Regulation of algorithms, or algorithmic regulation, is the creation of laws, rules and public sector policies for promotion and regulation of algorithms, particularly in artificial intelligence and machine learning. For the subset of AI algorithms, the term regulation of artificial intelligence is used. The regulatory and policy landscape for artificial intelligence (AI) is an emerging issue in jurisdictions globally, including in the European Union. Regulation of AI is considered necessary to both encourage AI and manage associated risks, but challenging. Another emerging topic is the regulation of blockchain algorithms (Use of the smart contracts must be regulated) and is mentioned along with regulation of AI algorithms. Many countries have enacted regulations of high frequency trades, which is shifting due to technological progress into the realm of AI algorithms. The motivation for regulation of algorithms is the apprehension of losing control over the algorithms, whose impact on human life increases. Multiple countries have already introduced regulations in case of automated credit score calculation—right to explanation is mandatory for those algorithms. For example, The IEEE has begun developing a new standard to explicitly address ethical issues and the values of potential future users. Bias, transparency, and ethics concerns have emerged with respect to the use of algorithms in diverse domains ranging from criminal justice to healthcare—many fear that artificial intelligence could replicate existing social inequalities along race, class, gender, and sexuality lines. == Regulation of artificial intelligence == === Public discussion === In 2016, Joy Buolamwini founded Algorithmic Justice League after a personal experience with biased facial detection software in order to raise awareness of the social implications of artificial intelligence through art and research. In 2017 Elon Musk advocated regulation of algorithms in the context of the existential risk from artificial general intelligence. According to NPR, the Tesla CEO was "clearly not thrilled" to be advocating for government scrutiny that could impact his own industry, but believed the risks of going completely without oversight are too high: "Normally the way regulations are set up is when a bunch of bad things happen, there's a public outcry, and after many years a regulatory agency is set up to regulate that industry. It takes forever. That, in the past, has been bad but not something which represented a fundamental risk to the existence of civilisation." In response, some politicians expressed skepticism about the wisdom of regulating a technology that is still in development. Responding both to Musk and to February 2017 proposals by European Union lawmakers to regulate AI and robotics, Intel CEO Brian Krzanich has argued that artificial intelligence is in its infancy and that it is too early to regulate the technology. Instead of trying to regulate the technology itself, some scholars suggest to rather develop common norms including requirements for the testing and transparency of algorithms, possibly in combination with some form of warranty. One suggestion has been for the development of a global governance board to regulate AI development. In 2020, the European Union published its draft strategy paper for promoting and regulating AI. Algorithmic tacit collusion is a legally dubious antitrust practise committed by means of algorithms, which the courts are not able to prosecute. This danger concerns scientists and regulators in EU, US and beyond. European Commissioner Margrethe Vestager mentioned an early example of algorithmic tacit collusion in her speech on "Algorithms and Collusion" on March 16, 2017, described as follows: "A few years ago, two companies were selling a textbook called The Making of a Fly. One of those sellers used an algorithm which essentially matched its rival’s price. That rival had an algorithm which always set a price 27% higher than the first. The result was that prices kept spiralling upwards, until finally someone noticed what was going on, and adjusted the price manually. By that time, the book was selling – or rather, not selling – for 23 million dollars a copy." In 2018, the Netherlands employed an algorithmic system SyRI (Systeem Risico Indicatie) to detect citizens perceived being high risk for committing welfare fraud, which quietly flagged thousands of people to investigators. This caused a public protest. The district court of Hague shut down SyRI referencing Article 8 of the European Convention on Human Rights (ECHR). In 2020, algorithms assigning exam grades to students in the UK sparked open protest under the banner "Fuck the algorithm." This protest was successful and the grades were taken back. In 2024, the Munich Convention on AI, Data and Human Rights was introduced as part of growing international efforts to regulate artificial intelligence through a human rights lens. Developed through a collaborative drafting process involving scholars from the Technical University of Munich, Stellenbosch University, Ulster University, and KNUST, the initiative calls for an international conversation on a binding treaty to safeguard human rights and the principles enshrined in the UN Charter in the age of AI. === Implementation === AI law and regulations can be divided into three main topics, namely governance of autonomous intelligence systems, responsibility and accountability for the systems, and privacy and safety issues. The development of public sector strategies for management and regulation of AI has been increasingly deemed necessary at the local, national, and international levels and in fields from public service management to law enforcement, the financial sector, robotics, the military, and international law. There are many concerns that there is not enough visibility and monitoring of AI in these sectors. In the United States financial sector, for example, there have been calls for the Consumer Financial Protection Bureau to more closely examine source code and algorithms when conducting audits of financial