Jaime Guillermo Carbonell (July 29, 1953 – February 28, 2020) was a computer scientist who made seminal contributions to the development of natural language processing tools and technologies. His research in machine translation resulted in the development of several state-of-the-art language translation and artificial intelligence systems. He earned his B.S. degrees in Physics and in Mathematics from MIT in 1975 and did his Ph.D. under Dr. Roger Schank at Yale University in 1979. He joined Carnegie Mellon University as an assistant professor of computer science in 1979 and moved to Pittsburgh. He was affiliated with the Language Technologies Institute, Computer Science Department, Machine Learning Department, and Computational Biology Department at Carnegie Mellon. His interests spanned several areas of artificial intelligence, language technologies and machine learning. In particular, his research focused on areas such as text mining (extraction, categorization, novelty detection) and in new theoretical frameworks such as a unified utility-based theory bridging information retrieval, summarization, free-text question-answering and related tasks. He also worked on machine translation, both high-accuracy knowledge-based MT and machine learning for corpus-based MT (such as generalized example-based MT). == Career == Carbonell was the Allen Newell Professor of Computer Science and head of the Language Technologies Institute at Carnegie Mellon University. He joined Carnegie Mellon in 1979, and became a key faculty member in the artificial intelligence area. He was appointed full professor in 1987, Newell Chair in 1995, and University Professor in 2012. He completed his undergraduate studies at MIT. He received dual degrees in Mathematics and Physics. He received his Ph.D. in computer science from Yale University in 1979. At the time of his appointment, Carbonell was the youngest chaired professor in the School of Computer Science at CMU. His research spanned several areas of computer science, mostly in artificial intelligence, including: machine learning, data and text mining, natural language processing, very-large-scale knowledge bases, translingual information retrieval and automated summarization. He wrote more than 300 technical papers and gave over 500 invited or refereed-paper presentations (colloquia, seminars, panels, conferences, keynotes, etc.). He died following a long illness on February 28, 2020. Mona Talat Diab became the director of CMU's Language Technologies Institute in 2023. == Research == Carbonell created MMR (maximal marginal relevance) technology for text summarization and informational novelty detection in search engines, invention of transformational analogy, a generalized method for case-based reasoning (CBR) to re-use, modify and compose past successful plans for increasingly complex problems and knowledge-based interlingual machine translation. He was instrumental in setting up the Computational Biolinguistics Program, a joint venture between Carnegie Mellon and the University of Pittsburgh, which combines Language Technologies and Machine Learning to model and predict genomic, proteomic and glycomic 3D structures. Carbonell also did work in machine learning. He organized the first four machine learning conferences, starting with CMU in 1981. The Language Technologies Institute (LTI), founded and directed by Carbonell, achieved top honors in multiple areas. These areas include machine translation, search engines (including founding of Lycos by Michael Mauldin, one of Carbonell’s PhD students), speech synthesis, and education. LTI remains the original, largest and best-known institute for language technologies, with over $12M in annual funding and 200 researchers (faculty, staff, PhD students, MS students, visiting scholars etc.). Carbonell made major technical contributions in several fields, including (1) Creation of MMR (maximal marginal relevance) technology for text summarization and informational novelty detection in search engines,(2) Proactive machine learning for multi-source cost-sensitive active learning, (3) Linked conditional random fields for predicting tertiary and quaternary protein folds, (4) Symmetric optimal phrasal alignment method for trainable example-based and statistical machine translation, (5) Series- anomaly modeling for financial fraud detection and syndromic surveillance, (6) Knowledge-based interlingual machine translation, (7) Robust case-frame parsing, (8) Seeded version-space