Similarity learning is an area of supervised machine learning in artificial intelligence. It is closely related to regression and classification, but the goal is to learn a similarity function that measures how similar or related two objects are. It has applications in ranking, in recommendation systems, visual identity tracking, face verification, and speaker verification. == Learning setup == There are four common setups for similarity and metric distance learning. Regression similarity learning In this setup, pairs of objects are given ( x i 1 , x i 2 ) {\displaystyle (x_{i}^{1},x_{i}^{2})} together with a measure of their similarity y i ∈ R {\displaystyle y_{i}\in R} . The goal is to learn a function that approximates f ( x i 1 , x i 2 ) ∼ y i {\displaystyle f(x_{i}^{1},x_{i}^{2})\sim y_{i}} for every new labeled triplet example ( x i 1 , x i 2 , y i ) {\displaystyle (x_{i}^{1},x_{i}^{2},y_{i})} . This is typically achieved by minimizing a regularized loss min W ∑ i l o s s ( w ; x i 1 , x i 2 , y i ) + r e g ( w ) {\displaystyle \min _{W}\sum _{i}loss(w;x_{i}^{1},x_{i}^{2},y_{i})+reg(w)} . Classification similarity learning Given are pairs of similar objects ( x i , x i + ) {\displaystyle (x_{i},x_{i}^{+})} and non similar objects ( x i , x i − ) {\displaystyle (x_{i},x_{i}^{-})} . An equivalent formulation is that every pair ( x i 1 , x i 2 ) {\displaystyle (x_{i}^{1},x_{i}^{2})} is given together with a binary label y i ∈ { 0 , 1 } {\displaystyle y_{i}\in \{0,1\}} that determines if the two objects are similar or not. The goal is again to learn a classifier that can decide if a new pair of objects is similar or not. Ranking similarity learning Given are triplets of objects ( x i , x i + , x i − ) {\displaystyle (x_{i},x_{i}^{+},x_{i}^{-})} whose relative similarity obey a predefined order: x i {\displaystyle x_{i}} is known to be more similar to x i + {\displaystyle x_{i}^{+}} than to x i − {\displaystyle x_{i}^{-}} . The goal is to learn a function f {\displaystyle f} such that for any new triplet of objects ( x , x + , x − ) {\displaystyle (x,x^{+},x^{-})} , it obeys f ( x , x + ) > f ( x , x − ) {\displaystyle f(x,x^{+})>f(x,x^{-})} (contrastive learning). This setup assumes a weaker form of supervision than in regression, because instead of providing an exact measure of similarity, one only has to provide the relative order of similarity. For this reason, ranking-based similarity learning is easier to apply in real large-scale applications. Locality sensitive hashing (LSH) Hashes input items so that similar items map to the same "buckets" in memory with high probability (the number of buckets being much smaller than the universe of possible input items). It is often applied in nearest neighbor search on large-scale high-dimensional data, e.g., image databases, document collections, time-series databases, and genome databases. A common approach for learning similarity is to model the similarity function as a bilinear form. For example, in the case of ranking similarity learning, one aims to learn a matrix W that parametrizes the similarity function f W ( x , z ) = x T W z {\displaystyle f_{W}(x,z)=x^{T}Wz} . When data is abundant, a common approach is to learn a siamese network – a deep network model with parameter sharing. == Metric learning == Similarity learning is closely related to distance metric learning. Metric learning is the task of learning a distance function over objects. A metric or distance function has to obey four axioms: non-negativity, identity of indiscernibles, symmetry and subadditivity (or the triangle inequality). In practice, metric learning algorithms ignore the condition of identity of indiscernibles and learn a pseudo-metric. When the objects x i {\displaystyle x_{i}} are vectors in R d {\displaystyle R^{d}} , then any matrix W {\displaystyle W} in the symmetric positive semi-definite cone S + d {\displaystyle S_{+}^{d}} defines a distance pseudo-metric of the space of x through the form D W ( x 1 , x 2 ) 2 = ( x 1 − x 2 ) ⊤ W ( x 1 − x 2 ) {\displaystyle D_{W}(x_{1},x_{2})^{2}=(x_{1}-x_{2})^{\top }W(x_{1}-x_{2})} . When W {\displaystyle W} is a symmetric positive definite matrix, D W {\displaystyle D_{W}} is a metric. Moreover, as any symmetric positive semi-definite matrix W ∈ S + d {\displaystyle W\in S_{+}^{d}} can be decomposed as W = L ⊤ L {\displaystyle W=L^{\top }L} where L ∈ R e × d {\displaystyle L\in R^{e\times d}} and e ≥ r a n k ( W ) {\displaystyle e\geq rank(W)} , the distance function D W {\displaystyle D_{W}} can be rewritten equivalently D W ( x 1 , x 2 ) 2 = ( x 1 − x 2 ) ⊤ L ⊤ L ( x 1 − x 2 ) = ‖ L ( x 1 − x 2 ) ‖ 2 2 {\displaystyle D_{W}(x_{1},x_{2})^{2}=(x_{1}-x_{2})^{\top }L^{\top }L(x_{1}-x_{2})=\|L(x_{1}-x_{2})\|_{2}^{2}} . The distance D W ( x 1 , x 2 ) 2 = ‖ x 1 ′ − x 2 ′ ‖ 2 2 {\displaystyle D_{W}(x_{1},x_{2})^{2}=\|x_{1}'-x_{2}'\|_{2}^{2}} corresponds to the Euclidean distance between the transformed feature vectors x 1 ′ = L x 1 {\displaystyle x_{1}'=Lx_{1}} and x 2 ′ = L x 2 {\displaystyle x_{2}'=Lx_{2}} . Many formulations for metric learning have been proposed. Some well-known approaches for metric learning include learning from relative comparisons, which is based on the triplet loss, large margin nearest