institutions' non-public data. In the United States, on January 7, 2019, following an Executive Order on 'Maintaining American Leadership in Artificial Intelligence', the White House's Office of Science and Technology Policy released a draft Guidance for Regulation of Artificial Intelligence Applications, which includes ten principles for United States agencies when deciding whether and how to regulate AI. In response, the National Institute of Standards and Technology has released a position paper, the National Security Commission on Artificial Intelligence has published an interim report, and the Defense Innovation Board has issued recommendations on the ethical use of AI. In April 2016, for the first time in more than two decades, the European Parliament adopted a set of comprehensive regulations for the collection, storage, and use of personal information, the General Data Protection Regulation (GDPR)1 (European Union, Parliament and Council 2016). The GDPR's policy on the right of citizens to receive an explanation for algorithmic decisions highlights the pressing importance of human interpretability in algorithm design. In 2016, China published a position paper questioning the adequacy of existing international law to address the eventuality of fully autonomous weapons, becoming the first permanent member of the U.N. Security Council to broach the issue, and leading to proposals for global regulation. In the United States, steering on regulating security-related AI is provided by the National Security Commission on Artificial Intelligence. In 2017, the U.K. Vehicle Technology and Aviation Bill imposes liability on the owner of an uninsured automated vehicle when driving itself and makes provisions for cases where the owner has made "unauthorized alterations" to the vehicle or failed to update its software. Further ethical issues arise when, e.g., a self-driving car swerves to avoid a pedestrian and causes a fatal accident. In 2021, the European Commission proposed the Artificial Intelligence Act. == Algorithm certification == There is a concept of algorithm certification emerging as a method of regulating algorithms. Algorithm certification involves auditing whether the algorithm used during the life cycle 1) conforms to the protocoled requirements (e.g., for correctness, completeness, consistency, and accuracy); 2) satisfies the standards, practices, and conventions; and 3) solves the right problem (e.g., correctly model physical laws), and satisfies the intended use and user needs in the operational environment. == Regulation of blockchain algorithms == Blockchain systems provide transparent and fixed records of transactions and hereby contradict the goal of the European GDPR, which is to give individuals full control of their private data. By implementing the Decree on Development of Digital Economy, Bel
Knowledge graph embedding
In representation learning, knowledge graph embedding (KGE), also called knowledge representation learning (KRL), or multi-relation learning, is a machine learning task of learning a low-dimensional representation of a knowledge graph's entities and relations while preserving their semantic meaning. Leveraging their embedded representation, knowledge graphs can be used for various applications such as link prediction, triple classification, entity recognition, clustering, and relation extraction. == Definition == A knowledge graph G = { E , R , F } {\displaystyle {\mathcal {G}}=\{E,R,F\}} is a collection of entities E {\displaystyle E} , relations R {\displaystyle R} , and facts F {\displaystyle F} . A fact is a triple ( h , r , t ) ∈ F {\displaystyle (h,r,t)\in F} that denotes a link r ∈ R {\displaystyle r\in R} between the head h ∈ E {\displaystyle h\in E} and the tail t ∈ E {\displaystyle t\in E} of the triple. Another notation that is often used in the literature to represent a triple (or fact) is ⟨ head , relation , tail ⟩ {\displaystyle \langle {\text{head}},{\text{relation}},{\text{tail}}\rangle } . This notation is called the Resource Description Framework (RDF). A knowledge graph represents the knowledge related to a specific domain; leveraging this structured representation, it is possible to infer a piece of new knowledge from it after some refinement steps. However, nowadays, people have to deal with the sparsity of data and the computational inefficiency to use them in a real-world application. The embedding of a knowledge graph is a function that translates each entity and each relation into a vector of a given dimension d {\displaystyle d} , called embedding dimension. It is even possible to embed the entities and relations with different dimensions. The embedding vectors can then be used for other tasks. A knowledge graph embedding is characterized by four aspects: Representation space: The low-dimensional space in which the entities and relations are represented. Scoring function: A measure of the goodness of a triple-embedded representation. Encoding models: The modality in which the embedded representation of the entities and relations interact with each other. Additional information: Any additional information coming from the knowledge graph that can enrich the embedded representation. Usually, an ad hoc scoring function is integrated into the general scoring function for each additional piece of information. == Embedding procedure == All algorithms for creating a knowledge graph embedding follow the same approach. First, the embedding vectors are initialized to random values. Then, they are iteratively optimized using a training set of triples. In each iteration, a batch of size b {\displaystyle b} triples is sampled from the training set, and a triple from it is sampled and corrupted—i.e., a triple that does not represent a true fact in the knowledge graph. The corruption of a triple involves substituting the head or the tail (or both) of the triple with another entity that makes the fact false. The original triple and the corrupted triple are added in the training batch, and then the embeddings are updated, optimizing a scoring function. Iteration stops when a stop condition is