the Internet via natural language voice commands, while others use touch buttons. Like other computers, smartglasses may collect information from internal or external sensors. It may control or retrieve data from other instruments or computers. In most cases, it supports wireless technologies like Bluetooth, Wi-Fi, and GPS. A small number of models run a mobile operating system and function as portable media players to send audio and video files to the user via a Bluetooth or WiFi headset. Some smartglasses models also feature full lifelogging and activity tracker capability. Smartglasses devices may also have features found on a smartphone. Some have activity tracker functionality features (also known as "fitness tracker") as seen in some GPS watches. == Features and applications == As with other lifelogging and activity tracking devices, the GPS tracking unit and digital camera of some smartglasses can be used to record historical data. For example, after the completion of a workout, data can be uploaded into a computer or online to create a log of exercise activities for analysis. Some smart watches can serve as full GPS navigation devices, displaying maps and current coordinates. Users can "mark" their current location and then edit the entry's name and coordinates, which enables navigation to those new coordinates. Although some smartglasses models manufactured in the 21st century are completely functional as standalone products, most manufacturers recommend or even require that consumers purchase mobile phone handsets that run the same operating system so that the two devices can be synchronized for additional and enhanced functionality. The smartglasses can work as an extension, for head-up display (HUD) or remote control of the phone and alert the user to communication data such as calls, SMS messages, emails, and calendar invites. === Security applications === Smart glasses could be used as a body camera. In 2018, Chinese police in Zhengzhou and Beijing were using smart glasses to take photos which are compared against a government database using facial recognition to identify suspects, retrieve an address, and track people moving beyond their home areas. === Sport applications === Smart glasses are used in sports like cycling, running, skiing, golf, tennis, or sailing, giving athletes real-time, heads-up data without looking down at the screen of a watch or smartphone. In 2025, Meta has announced a new partnership with sports eyewear brand Oakley. === Healthcare applications === Several proofs of concept for Google Glasses have been proposed in healthcare. In July 2013, Lucien Engelen started research on the usability and impact of Google Glass in health care. Engelen, who is based at Singularity University and in Europe at Radboud University Medical Center, is participating in the Glass Explorer program. Key findings of Engelen's research included: The quality of pictures and video are usable for healthcare education, reference, and remote consultation. The camera needs to be tilted to different angle for most of the operative procedures Tele-consultation is possible—depending on the available bandwidth—during operative procedures. A stabilizer should be added to the video function to prevent choppy transmission when a surgeon looks to screens or colleagues. Battery life can be easily extended with the use of an external battery. Controlling the device and/or programs from another device is needed for some features because of a sterile environment. Text-to-speech ("Take a Note" to Evernote) exhibited a correction rate of 60 percent, without the addition of a medical thesaurus. A protocol or checklist displayed on the screen of Google Glass can be helpful during procedures. Dr. Phil Haslam and Dr. Sebastian Mafeld demonstrated the first concept for Google Glass in the field of interventional radiology. They demonstrated the manner in which the concept of Google Glass could assist a liver biopsy and fistulaplasty, and the pair stated that Google Glass has the potential to improve patient safety, operator comfort, and procedure efficiency in the field of interventional radiology. In June 2013, surgeon