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Boosting (machine learning) : Robert E. Schapire (2003); The Boosting Approach to Machine Learning: An Overview, MSRI (Mathematical Sciences Research Institute) Workshop on Nonlinear Estimation and Classification Boosting: Foundations and Algorithms by Robert E. Schapire and Yoav Freund
Constellation model : The constellation model is a probabilistic, generative model for category-level object recognition in computer vision. Like other part-based models, the constellation model attempts to represent an object class by a set of N parts under mutual geometric constraints. Because it considers the geometric relationship between different parts, the constellation model differs significantly from appearance-only, or "bag-of-words" representation models, which explicitly disregard the location of image features. The problem of defining a generative model for object recognition is difficult. The task becomes significantly complicated by factors such as background clutter, occlusion, and variations in viewpoint, illumination, and scale. Ideally, we would like the particular representation we choose to be robust to as many of these factors as possible. In category-level recognition, the problem is even more challenging because of the fundamental problem of intra-class variation. Even if two objects belong to the same visual category, their appearances may be significantly different. However, for structured objects such as cars, bicycles, and people, separate instances of objects from the same category are subject to similar geometric constraints. For this reason, particular parts of an object such as the headlights or tires of a car still have consistent appearances and relative positions. The Constellation Model takes advantage of this fact by explicitly modeling the relative location, relative scale, and appearance of these parts for a particular object category. Model parameters are estimated using an unsupervised learning algorithm, meaning that the visual concept of an object class can be extracted from an unlabeled set of training images, even if that set contains "junk" images or instances of objects from multiple categories. It can also account for the absence of model parts due to appearance variability, occlusion, clutter, or detector error.
Constellation model : The idea for a "parts and structure" model was originally introduced by Fischler and Elschlager in 1973. This model has since been built upon and extended in many directions. The Constellation Model, as introduced by Dr. Perona and his colleagues, was a probabilistic adaptation of this approach. In the late '90s, Burl et al. revisited the Fischler and Elschlager model for the purpose of face recognition. In their work, Burl et al. used manual selection of constellation parts in training images to construct a statistical model for a set of detectors and the relative locations at which they should be applied. In 2000, Weber et al. made the significant step of training the model using a more unsupervised learning process, which precluded the necessity for tedious hand-labeling of parts. Their algorithm was particularly remarkable because it performed well even on cluttered and occluded image data. Fergus et al. then improved upon this model by making the learning step fully unsupervised, having both shape and appearance learned simultaneously, and accounting explicitly for the relative scale of parts.
Constellation model : In the first step, a standard interest point detection method, such as Harris corner detection, is used to generate interest points. Image features generated from the vicinity of these points are then clustered using k-means or another appropriate algorithm. In this process of vector quantization, one can think of the centroids of these clusters as being representative of the appearance of distinctive object parts. Appropriate feature detectors are then trained using these clusters, which can be used to obtain a set of candidate parts from images. As a result of this process, each image can now be represented as a set of parts. Each part has a type, corresponding to one of the aforementioned appearance clusters, as well as a location in the image space.
Constellation model : In Weber et al., shape and appearance models are constructed separately. Once the set of candidate parts had been selected, shape is learned independently of appearance. The innovation of Fergus et al. is to learn not only two, but three model parameters simultaneously: shape, appearance, and relative scale. Each of these parameters are represented by Gaussian densities.