reached. Usually, the stop condition depends on the overfitting of the training set. At the end, the learned embeddings should have extracted semantic meaning from the training triples and should correctly predict unseen true facts in the knowledge graph. === Pseudocode === The following is the pseudocode for the general embedding procedure. algorithm Compute entity and relation embeddings input: The training set S = { ( h , r , t ) } {\displaystyle S=\{(h,r,t)\}} , entity set E {\displaystyle E} , relation set R {\displaystyle R} , embedding dimension k {\displaystyle k} output: Entity and relation embeddings initialization: the entities e {\displaystyle e} and relations r {\displaystyle r} embeddings (vectors) are randomly initialized while stop condition do S b a t c h ← s a m p l e ( S , b ) {\displaystyle S_{batch}\leftarrow sample(S,b)} // Sample a batch from the training set for each ( h , r , t ) {\displaystyle (h,r,t)} in S b a t c h {\displaystyle S_{batch}} do ( h ′ , r , t ′ ) ← s a m p l e ( S ′ ) {\displaystyle (h',r,t')\leftarrow sample(S')} // Sample a corrupted fact T b a t c h ← T b a t c h ∪ { ( ( h , r , t ) , ( h ′ , r , t ′ ) ) } {\displaystyle T_{batch}\leftarrow T_{batch}\cup \{((h,r,t),(h',r,t'))\}} end for Update embeddings by minimizing the loss function end while == Performance indicators == These indexes are often used to measure the embedding quality of a model. The simplicity of the indexes makes them very suitable for evaluating the performance of an embedding algorithm even on a large scale. Given Q {\displaystyle {\ce {Q}}} as the set of all ranked predictions of a model, it is possible to define three different performance indexes: Hits@K, MR, and MRR. === Hits@K === Hits@K or in short, H@K, is a performance index that measures the probability to find the correct prediction in the first top K model predictions. Usually, it is used k = 10 {\displaystyle k=10} . Hits@K reflects the accuracy of an embedding model to predict the relation between two given triples correctly. Hits@K = | { q ∈ Q : q < k } | | Q | ∈ [ 0 , 1 ] {\displaystyle ={\frac {|\{q\in Q:q In computational learning theory, Occam learning is a model of algorithmic learning where the objective of the learner is to output a succinct representation of received training data. This is closely related to probably approximately correct (PAC) learning, where the learner is evaluated on its predictive power of a test set. Occam learnability implies PAC learning, and for a wide variety of concept classes, the converse is also true: PAC learnability implies Occam learnability. == Introduction == Occam Learning is named after Occam's razor, which is a principle stating that, given all other things being equal, a shorter explanation for observed data should be favored over a lengthier explanation. The theory of Occam learning is a formal and mathematical justification for this principle. It was first shown by Blumer, et al. that Occam learning implies PAC learning, which is the standard model of learning in computational learning theory. In other words, parsimony (of the output hypothesis) implies predictive power. == Definition of Occam learning == The succinctness of a concept c {\displaystyle c} in concept class C {\displaystyle {\mathcal {C}}} can be expressed by the length s i z e ( c ) {\displaystyle size(c)} of the shortest bit string that can represent c {\displaystyle c} in C {\displaystyle {\mathcal {C}}} . Occam learning connects the succinctness of a learning algorithm's output to its predictive power on unseen data. Let C {\displaystyle {\mathcal {C}}} and H {\displaystyle {\mathcal {H}}} be concept classes containing target concepts and hypotheses respectively. Then, for constants α ≥ 0 {\displaystyle \alpha \geq 0} and 0 ≤ β < 1 {\displaystyle 0\leq \beta <1} , a learning algorithm L {\displaystyle L} is an ( α , β ) {\displaystyle (\alpha ,\beta )} -Occam algorithm for C {\displaystyle {\mathcal {C}}} using H {\displaystyle {\mathcal {H}}} iff, given a set S = { x 1 , … , x m } {\displaystyle S=\{x_{1},\dots ,x_{m}\}} of m {\displaystyle m} samples labeled according to a concept c ∈ C {\displaystyle c\in {\mathcal {C}}} , L {\displaystyle L} outputs a hypothesis h ∈ H {\displaystyle h\in {\mathcal {H}}} such that h {\displaystyle h} is consistent with c {\displaystyle c} on S {\displaystyle S} (that is, h ( x ) = c ( x ) , ∀ x ∈ S {\displaystyle h(x)=c(x),\forall x\in S} ), and s i z e ( h ) ≤ ( n ⋅ s i z e ( c ) ) α m β {\displaystyle size(h)\leq (n\cdot size(c))^{\alpha }m^{\beta }} where n {\displaystyle n} is the maximum length of any sample x ∈ S {\displaystyle x\in S} . An Occam algorithm is called efficient if it runs in time polynomial in n {\displaystyle n} , m {\displaystyle m} , and s i z e ( c ) . {\displaystyle size(c).} We say a concept class C {\displaystyle {\mathcal {C}}} is Occam learnable with respect to a hypothesis class H {\displaystyle {\mathcal {H}}} if there exists an efficient Occam algorithm for C {\displaystyle {\mathcal {C}}} using H . {\displaystyle {\mathcal {H}}.} == The relation between Occam and PAC learning == Occam learnability implies PAC learnability, as the following theorem of Blumer, et al. shows: === Theorem (Occam learning implies PAC learning) === Let L {\displaystyle L} be an efficient ( α , β ) {\displaystyle (\alpha ,\beta )} -Occam algorithm for C {\displaystyle {\mathcal {C}}} using H {\displaystyle {\mathcal {H}}} . Then there exists a constant a > 0 {\displaystyle a>0} such that for any 0 < ϵ , δ < 1 {\displaystyle 0<\epsilon ,\delta <1} , for any distribution D {\displaystyle {\mathcal {D}}} , given m ≥ a ( 1 ϵ log 1 δ + ( ( n ⋅ s i z e ( c ) ) α ϵ ) 1 1 − β ) {\displaystyle m\geq a\left({\frac {1}{\epsilon }}\log {\frac {1}{\delta }}+\left({\frac {(n\cdot size(c))^{\alpha }}{\epsilon }}\right)^{\frac {1}{1-\beta }}\right)} samples drawn from D {\displaystyle {\mathcal {D}}} and labelled according to a concept c ∈ C {\displaystyle c\in {\mathcal {C}}} of length n {\displaystyle n} bits each, the algorithm L {\displaystyle L} will output a hypothesis h ∈ H {\displaystyle h\in {\mathcal {H}}} such that e r r o r ( h ) ≤ ϵ {\displaystyle error(h)\leq \epsilon } with probability at least 1 − δ {\displaystyle 1-\delta } .Here, e r r o r ( h ) {\displaystyle error(h)} is with respect to the concept c {\displaystyle c} and distribution D {\displaystyle {\mathcal {D}}} . This implies that the algorithm L {\displaystyle L} is also a PAC learner for the concept class C {\displaystyle {\mathcal {C}}} using hypothesis class H {\displaystyle {\mathcal {H}}} . A slightly more general formulation is as follows: === Theorem (Occam learning implies PAC learning, cardinality version) === Let 0 < ϵ , δ < 1 {\displaystyle 0<\epsilon ,\delta <1} . Let L {\displaystyle L} be an algorithm such that, given m {\displaystyle m} samples drawn from a fixed but unknown distribution D {\displaystyle {\mathcal {D}}} and labeled according to a concept c ∈ C {\displaystyle c\in {\mathcal {C}}} of length n {\displaystyle n} bits each, outputs a hypothesis h ∈ H n , m {\displaystyle h\in {\mathcal {H}}_{n,m}} that is consistent with the labeled samples. Then, there exists a constant b {\displaystyle b} such that if log | H n , m | ≤ b ϵ m − log 1 δ {\displaystyle \log |{\mathcal {H}}_{n,m}|\leq b\epsilon m-\log {\frac {1}{\delta }}} , then L {\displaystyle L} is guaranteed to output a hypothesis h ∈ H n , m {\displaystyle h\in {\mathcal {H}}_{n,m}} such that e r r o r ( h ) ≤ ϵ {\displaystyle error(h)\leq \epsilon } with probability at least 1 − δ {\displaystyle 1-\delta } . While the above theorems show that Occam learning is sufficient for PAC learning, it doesn't say anything about necessity. Board and Pitt show that, for a wide variety of concept classes, Occam learning is in fact necessary for PAC learning. They proved that for any concept class that is polynomially closed under exception lists, PAC learnability implies the existence of an Occam algorithm for that concept class. Concept classes that are polynomially closed under exception lists include Boolean formulas, circuits, deterministic finite automata, decision-lists, decision-trees, and other geometrically defined concept classes. A concept class C {\displaystyle {\mathcal {C}}} is polynomially closed under exception lists if there exists a polynomial-time algorithm A {\displaystyle A} such that, when given the representation of a concept c ∈ C {\displaystyle c\in {\mathcal {C}}} and a finite list E {\displaystyle E} of exceptions, outputs a representation of a concept c ′ ∈ C {\displaystyle c'\in {\mathcal {C}}} such that the concepts c {\displaystyle c} and c ′ {\displaystyle c'} agree except on the set E {\displaystyle E} . == Proof that Occam learning implies PAC learning == We first prove the Cardinality version. Call a hypothesis h ∈ H {\displaystyle h\in {\mathcal {H}}} bad if e r r o r ( h ) ≥ ϵ {\displaystyle error(h)\geq \epsilon } , where again e r r o r ( h ) {\displaystyle error(h)} is with respect to the true concept c {\displaystyle c} and the underlying distribution D {\displaystyle {\mathcal {D}}} . The probability that a set of samples S {\displaystyle S} is consistent with h {\displaystyle h} is at most ( 1 − ϵ ) m {\displaystyle (1-\epsilon )^{m}} , by the independence of the samples. By the union bound, the probability that there exists a bad hypothesis in H n , m {\displaystyle {\mathcal {H}}_{n,m}} is at most | H n , m | ( 1 − ϵ ) m {\displaystyle |{\mathcal {H}}_{n,m}|(1-\epsilon )^{m}} , which is less than δ {\displaystyle \delta } if log | H n , m | ≤ O ( ϵ m ) − log 1 δ {\displaystyle \log |{\mathcal {H}}_{n,m}|\leq O(\epsilon m)-\log {\frac {1}{\delta }}} . This concludes the proof of the second theorem above. Using the second theorem, we can prove the first theorem. Since we have a ( α , β ) {\displaystyle (\alpha ,\beta )} -Occam algorithm, this means that any hypothesis output by L {\displaystyle L} can be represented by at most ( n ⋅ s i z e ( c ) ) α m β {\displaystyle (n\cdot size(c))^{\alpha }m^{\beta }} bits, and thus log | H n , m | ≤ ( n ⋅ s i z e ( c ) ) α m β {\displaystyle \log |{\mathcal {H}}_{n,m}|\leq (n\cdot size(c))^{\alpha }m^{\beta }} . This is less than O ( ϵ m ) − log 1 δ {\displaystyle O(\epsilon m)-\log {\frac {1}{\delta }}} if we set m ≥ a ( 1 ϵ log 1 δ + ( ( n ⋅ s i z e ( c ) ) α ) ϵ ) 1 1 − β ) {\displaystyle m\geq a\left({\frac {1}{\epsilon }}\log {\frac {1}{\delta }}+\left({\frac {(n\cdot size(c))^{\alpha })}{\epsilon }}\right)^{\frac {1}{1-\beta }}\right)} for some constant a > 0 {\displaystyle a>0} . Thus, by the Cardinality version Theorem, L {\displaystyle L} will output a consistent hypothesis h {\displaystyle h} with probability at least 1 − δ {\displaystyle 1-\delta } . This concludes the proof of the first theorem above. == Improving sample complexity for common problems == Though Occam and PAC learnability are equivalent, the Occam framework can be used to produce tighter bounds on the sample complexity of classical problems including conjunctions, co In statistics and, in particular, in the fitting of linear or logistic regression models, the elastic net is a regularized regression method that linearly combines the L1 and L2 penalties of the lasso and ridge methods. Nevertheless, elastic net regularization is typically more accurate than both methods with regard to reconstruction. == Specification == The elastic net method overcomes the limitations of the LASSO (least absolute shrinkage and selection operator) method which uses a penalty function based on ‖ β ‖ 1 = ∑ j = 1 p | β j | . {\displaystyle \|\beta \|_{1}=\textstyle \sum _{j=1}^{p}|\beta _{j}|.} Use of this penalty function has several limitations. For example, in the "large p, small n" case (high-dimensional data with few examples), the LASSO selects at most n variables before it saturates. Also if there is a group of highly correlated variables, then the LASSO tends to select one variable from a group and ignore the others. To overcome these limitations, the elastic net adds a quadratic part ( ‖ β ‖ 2 {\displaystyle \|\beta \|^{2}} ) to the penalty, which when used alone is ridge regression (known also as Tikhonov regularization). The estimates from the elastic net method are defined by β ^ ≡ argmin β ( ‖ y − X β ‖ 2 + λ 2 ‖ β ‖ 2 + λ 1 ‖ β ‖ 1 ) . {\displaystyle {\hat {\beta }}\equiv {\underset {\beta }{\operatorname {argmin} }}(\|y-X\beta \|^{2}+\lambda _{2}\|\beta \|^{2}+\lambda _{1}\|\beta \|_{1}).