learning and (9) Invention of transformational and derivational analogy, generalized methods for case-based reasoning (CBR) to re-use, modify and compose past successful plans for increasingly complex problems. The teams led by Carbonell achieved top honors in many areas such as first scalable high-accuracy interlingual machine translation (1991), first speech-to-speech machine translation (1992), first large-scale spider and search engine (1994), and first trainable, large-scale protein-structure topology predictor (2005). Modern machine learning, co-founded by Carbonell, Michalski and Mitchell, is a fundamental enabling technology in search engines, data mining and social networking. Starting in 1980, he co-edited the first three books on ML, launched the ML conferences and was a co-founder and editor-in-chief of ML Journal. Carbonell’s innovations have led to several successful start-ups: Carnegie Group (AI expertsystems), Lycos (web search), Wisdom (financial optimization & ML), Carnegie Speech (spoken-language tutoring), Dynamix (data mining and pattern discovery), and Meaningful Machines (context-based machine translation). Carbonell was the founding director of The Language Technology Institute, the preeminent global institution in language studies, unparalleled in size and scope and has since been adopted/imitated in Germany (DFKI), Japan (Tokyo Univ.), and the US (Johns Hopkins). == Awards and honors == Okawa Prize, 2015 Best paper award, “Translingual Search” w/Yang, International Joint Conference on AI, 1997 Allen Newell endowed chair, Carnegie Mellon University, 1995 Elected fellow of AAAI, 1991 Computer Science teaching award, Carnegie Mellon University, 1987 Sperry Fellowship for excellence in AI research, 1986 Herbert Simon teaching award, 1986 "Recognition of Service" award from the ACM for the SIGART presidency, 1983–1985 Provided congressional testimony on machine translation, 1990 == Selected works == === Books === 1983. (with Ryszard S. Michalski & Tom M. Mitchell, Eds.) Machine learning: An artificial intelligence approach. Los Altos, CA: Morgan Kaufmann. 1986. (with Ryszard S. Michalski & Tom Mitchell, Eds.) Machine learning: An artificial intelligence approach. Vol. II. Los Altos, CA: Morgan-Kaufmann. 1986. (with Ryszard S. Michalski & Tom Mitchell, Eds.) Machine Learning: A Guide to Current Research. Kluwer Academic Publishers. == Contributions == “Protein Quaternary Fold Recognition Using Conditional Graphical Models” IJCAI 2007 (w/Liu et al.) “Context-Based Machine Translation” AMTA 2006 (w/Klein et al.) “SCRFs: A New Approach for Protein Fold Recognition,’’ Journal of Computational Biology, 13,2, 2006 (w/Liu et al) “MT for Resource-Poor Languages Using Elicitation-Based Learning” Machine Translation, 2004 ‘‘Learning Approaches for Detecting and Tracking News Events,’’ IEEE Trans I.S., 14, 4, 2000 (w/Yang)
Normal distributions transform
The normal distributions transform (NDT) is a point cloud registration algorithm introduced by Peter Biber and Wolfgang Straßer in 2003, while working at University of Tübingen. The algorithm registers two point clouds by first associating a piecewise normal distribution to the first point cloud, that gives the probability of sampling a point belonging to the cloud at a given spatial coordinate, and then finding a transform that maps the second point cloud to the first by maximising the likelihood of the second point cloud on such distribution as a function of the transform parameters. Originally introduced for 2D point cloud map matching in simultaneous localization and mapping (SLAM) and relative position tracking, the algorithm was extended to 3D point clouds and has wide applications in computer vision and robotics. NDT is very fast and accurate, making it suitable for application to large scale data, but it is also sensitive to initialisation, requiring a sufficiently accurate initial guess, and for this reason it is typically used in a coarse-to-fine alignment strategy. == Formulation == The NDT function associated to a point cloud is constructed by partitioning the space in regular cells. For each cell, it is possible to define the mean q = 1 n ∑ i x i {\displaystyle \textstyle \mathbf {q} ={\frac {1}{n}}\sum _{i}\mathbf {x_{i}} } and covariance S = 1 n ∑ i ( x i − q ) ( x i − q ) ⊤ {\displaystyle \textstyle \mathbf {S} ={\frac {1}{n}}\sum _{i}\left(\mathbf {x} _{i}-\mathbf {q} \right)\left(\mathbf {x} _{i}-\mathbf {q} \right)^{\top }} of the n {\displaystyle n} points