neighbor, and information theoretic metric learning (ITML). In statistics, the covariance matrix of the data is sometimes used to define a distance metric called Mahalanobis distance. == Applications == Similarity learning is used in information retrieval for learning to rank, in face verification or face identification, and in recommendation systems. Also, many machine learning approaches rely on some metric. This includes unsupervised learning such as clustering, which groups together close or similar objects. It also includes supervised approaches like K-nearest neighbor algorithm which rely on labels of nearby objects to decide on the label of a new object. Metric learning has been proposed as a preprocessing step for many of these approaches. == Scalability == Metric and similarity learning scale quadratically with the dimension of the input space, as can easily see when the learned metric has a bilinear form f W ( x , z ) = x T W z {\displaystyle f_{W}(x,z)=x^{T}Wz} . Scaling to higher dimensions can be achieved by enforcing a sparseness structure over the matrix model, as done with HDSL, and with COMET. == Software == metric-learn is a free software Python library which offers efficient implementations of several supervised and weakly-supervised similarity and metric learning algorithms. The API of metric-learn is compatible with scikit-learn. OpenMetricLearning is a Python framework to train and validate the models producing high-quality embeddings. == Further information == For further information on this topic, see the surveys on metric and similarity learning by Bellet et al. and Kulis.
Artificial general intelligence
Artificial general intelligence (AGI) is a hypothetical type of artificial intelligence that matches or surpasses human capabilities across virtually all cognitive tasks. Beyond AGI, artificial superintelligence (ASI) would outperform the best human abilities across every domain by a wide margin. Unlike artificial narrow intelligence (ANI), whose competence is confined to well‑defined tasks, an AGI system can generalise knowledge, transfer skills between domains, and solve novel problems without task‑specific reprogramming. Creating AGI is a stated goal of technology companies such as OpenAI, Google, xAI, and Meta. A 2020 survey identified 72 active AGI research and development projects across 37 countries. AGI is a common topic in science fiction and futures studies. Contention exists over whether AGI represents an existential risk. Some AI experts and industry figures have stated that mitigating the risk of human extinction posed by AGI should be a global priority. Others find the development of AGI to be in too remote a stage to present such a risk. == Terminology == AGI is also known as strong AI, full AI, human-level AI, human-level intelligent AI, or general intelligent action. The term "artificial general intelligence" was used in 1997 by Mark Gubrud in a discussion of the implications of fully automated military production and operations. A mathematical formalism of AGI named AIXI was proposed in 2000 by Marcus Hutter, who defines intelligence as "an agent’s ability to achieve goals or succeed in a wide range of environments". This type of AGI has also been called "universal artificial intelligence". The term AGI was re-introduced and popularized by Shane Legg and Ben Goertzel around 2002. Some academic sources reserve the term "strong AI" for computer programs that will experience sentience or consciousness. In contrast, weak AI (or narrow AI) can solve a specific problem but lacks general cognitive abilities. Some academic sources use "weak AI" to refer more broadly to any programs that neither experience consciousness nor have a mind in the same sense as humans. Related concepts include artificial superintelligence and transformative AI. An artificial superintelligence (ASI) is a hypothetical type of AGI that is much more generally intelligent than humans, while the notion of transformative AI relates to AI having a large impact on society, for example, similar to the agricultural or industrial revolution. A framework for classifying AGI was proposed in 2023 by Google DeepMind researchers. They define five performance levels of AGI: emerging, competent, expert, virtuoso, and superhuman. For example, a competent AGI is defined as an AI that outperforms 50% of skilled adults in a wide range of non-physical tasks, and a superhuman AGI (i.e., an artificial superintelligence) is similarly defined but with a threshold of 100%. They consider large language models like ChatGPT or LLaMA 2 to be instances of emerging AGI (comparable to unskilled humans). Regarding the autonomy of AGI and associated risks, they define five levels: tool (fully in human control), consultant, collaborator, expert, and agent (fully autonomous). == Characteristics == There is no single agreed-upon definition of intelligence as applied to computers. Computer scientist John McCarthy wrote in 2007: "We cannot yet characterize in general what kinds of computational procedures we want to call intelligent." === Intelligence traits === Researchers generally hold that a system is required to do all of the following to be regarded as an AGI: reason, use strategy, solve puzzles, and make judgments