Constellation model : The Constellation Model as conceived by Fergus et al. achieves successful categorization rates consistently above 90% on large datasets of motorbikes, faces, airplanes, and spotted cats. For each of these datasets, the Constellation Model is able to capture the "essence" of the object class in terms of appearance and/or shape. For example, face and motorbike datasets generate very tight shape models because objects in those categories have very well-defined structure, whereas spotted cats vary significantly in pose, but have a very distinctive spotted appearance. Thus, the model succeeds in both cases. It is important to note that the Constellation Model does not generally account for significant changes in orientation. Thus, if the model is trained on images of horizontal airplanes, it will not perform well on, for instance, images of vertically oriented planes unless the model is extended to account for this sort of rotation explicitly. In terms of computational complexity, the Constellation Model is very expensive. If N is the number of feature detections in the image, and P the number of parts in the object model, then the hypothesis space H is O ( N P ) )\, . Because the computation of sufficient statistics in the E-step of expectation maximization necessitates evaluating the likelihood for every hypothesis, learning becomes a major bottleneck operation. For this reason, only values of P ≤ 6 have been used in practical applications, and the number of feature detections N is usually kept within the range of about 20-30 per image.
Constellation model : One variation that attempts to reduce complexity is the star model proposed by Fergus et al. The reduced dependencies of this model allows for learning in O ( N 2 P ) P)\, time instead of O ( N P ) )\, . This allows for a greater number of model parts and image features to be used in training. Because the star model has fewer parameters, it is also better at avoiding the problem of over-fitting when trained on fewer images.
Constellation model : L. Fei-fei. Object categorization: the constellation models. Lecture Slides. (2005) (link not working)
Constellation model : Part-based models One-shot learning in computer vision
ImageNets : ImageNets is an open source framework for rapid prototyping of machine vision algorithms, developed by the Institute of Automation.
ImageNets : ImageNets is an open source and platform independent (Windows & Linux) framework for rapid prototyping of machine vision algorithms. With the GUI ImageNet Designer, no programming knowledge is required to perform operations on images. A configured ImageNet can be loaded and executed from C++ code without the need for loading the ImageNet Designer GUI to achieve higher execution performance.
ImageNets : ImageNets was developed by the Institute of Automation, University of Bremen, Germany. The software was first publicly released in 2010. Originally, ImageNets was developed for the Care-Providing Robot FRIEND but it can be used for a wide range of computer vision applications.
ImageNets : ImageNets homepage Download ImageNets
One-shot learning (computer vision) : One-shot learning is an object categorization problem, found mostly in computer vision. Whereas most machine learning-based object categorization algorithms require training on hundreds or thousands of examples, one-shot learning aims to classify objects from one, or only a few, examples. The term few-shot learning is also used for these problems, especially when more than one example is needed.
One-shot learning (computer vision) : The ability to learn object categories from few examples, and at a rapid pace, has been demonstrated in humans. It is estimated that a child learns almost all of the 10 ~ 30 thousand object categories in the world by age six. This is due not only to the human mind's computational power, but also to its ability to synthesize and learn new object categories from existing information about different, previously learned categories. Given two examples from two object categories: one, an unknown object composed of familiar shapes, the second, an unknown, amorphous shape; it is much easier for humans to recognize the former than the latter, suggesting that humans make use of previously learned categories when learning new ones. The key motivation for solving one-shot learning is that systems, like humans, can use knowledge about object categories to classify new objects.
One-shot learning (computer vision) : As with most classification schemes, one-shot learning involves three main challenges: Representation: How should objects and categories be described? Learning: How can such descriptions be created? Recognition: How can a known object be filtered from enveloping clutter, irrespective of occlusion, viewpoint, and lighting? One-shot learning differs from single object recognition and standard category recognition algorithms in its emphasis on knowledge transfer, which makes use of previously learned categories. Model parameters: Reuses model parameters, based on the similarity between old and new categories. Categories are first learned on numerous training examples, then new categories are learned using transformations of model parameters from those initial categories or selecting relevant parameters for a classifier. Feature sharing: Shares parts or features of objects across categories. One algorithm extracts "diagnostic information" in patches from already learned categories by maximizing the patches' mutual information, and then applies these features to the learning of a new category. A dog category, for example, may be learned in one shot from previous knowledge of horse and cow categories, because dog objects may contain similar distinguishing patches. Contextual information: Appeals to global knowledge of the scene in which the object appears. Such global information can be used as frequency distributions in a conditional random field framework to recognize objects. Alternatively context can consider camera height and scene geometry. Algorithms of this type have two advantages. First, they learn object categories that are relatively dissimilar; and second, they perform well in ad hoc situations where an image has not been hand-cropped and aligned.