} The quadratic penalty term makes the loss function strongly convex, and it therefore has a unique minimum. The elastic net method includes the LASSO and ridge regression: in other words, each of them is a special case where λ 1 = λ , λ 2 = 0 {\displaystyle \lambda _{1}=\lambda ,\lambda _{2}=0} or λ 1 = 0 , λ 2 = λ {\displaystyle \lambda _{1}=0,\lambda _{2}=\lambda } . Meanwhile, the naive version of elastic net method finds an estimator in a two-stage procedure : first for each fixed λ 2 {\displaystyle \lambda _{2}} it finds the ridge regression coefficients, and then does a LASSO type shrinkage. This kind of estimation incurs a double amount of shrinkage, which leads to increased bias and poor predictions. To improve the prediction performance, sometimes the coefficients of the naive version of elastic net is rescaled by multiplying the estimated coefficients by ( 1 + λ 2 ) {\displaystyle (1+\lambda _{2})} . Examples of where the elastic net method has been applied are: Support vector machine Metric learning Portfolio optimization Cancer prognosis == Reduction to support vector machine == It was proven in 2014 that the elastic net can be reduced to the linear support vector machine. A similar reduction was previously proven for the LASSO in 2014. The authors showed that for every instance of the elastic net, an artificial binary classification problem can be constructed such that the hyper-plane solution of a linear support vector machine (SVM) is identical to the solution β {\displaystyle \beta } (after re-scaling). The reduction immediately enables the use of highly optimized SVM solvers for elastic net problems. It also enables the use of GPU acceleration, which is often already used for large-scale SVM solvers. The reduction is a simple transformation of the original data and regularization constants X ∈ R n × p , y ∈ R n , λ 1 ≥ 0 , λ 2 ≥ 0 {\displaystyle X\in {\mathbb {R} }^{n\times p},y\in {\mathbb {R} }^{n},\lambda _{1}\geq 0,\lambda _{2}\geq 0} into new artificial data instances and a regularization constant that specify a binary classification problem and the SVM regularization constant X 2 ∈ R 2 p × n , y 2 ∈ { − 1 , 1 } 2 p , C ≥ 0. {\displaystyle X_{2}\in {\mathbb {R} }^{2p\times n},y_{2}\in \{-1,1\}^{2p},C\geq 0.} Here, y 2 {\displaystyle y_{2}} consists of binary labels − 1 , 1 {\displaystyle {-1,1}} . When 2 p > n {\displaystyle 2p>n} it is typically faster to solve the linear SVM in the primal, whereas otherwise the dual formulation is faster. Some authors have referred to the transformation as Support Vector Elastic Net (SVEN), and provided the following MATLAB pseudo-code: == Software == "Glmnet: Lasso and elastic-net regularized generalized linear models" is a software which is implemented as an R source package and as a MATLAB toolbox. This includes fast algorithms for estimation of generalized linear models with ℓ1 (the lasso), ℓ2 (ridge regression) and mixtures of the two penalties (the elastic net) using cyclical coordinate descent, computed along a regularization path. JMP Pro 11 includes elastic net regularization, using the Generalized Regression personality with Fit Model. "pensim: Simulation of high-dimensional data and parallelized repeated penalized regression" implements an alternate, parallelised "2D" tuning method of the ℓ parameters, a method claimed to result in improved prediction accuracy. scikit-learn includes linear regression and logistic regression with elastic net regularization. SVEN, a Matlab implementation of Support Vector Elastic Net. This solver reduces the Elastic Net problem to an instance of SVM binary classification and uses a Matlab SVM solver to find the solution. Because SVM is easily parallelizable, the code can be faster than Glmnet on modern hardware. SpaSM, a Matlab implementation of sparse regression, classification and principal component analysis, including elastic net regularized regression. Apache Spark provides support for Elastic Net Regression in its MLlib machine learning library. The method is available as a parameter of the more general LinearRegression class. SAS (software) The SAS procedure Glmselect and SAS Viya procedure Regselect support the use of elastic net regularization for model selection. Multispectral remote sensing is the collection and analysis of reflected, emitted, or back-scattered energy from an object or an area of interest in multiple bands of regions of the electromagnetic spectrum (Jensen, 2005). Subcategories of multispectral remote sensing include hyperspectral, in which hundreds of bands are collected and analyzed, and ultraspectral remote sensing where many hundreds of bands are used (Logicon, 1997). The main purpose of multispectral imaging is the potential to classify the image using multispectral classification. This is a much faster method of image analysis than is possible by human interpretation. == Multispectral remote sensing systems == Remote sensing systems gather data via instruments typically carried on satellites in orbit around the Earth. The remote sensing scanner detects the energy that radiates from the object or area of interest. This energy is recorded as an analog electrical signal and converted into a digital value though an A-to-D conversion. There are several multispectral remote sensing systems that can be categorized in the following way: === Multispectral imaging using discrete detectors and scanning mirrors === Landsat Multispectral Scanner (MSS) Landsat Thematic Mapper (TM) NOAA Geostationary Operational Environmental Satellite (GOES) NOAA Advanced Very High Resolution Radiometer (AVHRR) NASA and ORBIMAGE, Inc., Sea-viewing Wide field-of-view Sensor (SeaWiFS) Daedalus, Inc., Aircraft Multispectral Scanner (AMS) NASA Airborne Terrestrial Applications Sensor (ATLAS) === Multispectral imaging using linear arrays === SPOT 1, 2, and 3 High Resolution