under uncertainty, represent knowledge, including common sense knowledge, plan, learn, communicate in natural language, if necessary, integrate these skills in completion of any given goal. Many interdisciplinary approaches (e.g. cognitive science, computational intelligence, and decision making) consider additional traits such as imagination (the ability to form novel mental images and concepts) and autonomy. Computer-based systems exhibiting these capabilities are now widespread, with modern large language models demonstrating computational creativity, automated reasoning, and decision support simultaneously across domains. === Physical traits === Other capabilities are considered desirable in intelligent systems, as they may affect intelligence or aid in its expression. These include: the ability to sense (e.g. see, hear, etc.), and the ability to act (e.g. move and manipulate objects, change location to explore, etc.) This includes the ability to detect and respond to hazard. === Tests for human-level AGI === Several tests meant to confirm human-level AGI have been considered. ==== Turing test ==== The Turing test was proposed by Alan Turing in his 1950 paper "Computing Machinery and Intelligence". This test involves a human judge engaging in natural language conversations with both a human and a machine designed to generate human-like responses. The machine passes the test if it can convince the judge that it is human a significant fraction of the time. Turing proposed this as a practical measure of machine intelligence, focusing on the ability to produce human-like responses rather than on the internal workings of the machine. The idea of the test is that the machine has to try and pretend to be a man, by answering questions put to it, and it will only pass if the pretence is reasonably convincing. A considerable portion of a jury, who should not be experts about machines, must be taken in by the pretence. In 2014, a chatbot named Eugene Goostman, designed to imitate a 13-year-old Ukrainian boy, reportedly passed a Turing Test event by convincing 33% of judges that it was human. However, this claim was met with significant skepticism from the AI research community, who questioned the test's implementation and its relevance to AGI. A 2025 pre‑registered, three‑party Turing‑test study by Cameron R. Jones and Benjamin K. Bergen showed that GPT-4.5 was judged to be the human in 73% of five‑minute text conversations—surpassing the 67% humanness rate of real confederates and meeting the researchers' criterion for having passed the test. ==== Ikea test ==== The "Ikea test", also known as the Flat Pack Furniture Test, involves an AI controlling a robot which attempts to assemble an Ikea flat-pack furniture product after having been shown the parts and instructions. As early as 2013, MIT's IkeaBot demonstrated fully autonomous multi-robot assembly of an IKEA Lack table in ten minutes, with no human intervention and no pre-programmed assembly instructions. The robots inferred the assembly sequence from the geometry of the parts alone. ==== Coffee test ==== Steve Wozniak proposed a test where a machine is required to enter an average American home and figure out how to make coffee. It must find the coffee machine, find the coffee, add water, find a mug, and brew the coffee by pushing the proper buttons. This test has been substantially approached across multiple systems. In January 2024, Figure AI's Figure 01 humanoid learned to operate a Keurig coffee machine autonomously after watching video demonstrations, using end-to-end neural networks to translate visual input into motor actions. In 2025, researchers at the University of Edinburgh published the ELLMER framework in Nature Machine Intelligence, demonstrating a robotic arm that interprets verbal instructions, analyses its surroundings, and autonomously makes coffee in dynamic kitchen environments — adapting to unforeseen obstacles in real time rather than following pre-programmed sequences. ==== Suleyman's test ==== Mustafa Suleyman's test proposes giving an AI model US$100,000 and asking it to obtain US$1 million. ==== Use of video-games ==== Adams, et al. propose that the ability to learn and succeed in a wide range of video games can be used to test AI intelligence. This range would include games unknown to the AGI developers before the test is administered. === AI-complete problems === A problem is informally called "AI-complete" or "AI-hard" if it is believed that AGI would be needed to solve it, because the solution is beyond the capabilities of a purpose-specific algorithm. == History == === Classical AI === Modern AI research began in the mid-1950s. The first generation of AI researchers were convinced that artificial general intelligence was possible and that it would exist in just a few decades. AI pioneer Herbert A. Simon wrote in 1965: "machines will be capable, within twenty years, of doing any work a man can do". Their predictions were the inspiration for Stanley Kubrick and Arthur C. Clarke's fictional character HAL 9000, who embodied what AI researchers believed they could create by the year 2001. AI pioneer Marvin Minsky was a consultant on the project of making HAL 9000 as realistic as possible according to the consensus predictions of the time. He said in 1967, "Within a generation... the problem of