Dr. Rafael Grossmann was the first person to integrate Google Glass into the operating theater, when he wore the device during a PEG (percutaneous endoscopic gastrostomy) procedure. In August 2013, Google Glass was also used at Wexner Medical Center at Ohio State University. Surgeon Dr. Christopher Kaeding used Google Glass to consult with a colleague in a distant part of Columbus, Ohio. A group of students at The Ohio State University College of Medicine also observed the operation on their laptop computers. Following the procedure, Kaeding stated, "To be honest, once we got into the surgery, I often forgot the device was there. It just seemed very intuitive and fit seamlessly." 16 November 2013, in Santiago de Chile, the maxillofacial team led by Dr.gn Antonio Marino conducted the first orthognathic surgery assisted with Google Glass in Latin America, interacting with them and working with simultaneous three-dimensional navigation. The surgical team was interviewed by ADN radio. In January 2014, Indian Orthopedic Surgeon Selene G. Parekh conducted the foot and ankle surgery using Google Glass in Jaipur, which was broadcast live on Google website via the internet. The surgery was held during a three-day annual Indo-US conference attended by a team of experts from the US and co-organized by Ashish Sharma. Sharma said Google Glass allows looking at an X-Ray or MRI without taking the eye off of the patient and allows a doctor to communicate with a patient's family or friends during a procedure. In Australia, during January 2014, Melbourne tech startup Small World Social collaborated with the Australian Breastfeeding Association to create the first hands-free breastfeeding Google Glass application for new mothers. The application, named Google Glass Breastfeeding app trial, allows mothers to nurse their baby while viewing instructions about common breastfeeding issues (latching on, posture etc.) or call a lactation consultant via a secure Google Hangout, who can view the issue through the mother's Google Glass camera. The trial was successfully concluded in Melbourne in April 2014, and 100% of participants were breastfeeding confidently. == Display types == Various techniques have existed for see-through HMDs. Most of these techniques can be summarized into two main families: "Curved Mirror" (or Curved Combiner) based and "Waveguide" or "Light-guide" based. The mirror technique has been used in EyeTaps, by Meta in their Meta 1, by Vuzix in their Star 1200 product, by Olympus, and by Laster Technologies. Various waveguide techniques have existed for some time. These techniques include diffraction optics, holographic optics, polarized optics, reflective optics, and projection: Diffractive waveguide – slanted diffraction grating elements (nanometric 10E-9). Nokia technique now licensed to Vuzix. Holographic waveguide – 3 holographic optical elements (HOE) sandwiched together (RGB). Used by Sony and Konica Minolta. Reflective waveguide – A thick light guide with single semi-reflective mirror is used by Epson in their Moverio product. A curved light guide with partial-reflective segmented mirror array to out-couple the light is used by tooz technologies GmbH. Virtual retinal display (VRD) – Also known as a retinal scan display (RSD) or retinal projector (RP), is a display technology that draws a raster display (like a television) directly onto the retina of the eye - developed by MicroVision, Inc. OLED microdisplays for near-eye applications (outdoor optical equipment, night vision glasses, ocular equipment for medical devices, augme
Shadowrun