of the cloud x 1 , … , x n {\displaystyle \mathbf {x} _{1},\dots ,\mathbf {x} _{n}} that fall within the cell. The probability density of sampling a point at a given spatial location x {\displaystyle \mathbf {x} } within the cell is then given by the normal distribution e − 1 2 ( x − q ) ⊤ S − 1 ( x − q ) {\displaystyle e^{-{\frac {1}{2}}\left(\mathbf {x} -\mathbf {q} \right)^{\top }\mathbf {S} ^{-1}\left(\mathbf {x} -\mathbf {q} \right)}} . Two point clouds can be mapped by a Euclidean transformation f {\displaystyle f} with rotation matrix R {\displaystyle \mathbf {R} } and translation vector t {\displaystyle \mathbf {t} } f R , t ( x ) = R x + t {\displaystyle f_{\mathbf {R} ,\mathbf {t} }(\mathbf {x} )=\mathbf {R} \mathbf {x} +\mathbf {t} } that maps from the second cloud to the first, parametrised by the rotation angles and translation components. The algorithm registers the two point clouds by optimising the parameters of the transformation that maps the second cloud to the first, with respect to a loss function based on the NDT of the first point cloud, solving the following problem arg min R , t { − ∑ i NDT ( f R , t ( x i ) ) } {\displaystyle \arg \min _{\mathbf {R} ,\mathbf {t} }\left\{-\sum _{i}\operatorname {NDT} \left(f_{\mathbf {R} ,\mathbf {t} }\left(\mathbf {x_{i}} \right)\right)\right\}} where the loss function represents the negated likelihood, obtained by applying the transformation to all points in the second cloud and summing the value of the NDT at each transformed point f R , t ( x ) {\displaystyle f_{\mathbf {R} ,\mathbf {t} }(\mathbf {x} )} . The loss is piecewise continuous and differentiable, and can be optimised with gradient-based methods (in the original formulation, the authors use Newton's method). In order to reduce the effect of cell discretisation, a technique consists of partitioning the space into multiple overlapping grids, shifted by half cell size along the spatial directions, and computing the likelihood at a given location as the sum of the NDTs induced by each grid.
ACROSS Project
ACROSS is a Singular Strategic R&D Project led by Treelogic funded by the Spanish Ministry of Industry, Tourism and Trade activities in the field of Robotics and Cognitive Computing over an execution time-frame from 2009 to 2011. ACROSS project involves a number higher than 100 researchers from 13 Spanish entities. == ACROSS project objectives == ACROSS modifies the design of social robotics, blocked in providing predefined services, going further by means of intelligent systems. These systems are able to self-reconfigure and modify their behavior autonomously through the capacity for understanding, learning and software remote access. In order to provide an open framework for collaboration between universities, research centers and the Administration, ACROSS develops Open Source Services available to everybody. == Three application domains == ACROSS works in three application domains: Autonomous living: robots are used as technological tools to help handicapped person into daily tasks. Psycho-Affective Disorders (autism): robots are used to mitigate cognitive disorders. Marketing: robots are used to interact with humans in a recreational approach. == Consortium == Treelogic Alimerka Bizintek Universitat Politécnica de Catalunya University of Deusto European Centre for Soft Computing Fatronik - Tecnalia Fundació Hospital Comarcal Sant Antoni Abat Fundación Pública Andaluza para la Gestión de la Investigación en Salud de Sevilla, "Virgen del Rocío" University Hospitals m-BOT Omicron Electronic Universidad de Extremadura - RoboLab Verbio Technologies
Mira Murati
Ermira "Mira" Murati (born 16 December 1988) is an Albanian-American business executive. She launched an AI startup called Thinking Machines Lab in February 2025. Previously she was the chief technology officer of OpenAI, and a senior product manager at Tesla. == Early life and education == Murati was born on 16 December 1988 in Vlorë, Albania. She is fluent in Italian. At age 16, she won a United World Colleges (UWC) scholarship to study at Pearson College on Vancouver Island in Canada, from which she graduated in 2007 with an International Baccalaureate. After Pearson, she went to the United States to pursue further studies through a dual-degree program, earning a Bachelor of Arts from Colby College in 2011, and a Bachelor of Engineering degree from Dartmouth College's Thayer School of Engineering in 2012. == Career == === Early career === Murati interned