Modes of variation
In statistics, modes of variation are a continuously indexed set of vectors or functions that are centered at a mean and are used to depict the variation in a population or sample. Typically, variation patterns in the data can be decomposed in descending order of eigenvalues with the directions represented by the corresponding eigenvectors or eigenfunctions. Modes of variation provide a visualization of this decomposition and an efficient description of variation around the mean. Both in principal component analysis (PCA) and in functional principal component analysis (FPCA), modes of variation play an important role in visualizing and describing the variation in the data contributed by each eigencomponent. In real-world applications, the eigencomponents and associated modes of variation aid to interpret complex data, especially in exploratory data analysis (EDA). == Formulation == Modes of variation are a natural extension of PCA and FPCA. === Modes of variation in PCA === If a random vector X = ( X 1 , X 2 , ⋯ , X p ) T {\displaystyle \mathbf {X} =(X_{1},X_{2},\cdots ,X_{p})^{T}} has the mean vector μ p {\displaystyle {\boldsymbol {\mu }}_{p}} , and the covariance matrix Σ p × p {\displaystyle \mathbf {\Sigma } _{p\times p}} with eigenvalues λ 1 ≥ λ 2 ≥ ⋯ ≥ λ p ≥ 0 {\displaystyle \lambda _{1}\geq \lambda _{2}\geq \cdots \geq \lambda _{p}\geq 0} and corresponding orthonormal eigenvectors e 1 , e 2 , ⋯ , e p {\displaystyle \mathbf {e} _{1},\mathbf {e} _{2},\cdots ,\mathbf {e} _{p}} , by eigendecomposition of a real symmetric matrix, the covariance matrix Σ {\displaystyle \mathbf {\Sigma } } can be decomposed as Σ = Q Λ Q T , {\displaystyle \mathbf {\Sigma } =\mathbf {Q} \mathbf {\Lambda } \mathbf {Q} ^{T},} where Q {\displaystyle \mathbf {Q} } is an orthogonal matrix whose columns are the eigenvectors of Σ {\displaystyle \mathbf {\Sigma } } , and Λ {\displaystyle \mathbf {\Lambda } } is a diagonal matrix whose entries are the eigenvalues of Σ {\displaystyle \mathbf {\Sigma } } . By the Karhunen–Loève expansion for random vectors, one can express the centered random vector in the eigenbasis X − μ = ∑ k = 1 p ξ k e k , {\displaystyle \mathbf {X} -{\boldsymbol {\mu }}=\sum _{k=1}^{p}\xi _{k}\mathbf {e} _{k},} where ξ k = e k T ( X − μ ) {\displaystyle \xi _{k}=\mathbf {e} _{k}^{T}(\mathbf {X} -{\boldsymbol {\mu }})} is the principal component associated with the k {\displaystyle k} -th eigenvector e k {\displaystyle \mathbf {e} _{k}} , with the properties E ( ξ k ) = 0 , Var ( ξ k ) = λ k , {\displaystyle \operatorname {E} (\xi _{k})=0,\operatorname {Var} (\xi _{k})=\lambda _{k},} and E ( ξ k ξ l ) = 0 for l ≠ k . {\displaystyle \operatorname {E} (\xi _{k}\xi _{l})=0\ {\text{for}}\ l\neq k.} Then the k {\displaystyle k} -th mode of variation of X {\displaystyle \mathbf {X} } is the set of vectors, indexed by α {\displaystyle \alpha } , m k , α = μ ± α λ k e k , α ∈ [ − A , A ] , {\displaystyle \mathbf {m} _{k,\alpha }={\boldsymbol {\mu }}\pm \alpha {\sqrt {\lambda _{k}}}\mathbf {e} _{k},\alpha \in [-A,A],} where A {\displaystyle A} is typically selected as 2 or 3 {\displaystyle 2\ {\text{or}}\ 3} . === Modes of variation in FPCA === For a square-integrable random function X ( t ) , t ∈ T ⊂ R p {\displaystyle X(t),t\in {\mathcal {T}}\subset R^{p}} , where typically p = 1 {\displaystyle p=1} and T {\displaystyle {\mathcal {T}}} is an interval, denote the mean function by μ ( t ) = E ( X ( t ) ) {\displaystyle \mu (t)=\operatorname {E} (X(t))} , and the covariance function by G ( s , t ) = Cov ( X ( s ) , X ( t ) ) = ∑ k = 1 ∞ λ k φ k ( s ) φ k ( t ) , {\displaystyle G(s,t)=\operatorname {Cov} (X(s),X(t))=\sum _{k=1}^{\infty }\lambda _{k}\varphi _{k}(s)\varphi _{k}(t),} where λ 1 ≥ λ 2 ≥ ⋯ ≥ 0 {\displaystyle \lambda _{1}\geq \lambda _{2}\geq \cdots \geq 0} are the eigenvalues and { φ 1 , φ 2 , ⋯ } {\displaystyle \{\varphi _{1},\varphi _{2},\cdots \}} are the orthonormal eigenfunctions of the linear Hilbert–Schmidt operator G : L 2 ( T ) → L 2 ( T ) , G ( f ) = ∫ T G ( s , t ) f ( s ) d s . {\displaystyle G:L^{2}({\mathcal {T}})\rightarrow L^{2}({\mathcal {T}}),\,G(f)=\int _{\mathcal {T}}G(s,t)f(s)ds.} By the Karhunen–Loève theorem, one can express the centered function in the eigenbasis, X ( t ) − μ ( t ) = ∑ k = 1 ∞ ξ k φ k ( t ) , {\displaystyle X(t)-\mu (t)=\sum _{k=1}^{\infty }\xi _{k}\varphi _{k}(t),} where ξ k = ∫ T ( X ( t ) − μ ( t ) ) φ k ( t ) d t {\displaystyle \xi _{k}=\int _{\mathcal {T}}(X(t)-\mu (t))\varphi _{k}(t)dt} is the k {\displaystyle k} -th principal component with the properties E ( ξ k ) = 0 , Var ( ξ k ) = λ k , {\displaystyle \operatorname {E} (\xi _{k})=0,\operatorname {Var} (\xi _{k})=\lambda _{k},} and E ( ξ k ξ l ) = 0 for l ≠ k . {\displaystyle \operatorname {E} (\xi _{k}\xi _{l})=0{\text{ for }}l\neq k.