Visible (HRV) sensors and Spot 4 and 5 High Resolution Visible Infrared (HRVIR) and vegetation sensor Indian Remote Sensing System (IRS) Linear Imaging Self-scanning Sensor (LISS) Space Imaging, Inc. (IKONOS) Digital Globe, Inc. (QuickBird) ORBIMAGE, Inc. (OrbView-3) ImageSat International, Inc. (EROS A1) NASA Terra Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) NASA Terra Multiangle Imaging Spectroradiometer (MISR) === Imaging spectrometry using linear and area arrays === NASA Jet Propulsion Laboratory Airborne Visible/Infrared Imaging Spectrometer (AVIRIS) Compact Airborne Spectrographic Imager 3 (CASI 3) NASA Terra Moderate Resolution Imaging Spectrometer (MODIS) NASA Earth Observer (EO-1) Advanced Land Imager (ALI), Hyperion, and LEISA Atmospheric Corrector (LAC) === Satellite analog and digital photographic systems === Russian SPIN-2 TK-350, and KVR-1000 NASA Space Shuttle and International Space Station Imagery == Multispectral classification methods == A variety of methods can be used for the multispectral classification of images: Algorithms based on parametric and nonparametric statistics that use ratio-and interval-scaled data and nonmetric methods that can also incorporate nominal scale data (Duda et al., 2001), Supervised or unsupervised classification logic, Hard or soft (fuzzy) set classification logic to create hard or fuzzy thematic output products, Per-pixel or object-oriented classification logic, and Hybrid approaches == Supervised classification == In this classification method, the identity and location of some of the land-cover types are obtained beforehand from a combination of fieldwork, interpretation of aerial photography, map analysis, and personal experience. The analyst would locate sites that have similar characteristics to the known land-cover types. These areas are known as training sites because the known characteristics of these sites are used to train the classification algorithm for eventual land-cover mapping of the remainder of the image. Multivariate statistical parameters (means, standard deviations, covariance matrices, correlation matrices, etc.) are calculated for each training site. All pixels inside and outside of the training sites are evaluated and allocated to the class with the more similar characteristics. === Classification scheme === The first step in the supervised classification method is to identify the land-cover and land-use classes to be used. Land-cover refers to the type of material present on the site (e.g. water, crops, forest, wet land, asphalt, and concrete). Land-use refers to the modifications made by people to the land cover (e.g. agriculture, commerce, settlement). All classes should be selected and defined carefully to properly classify remotely sensed data into the correct land-use and/or land-cover information. To achieve this purpose, it is necessary to use a classification system that contains taxonomically correct definitions of classes. If a hard classification is desired, the following classes should be used: Mutually exclusive: there is not any taxonomic overlap of any classes (i.e., rain forest and evergreen forest are distinct classes). Exhaustive: all land-covers in the area have been included. Hierarchical: sub-level classes (e.g., single-family residential, multiple-family residential) are created, allowing that these classes can be included in a higher category (e.g., residential). Some examples of hard classification schemes are: American Planning Association Land-Based Classification System United States Geological Survey Land-use/Land-cover Classification System for Use with Remote Sensor Data U.S. Department of the Interior Fish and Wildlife Service U.S. National Vegetation and Classification System International Geosphere-Biosphere Program IGBP Land Cover Classification System === Training sites === Once the classification scheme is adopted, the image analyst may select training sites in the image that are representative of the land-cover or land-use of interest. If the environment where the data was collected is relatively homogeneous, the training data can be used. If different conditions are found in the site, it would not be possible to extend the remote sensing training data to the site. To solve this problem, a geographical stratification should be done during the preliminary stages of the project. All differences should be recorded (e.g. soil type, water turbidity, crop species, etc.). These differences should be recorded on the imagery and the selection training sites made based on the geographical stratification of this data. The final classification map would be a composite of the individual stratum classifications. After the data are organized in different training sites, a measurement vector is created. This vector would contain the brightness values for each pixel in each band in each training class. The mean, standard deviation, variance-covariance matrix, and correlation matrix are calculated from the measurement vectors. Once the statistics from each training site are determined, the most effective bands for each class should be selected. The objective of this discrimination is to eliminate the bands that can provide redundant information. Graphical and statistical methods can be used to achieve this objective. Some of the graphic methods are: Bar graph spectral plots Cospectral mean vector plots Feature space plots Cospectral parallelepiped or ellipse plots === Classification algorithm === The last step in supervised classification is selecting an appropriate algorithm. The choice of a specific algorithm depends on the input data and the desired output. Parametric algorithms are based on the fact that the data is normally distributed. If the data is not normally distributed, nonparametric algorithms should be used. The more common nonparametric algorithms are: One-dimensional density slicing Parallelipiped Minimum distance Nearest-neighbor Expert system analysis Convolutional neural network == Unsupervised classification == Unsupervised classification (also known as clustering) is a method of partitioning remote sensor