in 2011 as a summer analyst at Goldman Sachs in Tokyo, Japan. She then briefly worked for Zodiac Aerospace as an intern before joining the electric car company Tesla in 2013 as a product manager on the Model X. From 2016 to 2018, she worked for the augmented reality start-up Leap Motion (now Ultraleap). === OpenAI === In 2018, she joined OpenAI as the VP of Applied AI and partnerships. She became chief technology officer (CTO) in May 2022. She led OpenAI's work on ChatGPT, Dall-E, Codex and Sora, while overseeing its research, product and safety teams. She oversaw technical advancements and direction of OpenAI's various projects, including the development of advanced AI models and tools. Murati worked on several of OpenAI's notable products, such as the Generative Pretrained Transformer (GPT) series of language models. Commenting about the potential loss of creative jobs to AI, Murati said that "maybe [the jobs] shouldn’t have been there in the first place". In October 2023, Murati was ranked 57th on Fortune's list of "The 100 Most Powerful Women in Business of 2023". In November 2023, Murati became interim chief executive officer of OpenAI following the removal of Sam Altman from the job. She had collaborated with Ilya Sutskever, whose 52-page memo outlining concerns about Altman relied heavily on screenshots and information she provided, which contributed to the board's decision to oust him. Murati was replaced by Emmett Shear three days later, who left when Altman was reinstated five days later. Following these events, Murati returned to her role as CTO. In June 2024, Dartmouth College awarded Murati an honorary Doctor of Science for having "democratized technology and advanced a better, safer world for us all". In September 2024, Murati announced that she was stepping down as CTO to allow her the opportunity to "do my own exploration". This move came amid a wider executive exodus as OpenAI chief research officer Bob McGrew and a vice president of research, Barret Zoph, also announced their departures soon after. === Thinking Machines Lab === In February 2025, Murati launched Thinking Machines Lab, a new public benefit corporation aiming "to make AI systems more widely understood, customizable, and generally capable". She was reported to have hired "a team of about 30 leading researchers and engineers from competitors including Meta, Mistral, and OpenAI." People involved with the startup include OpenAI cofounder John Schulman, and advisors Alec Radford and Bob McGrew. The following month, Bloomberg reported that the company had reached an estimated valuation of $9 billion, with an "average founder stake value" of $1.4 billion. In April 2025, Thinking Machines Lab reportedly aimed for a $2 billion seed round (requiring a minimum investment of $50 million). The round was led by Andreessen Horowitz and included participation from the government of Albania, valuing the company at $12 billion. Thinking Machines Lab follows a governance structure wherein Mira Murati holds a deciding vote on board matters, weighted to provide her with a majority decision-making capability. In October 2025, Thinking Machines Lab announced its first product, Tinker, a tool used to create custom frontier AI models. == Publications == Murati, Ermira (Spring 2022). "Language & Coding Creativity". Daedalus. 151 (2). Cambridge, MA: American Academy of Arts and Sciences (AAAS): 156–167. doi:10.1162/daed_a_01907. Retrieved 25 September 2024.
Linear belief function
Linear belief functions are an extension of the Dempster–Shafer theory of belief functions to the case when variables of interest are continuous. Examples of such variables include financial asset prices, portfolio performance, and other antecedent and consequent variables. The theory was originally proposed by Arthur P. Dempster in the context of Kalman Filters and later was elaborated, refined, and applied to knowledge representation in artificial intelligence and decision making in finance and accounting by Liping Liu. == Concept == A linear belief function intends to represent our belief regarding the location of the true value as follows: We are certain that the truth is on a so-called certainty hyperplane but we do not know its exact location; along some dimensions of the certainty hyperplane, we believe the true value could be anywhere from –∞ to +∞ and the probability of being at a particular location is described by a normal distribution; along other dimensions, our knowledge is