} Then the k {\displaystyle k} -th mode of variation of X ( t ) {\displaystyle X(t)} is the set of functions, indexed by α {\displaystyle \alpha } , m k , α ( t ) = μ ( t ) ± α λ k φ k ( t ) , t ∈ T , α ∈ [ − A , A ] {\displaystyle m_{k,\alpha }(t)=\mu (t)\pm \alpha {\sqrt {\lambda _{k}}}\varphi _{k}(t),\ t\in {\mathcal {T}},\ \alpha \in [-A,A]} that are viewed simultaneously over the range of α {\displaystyle \alpha } , usually for A = 2 or 3 {\displaystyle A=2\ {\text{or}}\ 3} . == Estimation == The formulation above is derived from properties of the population. Estimation is needed in real-world applications. The key idea is to estimate mean and covariance. === Modes of variation in PCA === Suppose the data x 1 , x 2 , ⋯ , x n {\displaystyle \mathbf {x} _{1},\mathbf {x} _{2},\cdots ,\mathbf {x} _{n}} represent n {\displaystyle n} independent drawings from some p {\displaystyle p} -dimensional population X {\displaystyle \mathbf {X} } with mean vector μ {\displaystyle {\boldsymbol {\mu }}} and covariance matrix Σ {\displaystyle \mathbf {\Sigma } } . These data yield the sample mean vector x ¯ {\displaystyle {\overline {\mathbf {x} }}} , and the sample covariance matrix S {\displaystyle \mathbf {S} } with eigenvalue-eigenvector pairs ( λ ^ 1 , e ^ 1 ) , ( λ ^ 2 , e ^ 2 ) , ⋯ , ( λ ^ p , e ^ p ) {\displaystyle ({\hat {\lambda }}_{1},{\hat {\mathbf {e} }}_{1}),({\hat {\lambda }}_{2},{\hat {\mathbf {e} }}_{2}),\cdots ,({\hat {\lambda }}_{p},{\hat {\mathbf {e} }}_{p})} . Then the k {\displaystyle k} -th mode of variation of X {\displaystyle \mathbf {X} } can be estimated by m ^ k , α = x ¯ ± α λ ^ k e ^ k , α ∈ [ − A , A ] . {\displaystyle {\hat {\mathbf {m} }}_{k,\alpha }={\overline {\mathbf {x} }}\pm \alpha {\sqrt {{\hat {\lambda }}_{k}}}{\hat {\mathbf {e} }}_{k},\alpha \in [-A,A].} === Modes of variation in FPCA === Consider n {\displaystyle n} realizations X 1 ( t ) , X 2 ( t ) , ⋯ , X n ( t ) {\displaystyle X_{1}(t),X_{2}(t),\cdots ,X_{n}(t)} of a square-integrable random function X ( t ) , t ∈ T {\displaystyle X(t),t\in {\mathcal {T}}} with the mean function μ ( t ) = E ( X ( t ) ) {\displaystyle \mu (t)=\operatorname {E} (X(t))} and the covariance function G ( s , t ) = Cov ( X ( s ) , X ( t ) ) {\displaystyle G(s,t)=\operatorname {Cov} (X(s),X(t))} . Functional principal component analysis provides methods for the estimation of μ ( t ) {\displaystyle \mu (t)} and G ( s , t ) {\displaystyle G(s,t)} in detail, often involving point wise estimate and interpolation. Substituting estimates for the unknown quantities, the k {\displaystyle k} -th mode of variation of X ( t ) {\displaystyle X(t)} can be estimated by m ^ k , α ( t ) = μ ^ ( t ) ± α λ ^ k φ ^ k ( t ) , t ∈ T , α ∈ [ − A , A ] . {\displaystyle {\hat {m}}_{k,\alpha }(t)={\hat {\mu }}(t)\pm \alpha {\sqrt {{\hat {\lambda }}_{k}}}{\hat {\varphi }}_{k}(t),t\in {\mathcal {T}},\alpha \in [-A,A].} == Applications == Modes of variation are useful to visualize and describe the variation patterns in the data sorted by the eigenvalues. In real-world applications, modes of variation associated with eigencomponents allow to interpret complex data, such as the evolution of function traits and other infinite-dimensional data. To illustrate how modes of variation work in practice, two examples are shown in the graphs to the right, which display the first two modes of variation. The solid curve represents the sample mean function. The dashed, dot-dashed, and dotted curves correspond to modes of variation with α = ± 1 , ± 2 , {\displaystyle \alpha =\pm 1,\pm 2,} and ± 3 {\displaystyle \pm 3} , respectively. The first graph displays the first two modes of variation of female mortality data from 41 countries in 2003. The object of interest is log hazard function between ages 0 and 100 years. The first mode of variation suggests that the variation of female mortality is smaller for ages around 0 or 100, and larger for ages around 25. An appropriate and intuitive interpretation is that mortality around 25 is driven by accidental death, while around 0 or 100, mortality is related to congenital disease or natural death. Compared to female mortality
One-class classification
In machine learning, one-class classification (OCC), also known as unary classification or class-modelling, is an approach to the training of binary classifiers in which only examples of one of the two classes are used. Examples include the monitoring of helicopter gearboxes, motor failure prediction, or assessing the operational status of a nuclear plant as 'normal': In such scenarios, there are few, if any, examples of the catastrophic system states – rare outliers – that comprise the second class. Alternatively, the class that is being focused on may cover a small, coherent subset of the data and the training may rely on an information bottleneck approach. In practice, counter-examples from the second class may be used in later rounds of training to further refine the algorithm. == Overview == The term one-class classification (OCC) was coined by Moya & Hush (1996) and many applications can be found in scientific literature, for example outlier detection, anomaly detection, novelty detection. A feature of OCC is that it uses only sample points from the assigned class, so that a representative sampling is not strictly required for non-target classes. == Introduction == SVM based one-class classification (OCC) relies on identifying the smallest hypersphere (with radius r, and center c) consisting of all the data points. This method is called Support Vector Data Description (SVDD). Formally, the problem can be defined in