image data in multispectral feature space and extracting land-cover information. Unsupervised classification require less input information from the analyst compared to supervised classification because clustering does not require training data. This process consists in a series of numerical operations to search for the spectral properties of pixels. From this process, a map with m spectral classes is obtained. Using the map, the analyst tries to assign or transform the spectral classes into thematic information of interest (i.e. forest, agriculture, urban). This process may not be easy because some spectral clusters represent mixed classes of surface materials and may not be useful. The analyst has to understand the spectral characteristics of the terrain to be able to label clusters as a specific information class. There are hundreds of clustering algorithms. Two of the most conceptually simple algorithms are the chain method and the ISODATA method. === Chain method === The algorithm used in this method operates in a two-pass mode (it passes through the multispectral dataset two times. In the first pass, the program reads through the dataset and sequentially builds clusters (groups of p Croissant is a metadata format design to support sharing of datasets for machine learning applications. It is a platform-agnostic schema used to standardize metadata in data repositories like Hugging Face, kaggle, Dataverse and OpenML. == Structure == Croissant builds upon schema.org, uses primarily JSON-LD, and divides metadata in four "layers": Dataset Metadata, Resource, Structure and Semantic: The Dataset Metadata layer constrains which schema.org properties should be used, including additional properties, linking together the resources (files) of the dataset with general metadata, like licensing and citation information. The Resource layer describes the individual files and sets of those using two new classes, FileObject and FileSet. A FileSet may be a collection of related images. The Structure layer specifies how the files are organized in the dataset. A RecordSet class describes how resources are present, configurations that may very a lot between modality. This specification facilitates interoperability of the datasets. Finally, the Semantic layer adds information for practical reuse of the dataset, such as splits for train, test and validation subsets. It also provides a default extension for metadata related to responsible AI. The use of a standard machine-readable structure increases, for example, the discoverability of datasets in search engines such as Google Dataset Search. == History == Croissant was shared in arXiv in March 2024 and published in the proceedings of NeurIPS 2024. It started as community driven as a MLCommons Croissant Working Group, including stakeholders organizations from academia and industry, including Google, the open data institute, Sage Bionetworks and King's College London. Variations of Croissant are developed to support datasets in different areas of research, such as Geo-Croissant for geospatial datasets. Other technical extensions, such as support for RDF, soon followed. Julia is a dynamic general-purpose programming language. As a high-level language, distinctive aspects of Julia's design include a type system with parametric polymorphism, the use of multiple dispatch as a core programming paradigm, just-in-time compilation and a parallel garbage collection implementation. Notably, Julia does not support classes with encapsulated methods but instead relies on the types of all of a function's arguments to determine which method will be called. By default, Julia is run similarly to scripting languages, using its runtime, and allows for interactions, but Julia programs can also be compiled to small binary standalone executables (or to small libraries for e.g. Python), with e.g. the JuliaC.jl compiler. Julia programs can reuse libraries from other languages, and vice versa. Julia has interoperability with C, C++, Fortran, Rust, Python, and R. Additionally, some Julia packages have bindings to be used from Python and R as libraries. Julia is supported by programmer tools like IDEs (see below) and by notebooks like Pluto.jl, Jupyter, and since 2025, Google Colab officially supports Julia natively. Julia is sometimes used in embedded systems (e.g. has been used in a satellite in space on a Raspberry Pi Compute Module 4; 64-bit Pis work best with Julia, and Julia is supported in Raspbian). == History == Work on Julia began in 2009, when Jeff Bezanson, Stefan Karpinski, Viral B. Shah, and Alan Edelman set out to create a free language that was both high-level and fast. On 14 February 2012, the team launched a website with a blog post explaining the language's mission. In an interview with InfoWorld in April 2012, Karpinski said about the name of the language, Julia: "There's no good reason, really. It just seemed like a pretty name." Bezanson said he chose the name on the recommendation of a friend, then years later wrote: Maybe julia stands for "Jeff's uncommon lisp is automated"? Julia's syntax is stable, since version 1.0 in 2018, and Julia has a backward compatibility guarantee for 1.x and also a stability promise for the documented (stable) API, while in the years before in the early development prior to 0.7 the syntax (and semantics) was changed in new versions. All of the (registered package) ecosystem uses the new and improved syntax, and in most cases relies on new APIs that have been added regularly, and in some cases minor additional syntax added in a forward compatible way e.g. in Julia 1.7. In the 10 years since the 2012 launch of pre-1.0 Julia, the community has grown. The Julia package ecosystem has over 11.8 million lines of code (including docs and tests). The JuliaCon academic conference for Julia users and developers has been held annually since 2014 with JuliaCon2020 welcoming over 28,900 unique viewers, and then JuliaCon2021 breaking all previous records (with more than 300 JuliaCon2021 presentations available for free on YouTube, up from 162 the year before), and 43,000 unique viewers