Shadowrun is a science fantasy tabletop role-playing game set in an alternate future in which cybernetics, magic and fantasy creatures co-exist. It combines genres of cyberpunk, urban fantasy, and crime, with occasional elements of conspiracy, horror, and detective fiction. From its inception in 1989, it has spawned a franchise that includes a series of novels, a collectible card game, two miniature-based tabletop wargames, and multiple video games. The title is taken from the game's main premise – a near-future world damaged by a massive magical event, where industrial espionage and corporate warfare runs rampant. A shadowrun – a successful data theft or physical break-in at a rival corporation or organization – is one of the main tools employed by both corporate rivals and underworld figures. Deckers (futuristic hackers) can tap into an immersive, three-dimensional cyberspace on such missions as they seek access, physical or remote, to the power structures of rival groups. They are opposed by rival deckers and lethal, potentially brain-destroying artificial intelligences called "Intrusion Countermeasures" (IC), while they are protected by street fighters and/or mercenaries, often with cyborg implants (called cyberware), magicians, and other exotic figures. Magic has also returned to the world after a series of plagues; dragons who can take human form have returned as well, and are commonly found in high positions of corporate power. == Publication history == Shadowrun was developed and published by FASA from 1989 until early 2001, when the company closed and Shadowrun was transferred to WizKids, a company founded by former FASA employees. Two years before its closure, FASA sold its videogame branch, FASA Interactive, to Microsoft corporation, keeping rights to publishing novels and pen and paper RPGs. Since then, digital rights to Shadowrun IP have belonged to Microsoft. WizKids licensed the RPG rights to Fantasy Productions, who were already publishing a German version, until WizKids was acquired by Topps in 2003. Catalyst Game Labs, a publishing imprint of InMediaRes Productions, licensed the rights from Topps to publish new products. WizKids itself produced an unsuccessful collectible action figure game based on the property, called Shadowrun Duels. A fifth edition of Shadowrun was announced in December 2012. A limited-edition softcover was sold at the Origins Game Fair in June 2013, and the PDF in July 2013. A hardcover was published in August 2013. Shadowrun Anarchy was published in October 2016 It is a simplified version of the ruleset which allows focus more on the narration than on the rules. The sixth edition, called Shadowrun, Sixth World, was announced on May 1, 2019 to coincide with the game's 30th anniversary, along with a new website at shadowrunsixthworld.com. The game was published on August 26, 2019. The mechanics for this new version are generally similar to those of fifth edition, with some rules reworked for what line developer Jason Hardy describes as streamlining. This new version also progressed the in-game year to 2080. Since 2004, Shadowrun Missions (SRM) has offered fans "living campaigns" that allow for persistent character advancement. SRM is broken down into seasons which are made up of up to 24 individual missions that can be played at home, with special missions available to play exclusively at conventions. Each SRM season develops an overarching plot focused on a specific city from the Shadowrun setting. Missions settings have included the divided city of Denver, the corporate city-state of Manhattan, the Seattle Metroplex city-state, the formerly walled-off wastelands of Chicago, and Neo-Tokyo. For Shadowrun, Sixth World missions returned to Seattle, with twenty-four missions set in 2081, right after Seattle declared independence from the UCAS. The current Shadowrun Missions setting is 2083 New Orleans. The Shadowrun role-playing game has spawned several properties, including Shadowrun: The Trading Card Game, eight video games, an action figure game (Shadowrun Duels), two magazines, an art book and more than 50 novels, starting with the Secrets of Power series which introduces some of the original characters of Shadowrun and provides an introduction to this fictional universe. In addition to the main rule book there have been over 100 published supplements including adventures and expansions to both the rules and the game settings. Catalyst Game Labs announced that 2013 would be "The Year of Shadowrun," and in addition to the release of Shadowrun fifth edition that it has collaborated with publishers on the following properties: Shadowrun: Crossfire, The Adventure Deck-building Game; Shadowrun: Sprawl Gangers, a tactical miniatures wargame; and Shadowrun: Hostile Takeover, a board game designed by Bryan C.P. Steele was planned for release in late 2014/early 2015. Catalyst had been in collaboration with Nordic Games and Cliffhanger Studios to create Shadowrun Chronicles: Boston Lockdown online RPG, however it was shuttered November 30, 2018, with the producers citing lack of funding and the end of the license terms for use of the IP. == Fictional universe == Shadowrun takes place several decades in the future (2050 in the first edition, currently 2088). The end of the Mesoamerican Long Count calendar ushered in the "Sixth World", with once-mythological beings (e.g. dragons) appearing and forms of magic suddenly emerging. Large numbers of humans have "Goblinized" into orks and trolls, while many human children are born as elves, dwarves, and even more exotic creatures. In North America, indigenous peoples discovered that their traditional ceremonies allow them to command powerful spirits, and rituals associated with a new Ghost Dance movement let them take control of much of the western U.S. and Canada, where they formed a federation of Native American Nations. Seattle remains under U.S. control by treaty as a city-state enclave, and most game materials are set there and assume campaigns will use it as their setting. In parallel with these magical developments, the setting's 21st century features technological and social developments associated with cyberpunk science fiction. Megacorporations control the lives of their employees and command their own armies; many of the largest have extraterritoriality, such as currently enjoyed by foreign heads of state. Technological advances make cyberware (mechanical replacement body parts) and bioware (augmented vat-grown body parts implanted in place of or in tandem with natural organs) common. The Computer Crash of 2029 led to the creation of the Matrix, a worldwide computer network that users interact with via direct neural interface. When conflicts arise, corporations, governments, organized crime syndicates, and even wealthy individuals subcontract their dirty work to specialists, who then perform "shadowruns" or missions undertaken by deniable assets without identities or those that wish to remain unknown. The most skilled of these specialists, called shadowrunners, have earned a reputation for getting the job done. They have developed a knack for staying alive, and prospering, in the world of Shadowrun. The Shadowrun world is cross-genre, incorporating elements of both cyberpunk and urban fantasy. Unlike in a purely cyberpunk game, in the Shadowrun world, magic exists and has "worked" since 2011. Among other things, this split humankind into subtypes, also known as metatypes/metahumans. Some of these metatypes take the form of common fantasy races. Likewise, some animals have turned into familiar monsters of past fantasy and lore and both monsters and human magicians have regained magical powers. By the second half of the 21st century, in the time the game is set, these events are accepted as commonplace. Man, machine, and magic exist in a world where the amazing is among the most common and technology has entered into every facet of human (and metahuman) life. === Races === Characters in Shadowrun can be humans, orks, trolls, elves, dwarves, as well as certain diverging subspecies (known as metavariants) such as gnomes, giants, dryads, etc. In the early days, when magic returned to the world, humans began to either change into, or give birth to, elf and dwarf infants, a phenomenon called Unexplained Genetic Expression (UGE). Later, some juvenile and adult humans "goblinized" into other races (mostly orks, but also some trolls). The term "metahuman" is used either to refer to humanity as a whole, including all races, or to refer specifically to non-human races, depending on context. The return of Halley's Comet brought even further variation in the form of changelings, who have variation atypical to their metatype or even species, such as electroreception. Two of the metahuman races, elves and orks, have fictional languages. Additionally, a virus known as the Human Meta-Human Vampiric Virus (HMHVV), with many variant strains, has been known to cause f