the following constrained optimization form, min r , c r 2 subject to, | | Φ ( x i ) − c | | 2 ≤ r 2 ∀ i = 1 , 2 , . . . , n {\displaystyle \min _{r,c}r^{2}{\text{ subject to, }}||\Phi (x_{i})-c||^{2}\leq r^{2}\;\;\forall i=1,2,...,n} However, the above formulation is highly restrictive, and is sensitive to the presence of outliers. Therefore, a flexible formulation, that allow for the presence of outliers is formulated as shown below, min r , c , ζ r 2 + 1 ν n ∑ i = 1 n ζ i {\displaystyle \min _{r,c,\zeta }r^{2}+{\frac {1}{\nu n}}\sum _{i=1}^{n}\zeta _{i}} subject to, | | Φ ( x i ) − c | | 2 ≤ r 2 + ζ i ∀ i = 1 , 2 , . . . , n {\displaystyle {\text{subject to, }}||\Phi (x_{i})-c||^{2}\leq r^{2}+\zeta _{i}\;\;\forall i=1,2,...,n} From the Karush–Kuhn–Tucker conditions for optimality, we get c = ∑ i = 1 n α i Φ ( x i ) , {\displaystyle c=\sum _{i=1}^{n}\alpha _{i}\Phi (x_{i}),} where the α i {\displaystyle \alpha _{i}} 's are the solution to the following optimization problem: max α ∑ i = 1 n α i κ ( x i , x i ) − ∑ i , j = 1 n α i α j κ ( x i , x j ) {\displaystyle \max _{\alpha }\sum _{i=1}^{n}\alpha _{i}\kappa (x_{i},x_{i})-\sum _{i,j=1}^{n}\alpha _{i}\alpha _{j}\kappa (x_{i},x_{j})} subject to, ∑ i = 1 n α i = 1 and 0 ≤ α i ≤ 1 ν n for all i = 1 , 2 , . . . , n . {\displaystyle \sum _{i=1}^{n}\alpha _{i}=1{\text{ and }}0\leq \alpha _{i}\leq {\frac {1}{\nu n}}{\text{for all }}i=1,2,...,n.} The introduction of kernel function provide additional flexibility to the One-class SVM (OSVM) algorithm. === PU (Positive Unlabeled) learning === A similar problem is PU learning, in which a binary classifier is constructed by semi-supervised learning from only positive and unlabeled sample points. In PU learning, two sets of examples are assumed to be available for training: the positive set P {\displaystyle P} and a mixed set U {\displaystyle U} , which is assumed to contain both positive and negative samples, but without these being labeled as such. This contrasts with other forms of semisupervised learning, where it is assumed that a labeled set containing examples of both classes is available in addition to unlabeled samples. A variety of techniques exist to adapt supervised classifiers to the PU learning setting, including variants of the EM algorithm. PU learning has been successfully applied to text, time series, bioinformatics tasks, and remote sensing data. == Approaches == Several approaches have been proposed to solve one-class classification (OCC). The approaches can be distinguished into three main categories, density estimation, boundary methods, and reconstruction methods. === Density estimation methods === Density estimation methods rely on estimating the density of the data points, and set the threshold. These methods rely on assuming distributions, such as Gaussian, or a Poisson distribution. Following which discordancy tests can be used to test the new objects. These methods are robust to scale variance. Gaussian model is one of the simplest methods to create one-class classifiers. Due to Central Limit Theorem (CLT), these methods work best when large number of samples are present, and they are perturbed by small independent error values. The probability distribution for a d-dimensional object is given by: p N ( z ; μ ; Σ ) = 1 ( 2 π ) d 2 | Σ | 1 2 exp { − 1 2 ( z − μ ) T Σ − 1 ( z − μ ) } {\displaystyle p_{\mathcal {N}}(z;\mu ;\Sigma )={\frac {1}{(2\pi )^{\frac {d}{2}}|\Sigma |^{\frac {1}{2}}}}\exp \left\{-{\frac {1}{2}}(z-\mu )^{T}\Sigma ^{-1}(z-\mu )\right\}} Where, μ {\displaystyle \mu } is the mean and Σ {\displaystyle \Sigma } is the covariance matrix. Computing the inverse of covariance matrix ( Σ − 1 {\displaystyle \Sigma ^{-1}} ) is the costliest operation, and in the cases where the data is not scaled properly, or data has singular directions pseudo-inverse Σ + {\displaystyle \Sigma ^{+}} is used to approximate the inverse, and is calculated as Σ T ( Σ Σ T ) − 1 {\displaystyle \Sigma ^{T}(\Sigma \Sigma ^{T})^{-1}} . === Boundary methods === Boundary methods focus on setting boundaries around a few set of points, called target points. These methods attempt to optimize the volume. Boundary methods rely on distances, and hence are not robust to scale variance. K-centers method, NN-d, and SVDD are some of the key examples. K-centers In K-center algorithm, k {\displaystyle k} small balls with equal radius are placed to minimize the maximum distance of all minimum distances between training objects and the centers. Formally, the following error is minimized, ε k − c e n t e r = max i ( min k | | x i − μ k | | 2 ) {\displaystyle \varepsilon _{k-center}=\max _{i}(\min _{k}||x_{i}-\mu _{k}||^{2})} The algorithm uses forward search method with random initialization, where the radius is determined by the maximum distance of the object, any given ball should capture. After the centers are determined, for any given test object z {\displaystyle z} the distance can be calculated as, d k − c e n t r ( z ) = min k | | z − μ k | | 2 {\displaystyle d_{k-centr}(z)=\min _{k}||z-\mu _{k}||^{2}} === Reconstruction methods === Reconstruction methods use prior knowledge and generating process to build a generating model that best fits the data. New objects can be described in terms of a