vacuous, i.e., the true value is somewhere from –∞ to +∞ but the associated probability is unknown. A belief function in general is defined by a mass function over a class of focal elements, which may have nonempty intersections. A linear belief function is a special type of belief function in the sense that its focal elements are exclusive, parallel sub-hyperplanes over the certainty hyperplane and its mass function is a normal distribution across the sub-hyperplanes. Based on the above geometrical description, Shafer and Liu propose two mathematical representations of a LBF: a wide-sense inner product and a linear functional in the variable space, and as their duals over a hyperplane in the sample space. Monney proposes still another structure called Gaussian hints. Although these representations are mathematically neat, they tend to be unsuitable for knowledge representation in expert systems. == Knowledge representation == A linear belief function can represent both logical and probabilistic knowledge for three types of variables: deterministic such as an observable or controllable, random whose distribution is normal, and vacuous on which no knowledge bears. Logical knowledge is represented by linear equations, or geometrically, a certainty hyperplane. Probabilistic knowledge is represented by a normal distribution across all parallel focal elements. In general, assume X is a vector of multiple normal variables with mean μ and covariance Σ. Then, the multivariate normal distribution can be equivalently represented as a moment matrix: M ( X ) = ( μ Σ ) . {\displaystyle M(X)=\left({\begin{array}{{20}c}\mu \\\Sigma \end{array}}\right).} If the distribution is non-degenerate, i.e., Σ has a full rank and its inverse exists, the moment matrix can be fully swept: M ( X → ) = ( μ Σ − 1 − Σ − 1 ) {\displaystyle M({\vec {X}})=\left({\begin{array}{{20}c}\mu \Sigma ^{-1}\\-\Sigma ^{-1}\end{array}}\right)} Except for normalization constant, the above equation completely determines the normal density function for X. Therefore, M ( X → ) {\displaystyle M({\vec {X}})} represents the probability distribution of X in the potential form. These two simple matrices allow us to represent three special cases of linear belief functions. First, for an ordinary normal probability distribution M(X) represents it. Second, suppose one makes a direct observation on X and obtains a value μ. In this case, since there is no uncertainty, both variance and covariance vanish, i.e., Σ = 0. Thus, a direct observation can be represented as: M ( X ) = ( μ 0 ) {\displaystyle M(X)=\left({\begin{array}{{20}c}\mu \\0\end{array}}\right)} Third, suppose one is completely ignorant about X. This is a very thorny case in Bayesian statistics since the density function does not exist. By using the fully swept moment matrix, we represent the vacuous linear belief functions as a zero matrix in the swept form follows: M ( X → ) = [ 0 0 ] {\displaystyle M({\vec {X}})=\left[{\begin{array}{{20}c}0\\0\end{array}}\right]} One way to understand the representation is to imagine complete ignorance as the limiting case when the variance of X approaches to ∞, where one can show that Σ−1 = 0 and hence M ( X → ) {\displaystyle M({\vec {X}})} vanishes. However, the above equation is not the same as an improper prior or normal distribution with infinite variance. In fact, it does not correspond to any unique probability distribution. For this reason, a better way is to understand the vacuous linear belief functions as the neutral element for combination (see later). To represent the remaining three special cases, we need the concept of partial sweeping. Unlike a full sweeping, a partial sweeping is a transformation on a subset of variables. Suppose X and Y are two vectors of normal variables with the joint moment matrix: M ( X , Y ) = [ μ 1 Σ 11 Σ 21 μ 2 Σ 12 Σ 22 ] {\displaystyle M(X,Y)=\left[{\begin{array}{{20}c}{\begin{array}{{20}c}\mu _{1}\\\Sigma _{11}\\\Sigma _{21}\end{array}}&{\begin{array}{{20}c}\mu _{2}\\\Sigma _{12}\\\Sigma _{22}\end{array}}\end{array}}\right]} Then M(X, Y) may be partially swept. For example, we can define the partial sweeping on X as follows: M ( X → , Y ) = [ μ 1 ( Σ 11 ) − 1 − ( Σ 11 ) − 1 Σ 21 ( Σ 11 ) − 1 μ 2 − μ 1 ( Σ 11 ) − 1 Σ 12 ( Σ 11 ) − 1 Σ 12 Σ 22 − Σ 21 ( Σ 11 ) − 1 Σ 12 ] {\displaystyle M({\vec {X}},Y)=\left[{\begin{array}{{20}c}{\begin{array}{{20}c}\mu _{1}(\Sigma _{11})^{-1}\\-(\Sigma _{11})^{-1}\\\Sigma _{21}(\Sigma _{11})^{-1}\end{array}}&{\begin{array}{{20}c}\mu _{2}-\mu _{1}(\Sigma _{11})^{-1}\Sigma _{12}\\(\Sigma _{11})^{-1}\Sigma _{12}\\\Sigma _{22}-\Sigma _{21}(\Sigma _{11})^{-1}\Sigma _{12}\end{array}}\end{array}}\right]} If X is one-dimensional, a partial sweeping replaces the variance of X by its negative inverse and multiplies the inverse with other elements. If X is multidimensional, the operation involves the inverse of the covariance matrix of X and other multiplications. A swept matrix obtained from a partial sweeping on a subset of variables can be equivalently obtained by a sequence of partial sweepings on each individual variable in the subset and the order of the sequence does not matter. Similarly, a fully swept matrix is the result of partial sweepings on all variables. We can make two observations. First, after the partial sweeping on X, the mean vector and covariance matrix of X are respectively μ 1 ( Σ 11 ) − 1 {\displaystyle \mu _{1}(\Sigma _{11})^{-1}} and − ( Σ 11 ) − 1 {\displaystyle -(\Sigma _{11})^{-1}} , which are the same as that of a full sweeping of the marginal moment matrix of X. Thus, the elements corresponding to X in the above partial sweeping equation represent the marginal distribution of X in potential form. Second, according to statistics, μ 2 − μ 1 ( Σ 11 ) − 1 Σ 12 {\displaystyle \mu _{2}-\mu _{1}(\Sigma _{11})^{-1}\Sigma _{12}} is the conditional mean of Y given X = 0; Σ 22 − Σ 21 ( Σ 11 ) − 1 Σ 12 {\displaystyle \Sigma _{22}-\Sigma _{21}(\Sigma _{11})^{-1}\Sigma _{12}} is the conditional covariance matrix of Y given X = 0; and ( Σ 11 ) − 1 Σ 12 {\displaystyle (\Sigma _{11})^{-1}\Sigma _{12}} is the slope of the regression model of Y on X. Therefore, the elements corresponding to Y indices and the intersection of X and Y in M ( X → , Y ) {\displaystyle M({\vec {X}},Y)} represents the conditional distribution of Y given X = 0. These semantics render the partial sweeping operation a useful method for manipulating multivariate normal distributions. They also form the basis of the moment matrix representations for the three remaining important cases of linear belief functions, including proper belief functions, linear equations, and linear regression models. === Proper linear belief functions === For variables X and Y, assume there exists a piece of evidence justifying a normal distribution for variables Y while bearing no opinions for variables X. Also, assume that X and Y are not perfectly linearly related, i.e., their correlation is less than 1. This case involves a mix of an ordinary normal distribution for Y and a vacuous belief function for X. Thus, we represent it using a partially swept matrix as follows: M ( X → , Y ) = [ 0 0 0 μ 2 0 Σ 22 ] {\displaystyle M({\vec {X}},Y)=\left[{\begin{array}{{20}c}{\begin{array}{{20}c}0\\0\\0\end{array}}&{\begin{array}{{20}c}\mu _{2}\\0\\\Sigma _{22}\\\end{array}}\end{array}}\right]} This is how we could understand the representation. Since we are ignorant on X, we use its swept form and set μ 1 ( Σ 11 ) − 1 = 0 {\displaystyle \mu _{1}(\Sigma _{11})^{-1}=0} and − ( Σ 11 ) − 1 = 0 {\displaystyle -(\Sigma _{11})^{-1}=0} . Since the correlation between X and Y is less than 1, the regression coefficient of X on Y approaches to 0 when the variance of X approaches to ∞. Therefore, ( Σ 11 ) − 1 Σ 12 = 0 {\displaystyle (\Sigma _{11})^{-1}\Sigma _{12}=0} . Similarly, one can prove that μ 1 ( Σ 11 ) − 1 Σ 12 = 0 {\displaystyle \mu _{1}(\Sigma _{11})^{-1}\Sigma _{12}=0} and Σ 21 ( Σ 11 ) −
Clips (software)
Clips is a discontinued mobile video editing software application created by Apple Inc. It was released onto the iOS App Store on April 6, 2017, for free. Initially, it was only available on 64-bit devices running iOS 10.3 or later; as of version 3.1.3, it requires iOS 16.0 or later. Apple describes it as an app for "making and sharing fun videos with text, effects, graphics, and more.". Its final release was on May 9, 2024 before was removed from the App Store on October 10, 2025. == Features == After launching of the app, the user sees the view of the front-facing camera. The app allows the user to create a new clip by tapping on a red record button, or use photos or videos from the device's photo library. Once a clip is recorded, it can be added to a project timeline shown at the bottom of the screen. The user can share their project on social media platforms. The user can also add filters and effects to the project. "Live Titles" (available in several styles) can also be created by dictating to the device.