Tensor network
Tensor networks or tensor network states are a class of variational wave functions used in the study of many-body quantum systems and fluids. Tensor networks extend one-dimensional matrix product states to higher dimensions while preserving some of their useful mathematical properties. The wave function is encoded as a tensor contraction of a network of individual tensors. The structure of the individual tensors can impose global symmetries on the wave function (such as antisymmetry under exchange of fermions) or restrict the wave function to specific quantum numbers, like total charge, angular momentum, or spin. It is also possible to derive strict bounds on quantities like entanglement and correlation length using the mathematical structure of the tensor network. This has made tensor networks useful in theoretical studies of quantum information in many-body systems. They have also proved useful in variational studies of ground states, excited states, and dynamics of strongly correlated many-body systems. == Diagrammatic notation == In general, a tensor network diagram (Penrose diagram) can be viewed as a graph where nodes (or vertices) represent individual tensors, while edges represent summation over an index. Free indices are depicted as edges (or legs) attached to a single vertex only. Sometimes, there is also additional meaning to a node's shape. For instance, one can use trapezoids for unitary matrices or tensors with similar behaviour. This way, flipped trapezoids would be interpreted as complex conjugates to them. == History == Foundational research on tensor networks began in 1971 with a paper by Roger Penrose. In "Applications of negative dimensional tensors" Penrose developed tensor diagram notation, describing how the diagrammatic language of tensor networks could be used in applications in physics. In 1992, Steven R. White developed the density matrix renormalization group (DMRG) for quantum lattice systems. The DMRG was the first successful tensor network and associated algorithm. In 2002, Guifré Vidal and Reinhard Werner attempted to quantify entanglement, laying the groundwork for quantum resource theories. This was also the first description of the use of tensor networks as mathematical tools for describing quantum systems. In 2004, Frank Verstraete and Ignacio Cirac developed the theory of matrix product states, projected entangled pair states, and variational renormalization group methods for quantum spin systems. In 2006, Vidal developed the multi-scale entanglement renormalization ansatz (MERA). In 2007 he developed entanglement renormalization for quantum lattice systems. In 2010, Ulrich Schollwock developed the density-matrix renormalization group for the simulation of one-dimensional strongly correlated quantum lattice systems. In 2014, Román Orús introduced tensor networks for complex quantum systems and machine learning, as well as tensor network theories of symmetries, fermions, entanglement and holography. == Connection to machine learning == Tensor networks have been adapted for supervised learning, taking advantage of similar mathematical structure in variational studies in quantum mechanics and large-scale machine learning. This crossover has spurred collaboration between researchers in artificial intelligence and quantum information science. In June 2019, Google, the Perimeter Institute for Theoretical Physics, and X (company), released TensorNetwork, an open-source library for efficient tensor calculations. The main interest in tensor networks and their study from the perspective of machine learning is to reduce the number of trainable parameters (in a layer) by approximating a high-order tensor with a network of lower-order ones. Using the so-called tensor train technique (TT), one can reduce an N-order tensor (containing exponentially many trainable parameters) to a chain of N tensors of order 2 or 3, which gives us a polynomial number of parameters.
Kolmogorov–Arnold Networks