state of the generating model. Some examples of reconstruction methods for OCC are, k-means clustering, learning vector quantization, self-organizing maps, etc. == Applications == === Document classification === The basic Support Vector Machine (SVM) paradigm is trained using both positive and negative examples, however studies have shown there are many valid reasons for using only positive examples. When the SVM algorithm is modified to only use positive examples, the process is considered one-class classification. One situation where this type of classification might prove useful to the SVM paradigm is in trying to identify a web browser's sites of interest based only off of the user's browsing history. === Biomedical studies === One-class classification can be particularly useful in biomedical studies where often data from other classes can be difficult or impossible to obtain. In studying biomedical data it can be difficult and/or expensive to obtain the set of labeled data from the second class that would be necessary to perform a two-class classification. A study from The Scientific World Journal found that the typicality approach is the most useful in analysing biomedical data because it can be applied to any type of dataset (continuous, discrete, or nominal). The typicality approach is based on the clustering of data by examining data and placing it into new or existing clusters. To apply typicality to one-class classification for biomedical studies, each new observation, y 0 {\displaystyle y_{0}} , is compared to the target class, C {\displaystyle C} , and identified as an outlier or a member of the target class. === Unsupervised Concept Drift Detection === One-class classification has similarities with unsupervised concept drift detection, where both aim to identify whether the unseen data share similar characteristics to the initial data. A concept is referred to as the fixed probability distribution which data is drawn from. In unsupervised concept drift detection, the goal is to detect if the data distribution changes without utilizing class labels. In one-class classification, the flow of data is not important. Unseen data is classified as typical or outlier depending on its characteristics, whether it is from the initi
Synaptic weight
In neuroscience and computer science, synaptic weight refers to the strength or amplitude of a connection between two nodes, corresponding in biology to the amount of influence the firing of one neuron has on another. The term is typically used in artificial and biological neural network research. == Computation == In a computational neural network, a vector or set of inputs x {\displaystyle {\textbf {x}}} and outputs y {\displaystyle {\textbf {y}}} , or pre- and post-synaptic neurons respectively, are interconnected with synaptic weights represented by the matrix w {\displaystyle w} , where for a linear neuron y j = ∑ i w i j x i or y = w x {\displaystyle y_{j}=\sum _{i}w_{ij}x_{i}~~{\textrm {or}}~~{\textbf {y}}=w{\textbf {x}}} . where the rows of the synaptic matrix represent the vector of synaptic weights for the output indexed by j {\displaystyle j} . The synaptic weight is changed by using a learning rule, the most basic of which is Hebb's rule, which is usually stated in biological terms as Neurons that fire together, wire together. Computationally, this means that if a large signal from one of the input neurons results in a large signal from one of the output neurons, then the synaptic weight between those two neurons will increase. The rule is unstable, however, and is typically modified using such variations as Oja's rule, radial basis functions or the backpropagation algorithm. == Biology == For biological networks, the effect of synaptic weights is not as simple as for linear neurons or Hebbian learning. However, biophysical models such as BCM theory have seen some success in mathematically describing these networks. In the mammalian central nervous system, signal transmission is carried out by interconnected networks of nerve cells, or neurons. For the basic pyramidal neuron, the input signal is carried by the axon, which releases neurotransmitter chemicals into the synapse which is picked up by the dendrites of the next neuron, which can then generate an action potential which is analogous to the output signal in the computational case. The synaptic weight in this process is determined by several variable factors: How well the input signal propagates through the axon (see myelination), The amount of neurotransmitter released into the synapse and the amount that can be absorbed in the following cell (determined by the number of AMPA and NMDA receptors on the cell membrane and the amount of intracellular calcium and other ions), The number of such connections made by the axon to the dendrites, How well the signal propagates and integrates in the postsynaptic cell. The changes in synaptic weight that occur is known as synaptic plasticity, and the process behind long-term changes (long-term potentiation and depression) is still poorly understood. Hebb's original learning rule was originally applied to biological systems, but has had to undergo many modifications as a number of theoretical and experimental problems came to light.