Hyper basis function network
In machine learning, a Hyper basis function network, or HyperBF network, is a generalization of radial basis function (RBF) networks concept, where the Mahalanobis-like distance is used instead of the Euclidean distance measure. Hyper basis function networks were first introduced by Poggio and Girosi in the 1990 paper “Networks for Approximation and Learning”. == Network Architecture == The typical HyperBF network structure consists of a real input vector x ∈ R n {\displaystyle x\in \mathbb {R} ^{n}} , a hidden layer of activation functions and a linear output layer. The output of the network is a scalar function of the input vector, ϕ : R n → R {\displaystyle \phi :\mathbb {R} ^{n}\to \mathbb {R} } , is given by where N {\displaystyle N} is a number of neurons in the hidden layer, μ j {\displaystyle \mu _{j}} and a j {\displaystyle a_{j}} are the center and weight of neuron j {\displaystyle j} . The activation function ρ j ( | | x − μ j | | ) {\displaystyle \rho _{j}(||x-\mu _{j}||)} at the HyperBF network takes the following form where R j {\displaystyle R_{j}} is a positive definite d × d {\displaystyle d\times d} matrix. Depending on the application, the following types of matrices R j {\displaystyle R_{j}} are usually considered R j = 1 2 σ 2 I d × d {\displaystyle R_{j}={\frac {1}{2\sigma ^{2}}}\mathbb {I} _{d\times d}} , where σ > 0 {\displaystyle \sigma >0} . This case corresponds to the regular RBF network. R j = 1 2 σ j 2 I d × d {\displaystyle R_{j}={\frac {1}{2\sigma _{j}^{2}}}\mathbb {I} _{d\times d}} , where σ j > 0 {\displaystyle \sigma _{j}>0} . In this case, the basis functions are radially symmetric, but are scaled with different width. R j = d i a g ( 1 2 σ j 1 2 , . . . , 1 2 σ j z 2 ) I d × d {\displaystyle R_{j}=diag\left({\frac {1}{2\sigma _{j1}^{2}}},...,{\frac {1}{2\sigma _{jz}^{2}}}\right)\mathbb {I} _{d\times d}} , where σ j i > 0 {\displaystyle \sigma _{ji}>0} . Every neuron has an elliptic shape with a varying size. Positive definite matrix, but not diagonal. == Training == Training HyperBF networks involves estimation of weights a j {\displaystyle a_{j}} , shape and centers of neurons R j {\displaystyle R_{j}} and μ j {\displaystyle \mu _{j}} . Poggio and Girosi (1990) describe the training method with moving centers and adaptable neuron shapes. The outline of the method is provided below. Consider the quadratic loss of the network H [ ϕ ∗ ] = ∑ i = 1 N ( y i − ϕ ∗ ( x i ) ) 2 {\displaystyle H[\phi ^{}]=\sum _{i=1}^{N}(y_{i}-\phi ^{}(x_{i}))^{2}} . The following conditions must be satisfied at the optimum: where R j = W T W {\displaystyle R_{j}=W^{T}W} . Then in the gradient descent method the values of a j , μ j , W {\displaystyle a_{j},\mu _{j},W} that minimize H [ ϕ ∗ ] {\displaystyle H[\phi ^{}]} can be found as a stable fixed point of the following dynamic system: where ω {\displaystyle \omega } determines the rate of convergence. Overall, training HyperBF networks can be computationally challenging. Moreover, the high degree of freedom of HyperBF leads to overfitting and poor generalization. However, HyperBF networks have an important advantage that a small number of neurons is enough for learning complex functions.