Kolmogorov–Arnold Networks (KANs) are a type of artificial neural network architecture inspired by the Kolmogorov–Arnold representation theorem, also known as the superposition theorem. Unlike traditional multilayer perceptrons (MLPs), which rely on fixed activation functions and linear weights, KANs replace each weight with a learnable univariate function, often represented using splines. == History == KANs (Kolmogorov–Arnold Networks) were proposed by Liu et al. (2024) as a generalization of the Kolmogorov–Arnold representation theorem (KART), aiming to outperform MLPs in small-scale AI and scientific tasks. Before KANs, numerous studies explored KART's connections to neural networks or used it as a basis for designing new network architectures. In the 1980s and 1990s, early research applied KART to neural network design. Kůrková et al. (1992), Hecht-Nielsen (1987), and Nees (1994) established theoretical foundations for multilayer networks based on KART. Igelnik et al. (2003) introduced the Kolmogorov Spline Network using cubic splines to model complex functions. Sprecher (1996, 1997) introduced numerical methods for building network layers, while Nakamura et al. (1993) created activation functions with guaranteed approximation accuracy. These works linked KART's theoretical potential with practical neural network implementation. KART has also been used in other computational and theoretical fields. Coppejans (2004) developed nonparametric regression estimators using B-splines, Bryant (2008) applied it to high-dimensional image tasks, Liu (2015) investigated theoretical applications in optimal transport and image encryption, and more recently, Polar and Poluektov (2021) used Urysohn operators for efficient KART construction, while Fakhoury et al. (2022) introduced ExSpliNet, integrating KART with probabilistic trees and multivariate B-splines for improved function approximation. == Architecture == KANs are based on the Kolmogorov–Arnold representation theorem, which was linked to the 13th Hilbert problem. Given x = ( x 1 , x 2 , … , x n ) {\displaystyle x=(x_{1},x_{2},\dots ,x_{n})} consisting of n variables, a multivariate continuous function f ( x ) {\displaystyle f(x)} can be represented as: f ( x ) = f ( x 1 , … , x n ) = ∑ q = 1 2 n + 1 Φ q ( ∑ p = 1 n φ q , p ( x p ) ) {\displaystyle f(x)=f(x_{1},\dots ,x_{n})=\sum _{q=1}^{2n+1}\Phi _{q}\left(\sum _{p=1}^{n}\varphi _{q,p}(x_{p})\right)} (1) This formulation contains two nested summations: an outer and an inner sum. The outer sum ∑ q = 1 2 n + 1 {\displaystyle \sum _{q=1}^{2n+1}} aggregates 2 n + 1 {\displaystyle 2n+1} terms, each involving a function Φ q : R → R {\displaystyle \Phi _{q}:\mathbb {R} \to \mathbb {R} } . The inner sum ∑ p = 1 n {\displaystyle \sum _{p=1}^{n}} computes n terms for each q, where each term φ q , p : [ 0 , 1 ] → R {\displaystyle \varphi _{q,p}:[0,1]\to \mathbb {R} } is a continuous function of the single variable x p {\displaystyle x_{p}} . The inner continuous functions φ q , p {\displaystyle \varphi _{q,p}} are universal, independent of f {\displaystyle f} , while the outer functions Φ q {\displaystyle \Phi _{q}} depend on the specific function f {\displaystyle f} being represented. The representation (1) holds for all multivariate functions f {\displaystyle f} as proved in . If f {\displaystyle f} is continuous, then the outer functions Φ q {\displaystyle \Phi _{q}} are continuous; if f {\displaystyle f} is discontinuous, then the corresponding Φ q {\displaystyle \Phi _{q}} are generally discontinuous, while the inner functions φ q , p {\displaystyle \varphi _{q,p}} remain the same universal functions. Liu et al. proposed the name KAN. A general KAN network consisting of L layers takes x to generate the output as: K A N ( x ) = ( Φ L − 1 ∘ Φ L − 2 ∘ ⋯ ∘ Φ 1 ∘ Φ 0 ) x {\displaystyle \mathrm {KAN} (x)=(\Phi ^{L-1}\circ \Phi ^{L-2}\circ \cdots \circ \Phi ^{1}\circ \Phi ^{0})x} (3) Here, Φ l {\displaystyle \Phi ^{l}} is the function matrix of the l-th KAN layer or a set of pre-activations. Let i denote the neuron of the l-th layer and j the neuron of the (l+1)-th layer. The activation function φ j , i l {\displaystyle \varphi _{j,i}^{l}} connects (l, i) to (l+1, j): φ j , i l , l = 0 , … , L − 1 , i = 1 , … , n l , j = 1 , … , n l + 1 {\displaystyle \varphi _{j,i}^{l},\quad