during the conference. Three of the Julia co-creators are the recipients of the 2019 James H. Wilkinson Prize for Numerical Software (awarded every four years) "for the creation of Julia, an innovative environment for the creation of high-performance tools that enable the analysis and solution of computational science problems." Also, Alan Edelman, professor of applied mathematics at MIT, has been selected to receive the 2019 IEEE Computer Society Sidney Fernbach Award "for outstanding breakthroughs in high-performance computing, linear algebra, and computational science and for contributions to the Julia programming language." Version 0.3 was released in August 2014. Both Julia 0.7 and version 1.0 were released on 8 August 2018. Julia 1.4 added syntax for generic array indexing to handle e.g. 0-based arrays. The memory model was also changed. Julia 1.5 released in August 2020 added record and replay debugging support, for Mozilla's rr tool. The release changed the behavior in the REPL (to soft scope) to the one used in Jupyter, but keeps full compatible with non-REPL code (that retains hard scope). Julia 1.6 was the largest release since 1.0, and it was the long-term support (LTS) version for the longest time. Since Julia 1.7 development is back to time-based releases, and it was released in November 2021 with e.g. a new default random-number generator and Julia 1.7.3 fixed at least one security issue. Julia 1.8 added options for hiding source code when compiling Julia source code to executables. Julia 1.9 has added the ability to precompile packages to native machine code, done automatically; to improve precompilation of packages a new package PrecompileTools.jl was introduced, for use by package developers. Julia 1.10 was released on 25 December 2023 with new features such as parallel garbage collection. Julia 1.11 was released on 7 October 2024, and with it 1.10.5 became the next long-term support (LTS) version (i.e. those became the only two supported versions), since replaced by 1.10.10 released on 27 June, and 1.6 is no longer an LTS version. Julia 1.11 adds e.g. the new public keyword to signal safe public API (Julia users are advised to use such API, not internals, of Julia or packages, and package authors advised to use the keyword, generally indirectly, e.g. prefixed with the @compat macro, from Compat.jl, to also support older Julia versions, at least the LTS version). Julia 1.12 was released on 7 October 2025 (and 1.12.5 on 9 February 2026), and with it a JuliaC.jl package including the juliac compiler that works with it, for making rather small binary executables (much smaller than was possible before; through the use of new so-called trimming feature). Julia 1.10 LTS is an officially still-supported branch, but the 1.11 branch has also been maintained after 1.12 release, with 1.11.8 released and then 1.11.9 released on 8 February 2026. === JuliaCon === Since 2014, the Julia Community has hosted an annual Julia Conference focused on developers and users. The first JuliaCon took place in Chicago and kickstarted the annual occurrence of the conference. Since 2014, the conference has taken place across a number of locations including MIT and the University of Maryland, Baltimore. The event audience has grown from a few dozen people to over 28,900 unique attendees during JuliaCon 2020, which took place virtually. JuliaCon 2021 also took place virtually with keynote addresses from professors William Kahan, the primary architect of the IEEE 754 floating-point standard (which virtually all CPUs and languages, including Julia, use), Jan Vitek, Xiaoye Sherry Li, and Soumith Chintala, a co-creator of PyTorch. JuliaCon grew to 43,000 unique attendees and more than 300 presentations (still freely accessible, plus for older years). JuliaCon 2022 will also be virtual held between July 27 and July 29, 2022, for the first time in several languages, not just in English. === Sponsors === The Julia language became a NumFOCUS fiscally sponsored project in 2014 in an effort to ensure the project's long-term sustainability. Jeremy Kepner at MIT Lincoln Laboratory was the founding sponsor of the Julia project in its early days. In addition, funds from the Gordon and Betty Moore Foundation, the Alfred P. Sloan Foundation, Intel, and agencies such as NSF, DARPA, NIH, NASA, and FAA have been essential to the development of Julia. Mozilla, the maker of Firefox web browser, with its research grants for H1 2019, sponsored "a member of the official Julia team" for the project "Bringing Julia to the Browser", meaning to Firefox and other web browsers. The Julia language is also supported by individual donors on GitHub. === The Julia company === JuliaHub, Inc. was founded in 2015 as Julia Computing, Inc. by Viral B. Shah, Deepak Vinchhi, Alan Edelman, Jeff Bezanson, Stefan Karpinski and Keno Fischer. In June 2017, Julia Computing raised US$4.6 million in seed funding from General Catalyst and Founder Collective, the same month was "granted $910,000 by the Alfred P. Sloan Foundation to support open-source Julia development, including $160,000 to promote diversity in the Julia community", and in December 2019 the company got $1.1 million funding from the US government to "develop a neural component machine learning tool to reduce the total energy consumption of heating, ventilation, and air conditioning (HVAC) systems in buildings". In July 2021, Julia Computing announced they raised a $24 million Series A round led by Dorilton Ventures, which also owns Formula One team Williams Racing, that partnered with Julia Computing. Williams' Commercial Director said: "Investing in companies building best-in-class cloud technology is a strategic focus for Dorilton and Julia's versatile platform, with revolutionary capabilities in simulation and modelling, is hugely relevant to our business. We look forward to embedding Julia Computing in the world's most technologically advanced sport". In June 2023, JuliaHub received (again, now Occam learning
Elastic net regularization
Multispectral pattern recognition
Croissant (metadata format)
Julia (programming language)