Inverse consistency
In image registration, inverse consistency measures the consistency of mappings between images produced by a registration algorithm. The inverse consistency error, introduced by Christiansen and Johnson in 2001, quantifies the distance between the composition of the mappings from each image to the other, produced by the registration procedure, and the identity function, and is used as a regularisation constraint in the loss function of many registration algorithms to enforce consistent mappings. Inverse consistency is necessary for good image registration but it is not sufficient, since a mapping can be perfectly consistent but not register the images at all. == Definition == Image registration is the process of establishing a common coordinate system between two images, and given two images I 1 : Ω 1 → R I 2 : Ω 2 → R {\displaystyle {\begin{aligned}I_{1}:\Omega _{1}\to \mathbb {R} \\I_{2}:\Omega _{2}\to \mathbb {R} \end{aligned}}} registering a source image I 1 {\displaystyle I_{1}} to a target image I 2 {\displaystyle I_{2}} consists of determining a transformation f 1 : Ω 2 → Ω 1 {\displaystyle f_{1}:\Omega _{2}\to \Omega _{1}} that maps points from the target space to the source space. An ideal registration algorithm should not be sensitive to which image in the pair is used as source or target, and the registration operator should be antisymmetric such that the mappings f 1 : Ω 2 → Ω 1 f 2 : Ω 1 → Ω 2 {\displaystyle {\begin{aligned}f_{1}:\Omega _{2}\to \Omega _{1}\\f_{2}:\Omega _{1}\to \Omega _{2}\end{aligned}}} produced when registering I 1 {\displaystyle I_{1}} to I 2 {\displaystyle I_{2}} and I 2 {\displaystyle I_{2}} to I 1 {\displaystyle I_{1}} respectively should be the inverse of each other, i.e. f 2 = f 1 − 1 {\displaystyle f_{2}=f_{1}^{-1}} and f 1 = f 2 − 1 {\displaystyle f_{1}=f_{2}^{-1}} or, equivalently, f 2 ∘ f 1 = id Ω 2 {\displaystyle f_{2}\circ f_{1}=\operatorname {id} _{\Omega _{2}}} and f 1 ∘ f 2 = id Ω 1 {\displaystyle f_{1}\circ f_{2}=\operatorname {id} _{\Omega _{1}}} , where ∘ {\displaystyle \circ } denotes the function composition operator. Real algorithms are not perfect, and when swapping the role of source and target image in a registration problem the so obtained transformations are not the inverse of each other. Inverse consistency can be enforced by adding to the loss function of the registration a symmetric regularisation term that penalises inconsistent transformations ∫ Ω 2 ‖ f 2 ( f 1 ( x ) ) − x ‖ 2 d x + ∫ Ω 1 ‖ f 1 ( f 2 ( x ) ) − x ‖ 2 d x . {\displaystyle \int _{\Omega _{2}}\left\Vert f_{2}(f_{1}(x))-x\right\Vert ^{2}\mathrm {d} x+\int _{\Omega _{1}}\left\Vert f_{1}(f_{2}(x))-x\right\Vert ^{2}\mathrm {d} x.} Inverse consistency can be used as a quality metric to evaluate image registration results. The inverse consistency error ( I C E {\displaystyle ICE} ) measures the distance between the composition of the two transforms and the identity function, and it can be formulated in terms of both average ( I C E a {\displaystyle ICE_{a}} ) or maximum ( I C E m {\displaystyle ICE_{m}} ) over a region of interest Ω {\displaystyle \Omega } of the image: I C E a = 1 ∫ Ω d x ∫ Ω ‖ f 2 ( f 1 ( x ) ) − x ‖ d x I C E m = max x ∈ Ω ‖ f 2 ( f 1 ( x ) ) − x ‖ . {\displaystyle {\begin{aligned}ICE_{a}&={\frac {1}{\int _{\Omega }\mathrm {d} x}}\int _{\Omega }\left\Vert f_{2}(f_{1}(x))-x\right\Vert \mathrm {d} x\\ICE_{m}&=\max _{x\in \Omega }\left\Vert f_{2}(f_{1}(x))-x\right\Vert .\end{aligned}}} While inverse consistency is a necessary property of good registration algorithms, inverse consistency error alone is not a sufficient metric to evaluate the quality of image registration results, since a perfectly consistent mapping, with no other constraint, may be not even close to correctly register a pair of images.
Logic learning machine
Logic learning machine (LLM) is a machine learning method based on the generation of intelligible rules. LLM is an efficient implementation of the Switching Neural Network (SNN) paradigm, developed by Marco Muselli, Senior Researcher at the Italian National Research Council CNR-IEIIT in Genoa. LLM has been employed in many different sectors, including the field of medicine (orthopedic patient classification, DNA micro-array analysis and Clinical Decision Support Systems), financial services and supply chain management. == History == The Switching Neural Network approach was developed in the 1990s to overcome the drawbacks of the most commonly used machine learning methods. In particular, black box methods, such as multilayer perceptron and support vector machine, had good accuracy but could not provide deep insight into the studied phenomenon. On the other hand, decision trees were able to describe the phenomenon but often lacked accuracy. Switching Neural Networks made use of Boolean algebra to build sets of intelligible rules able to obtain very good performance. In 2014, an efficient version of Switching Neural Network was developed and implemented in the Rulex suite with the name Logic Learning Machine. Also, an LLM version devoted to regression problems was developed. == General == Like other machine learning methods, LLM uses data to build a model able to perform a good forecast about future behaviors. LLM starts from a table including a target variable (output) and some inputs and generates a set of rules that return the output value y {\displaystyle y} corresponding to a given configuration of inputs. A rule is written in the form: if premise then consequence where consequence contains the output value whereas premise includes one or more conditions on the inputs. According to the input type, conditions can have different forms: for categorical variables the input value must be in a given subset: x 1 ∈ { A , B , C , . . . } {\displaystyle x_{1}\in \{A,B,C,...\}} . for ordered variables the condition is written as an inequality or an interval: x 2 ≤ α {\displaystyle x_{2}\leq \alpha } or β ≤ x 3 ≤ γ {\displaystyle \beta \leq x_{3}\leq \gamma } A possible rule is therefore in the form if x 1 ∈ { A , B , C , . . . } {\displaystyle x_{1}\in \{A,B,C,...\}} AND x 2 ≤ α {\displaystyle x_{2}\leq \alpha } AND β ≤ x 3 ≤ γ {\displaystyle \beta \leq x_{3}\leq \gamma } then y = y ¯ {\displaystyle y={\bar {y}}} == Types == According to the output type, different versions of the Logic Learning Machine have been developed: Logic Learning Machine for classification, when the output is a categorical variable, which can assume values in a finite set Logic Learning Machine for regression, when the output is an integer or real number.