l=0,\dots ,L-1,\;i=1,\dots ,n_{l},\;j=1,\dots ,n_{l+1}} (4) where nl is the number of nodes of the l-th layer. Thus, the function matrix Φ l {\displaystyle \Phi ^{l}} can be represented as an n l + 1 × n l {\displaystyle n_{l+1}\times n_{l}} matrix of activations: x l + 1 = ( φ 1 , 1 l ( ⋅ ) φ 1 , 2 l ( ⋅ ) ⋯ φ 1 , n l l ( ⋅ ) φ 2 , 1 l ( ⋅ ) φ 2 , 2 l ( ⋅ ) ⋯ φ 2 , n l l ( ⋅ ) ⋮ ⋮ ⋱ ⋮ φ n l + 1 , 1 l ( ⋅ ) φ n l + 1 , 2 l ( ⋅ ) ⋯ φ n l + 1 , n l l ( ⋅ ) ) x l {\displaystyle x^{l+1}={\begin{pmatrix}\varphi _{1,1}^{l}(\cdot )&\varphi _{1,2}^{l}(\cdot )&\cdots &\varphi _{1,n_{l}}^{l}(\cdot )\\\varphi _{2,1}^{l}(\cdot )&\varphi _{2,2}^{l}(\cdot )&\cdots &\varphi _{2,n_{l}}^{l}(\cdot )\\\vdots &\vdots &\ddots &\vdots \\\varphi _{n_{l+1},1}^{l}(\cdot )&\varphi _{n_{l+1},2}^{l}(\cdot )&\cdots &\varphi _{n_{l+1},n_{l}}^{l}(\cdot )\end{pmatrix}}x^{l}} == Implementations == To make the KAN layers optimizable, the inner function is formed by the combination of spline and basic functions as the formula: φ ( x ) = w b b ( x ) + w s spline ( x ) {\displaystyle \varphi (x)=w_{b}\,b(x)+w_{s}\,{\text{spline}}(x)} where b ( x ) {\displaystyle b(x)} is the basic function, usually defined as s i l u ( x ) = x / ( 1 + e x ) {\displaystyle silu(x)=x/(1+e^{x})} and w b {\displaystyle w_{b}} is the base weight matrix. Also, w s {\displaystyle w_{s}} is the spline weight matrix and spline ( x ) {\displaystyle {\text{spline}}(x)} is the spline function. The spline function can be a sum of B-splines. spline ( x ) = ∑ i c i B i ( x ) {\displaystyle {\text{spline}}(x)=\sum _{i}c_{i}B_{i}(x)} Many studies suggested to use other polynomial and curve functions instead of B-spline to create new KAN variants. == Functions used == The choice of functional basis strongly influences the performance of KANs. Common function families include: B-splines: Provide locality, smoothness, and interpretability; they are the most widely used in current implementations. RBFs (include Gaussian RBFs): Capture localized features in data and are effective in approximating functions with non-linear or clustered structures. Chebyshev polynomials: Offer efficient approximation with minimized error in the maximum norm, making them useful for stable function representation. Rational function: Useful for approximating functions with singularities or sharp variations, as they can model asymptotic behavior better than polynomials. Fourier series: Capture periodic patterns effectively and are particularly useful in domains such as physics-informed machine learning. Wavelet functions (DoG, Mexican hat, Morlet, and Shannon): Used for feature extraction as they can capture both high-frequency and low-frequency data components. Piecewise linear functions: Provide efficient approximation for multivariate functions in KANs. == Usage == In some modern neural architectures like convolutional neural networks (CNNs), recurrent neural networks (RNNs), and Transformers, KANs are typically used as drop-in substitutes for MLP layers. Despite KANs' general-purpose design, researchers have created and used them for a number of tasks: Scientific machine learning (SciML): Function fitting, partial differential equations (PDEs) and physical/mathematical laws. Continual learning: KANs better preserve previously learned information during incremental updates, avoiding catastrophic forgetting due to the locality of spline adjustments. Graph neural networks: Extensions such as Kolmogorov–Arnold Graph Neural Networks (KA-GNNs) integrate KAN modules into message-passing architectures, showing improvements in molecular property prediction tasks. Sensor data processing: Kolmogorov–Arnold Networks (KANs) have recently been applied to sensor data processing due to their ability to model complex nonlinear relationships with relatively few parameters and improved interpretability compared to conventional multilayer perceptrons. Applications include industrial soft sensors, biomedical signal analysis, remote sensing, and environmental monitoring systems. == Drawbacks == KANs can be computationally intensive and require a large number of parameters due to their use of polynomial functions to capture data.
Nature Manifesto
Nature Manifesto is an Immersive sound piece and multimedia installation by Icelandic artist Björk and artist and curator Aleph Molinari, created in collaboration with the French Institute for Research and Coordination in Acoustics/Music (IRCAM). The installation was showcased at the Centre Pompidou in Paris, France from November 20, 2024 to December 9, 2024, as part of the museum's "Biodiversity: Which Culture for Which Future?" forum. It combines natural soundscapes, calls of extinct animals reconstructed through artificial intelligence, and Björk's narration to address damages to biodiversity and the collapse of ecosystems. == Background == Björk's work intricately weaves themes of nature and technology, reflecting her deep engagement with both realms. In 2008, she co-founded the Náttúra campaign to protest the construction of foreign-backed aluminum factories in Iceland, aiming to protect the country's natural landscapes. She released the single "Náttúra" featuring Thom Yorke, with all proceeds supporting this environmental initiative. Her 2011 album Biophilia further exemplifies this synthesis, exploring the relationships between music, nature, and technology through a multimedia project that included interactive apps, custom-made instruments, and educational workshops. Björk's Cornucopia tour (2019-2023) seamlessly integrates themes of nature preservation and environmental activism, and featured a recorded message by Swedish climate activist Greta Thunberg. The tour's fusion of music, technology, and natural imagery reflects Björk's vision of a harmonious coexistence between humanity and nature, advocating for sustainable futures. Björk has previously used artificial intelligence in her works. In 2020, she collaborated with Microsoft to create Kórsafn, a sound installation for the Sister City Hotel lobby in New York City which used an AI-powered model that elaborated choral recordings from her discography through a sensor on the rooftop of the building that would generate music according to data like the weather and the seasons. For her charity single "Oral", featuring Spanish singer Rosalía, she released a music video directed by photographer and visual artist Carlota Guerrero, who used AI-generated deepfake versions of the artists. == Concept == Nature Manifesto is a three-minute and forty-second immersive sound piece. The composition merges Björk's voice, as she articulates a manifesto on biodiversity and the climate crisis, with cries of extinct and endangered animals, harmonizing them with natural soundscapes. The installation was curated by Chloé Siganos and Aleph Molinari, with associate curator Delphine Le Gatt. The primary goal of Nature Manifesto is to foster a deeper understanding of humanity's impact on the natural world. Conceived as a "post-optimistic" manifesto, Aleph Molinari stated that the project's purpose was to "offer a voice to nature". He stated that "the modern concept of nature itself is problematic [...] because it’s a concept born in the Romantic period and, with the rise of the industrial era, became an antithesis to human civilisation and everything urban. Nature came to define what was outside, the savage Other... But nature is everything that we’re part of." The soundscape features recreated calls of extinct and endangered species, developed in collaboration with the French sound research institute IRCAM. Artificial intelligence was employed to simulate the vocalizations of animals that no longer exist in the wild. To save energy and lessen the ecological impact of the use of AI, the research institute developed a "frugal AI" model capable of generating audio in real-time on local servers without a graphics processing unit. The sounds were then produced and edited by Björk in collaboration with Robin Meier Wiratunga and Bergur Þórisson. The installation was located within the Centre Pompidou's escalator, known as the "caterpillar". The installation was further supported by videos created by visual artist Sam Balfus (also known as Balfua) by using artificial intelligence, and edited by Santiago Molinari. == Activism == To sustain and broaden the themes presented in Nature Manifesto, Björk publicly urged French President Emmanuel Macron to prohibit bottom trawling within France's marine protected areas (MPA). She criticized the French government's claim of protecting 30% of its marine territories, highlighting that over 90% of these MPAs exist only on paper, allowing destructive practices like bottom trawling to continue unchecked. She collaborated with non-governmental organizations Sustainable Ocean Alliance, Ungir umhverfissinnar and Bloom, to advocate for genuine ocean conservation. Björk promoted the cause through her social media profiles by sharing petitions. In November 2024, Björk lent her Instagram account to French environmental activists to directly address Macron. The activists used the platform to call for stronger protection of the ocean, urging Macron to impose stricter restrictions on harmful fishing practices, particularly bottom trawling. == Reception == Nature Manifesto received mixed to positive reviews from critics. Some critiques focused on the installation's setting, suggesting that the movement inherent to the escalator space diminished the immersive potential of the soundscape. The choice of using artificial intelligence was also questioned. Björk and Molinari defended this, as both see AI as a tool that can be used creatively and sustainably, with Björk focusing on the importance of human input to give AI a "soul", and Molinari stressing the need for sustainable technological practices in the broader context of digital life. After the exhibition ended, Björk further opinionated: "this is how we will work in the future. [...] if there is no soul in tomorrow's music made by AI it is because [no one] put it there and we have to speak out and guard this as listeners", further stating that there is already "soulless muzak" [sic] on Spotify, "mass manufactured without the attention of creativity".