One-shot learning (computer vision) : The Bayesian one-shot learning algorithm represents the foreground and background of images as parametrized by a mixture of constellation models. During the learning phase, the parameters of these models are learned using a conjugate density parameter posterior and Variational Bayesian Expectation–Maximization (VBEM). In this stage the previously learned object categories inform the choice of model parameters via transfer by contextual information. For object recognition on new images, the posterior obtained during the learning phase is used in a Bayesian decision framework to estimate the ratio of p(object \| test, train) to p(background clutter \| test, train) where p is the probability of the outcome.
One-shot learning (computer vision) : Another algorithm uses knowledge transfer by model parameters to learn a new object category that is similar in appearance to previously learned categories. An image is represented as either a texture and shape, or as a latent image that has been transformed, denoted by I = T ( I L ) ) . A Siamese neural network works in tandem on two different input vectors to compute comparable output vectors.
One-shot learning (computer vision) : Variational Bayesian methods Variational message passing Expectation–maximization algorithm Bayesian inference Feature detection Association rule learning Hopfield network Zero-shot learning
One-shot learning (computer vision) : Li, Fei Fei (2006). "Knowledge transfer in learning to recognize visual object classes" (PDF). International Conference on Development and Learning (ICDL). Li, Fei Fei; Fergus, R.; Perona, P. (2006). "One-Shot learning of object categories" (PDF). IEEE Transactions on Pattern Analysis and Machine Intelligence. 28 (4): 594–611. doi:10.1109/TPAMI.2006.79. PMID 16566508. S2CID 6953475. Miller; Matsakis; Viola (2000). "Learning from One Example through Shared Densities on Transforms" (PDF). Proc. Computer Vision and Pattern Recognition. Li, F.F.; VanRullen, R.; Coch, C.; Perona, P. (2002). "Rapid natural scene categorization in the near absence of attention". PNAS. 99 (14): 9596–9601. Bibcode:2002PNAS...99.9596L. doi:10.1073/pnas.092277599. PMC 123186. PMID 12077298. Thorpe, S.; Fize, D.; Marlot, C. (1996). "Speed of processing in the human visual system" (PDF). Nature. 381 (6582): 520–522, 1996. Bibcode:1996Natur.381..520T. doi:10.1038/381520a0. PMID 8632824. S2CID 4303570. Biederman, I. (1987). "Recognition-by-Components: a theory of human understanding" (PDF). Psychological Review. 94 (2): 115–147. doi:10.1037/0033-295X.94.2.115. PMID 3575582. Fink, M. (2004). "Object classification from a single example utilizing class relevance pseudo-metrics". NIPS. CiteSeerX 10.1.1.91.7461. Bart; Ullman (2005). "Cross-generalization: learning novel classes from a single example by feature replacement" (PDF). CVPR. Murphy, K.; Torralba, A.; Freeman, W.T. (2004). "Using the forest to see the trees: a graphical model relating features, objects, and scenes" (PDF). NIPS. Hoiem, D.; Efros, A.A.; Herbert, M. (2005). "Geometric context from a single image" (PDF). ICCV. Attias, H. (1999). "Inferring Parameters and Structure of Latent Variable Models by Variational Bayes". Proc. Of the 15th Conf. In Uncertainty in Artificial Intelligence: 21–30. arXiv:1301.6676. Burl, M.; Weber, M.; Perona, P. (1996). "A Probabilistic Approach to Object Recognition Using Local Photometry and Global Geometry" (PDF). Proc. European Conf. Computer Vision. Lecture Notes in Computer Science. 1407: 628–641. doi:10.1007/BFb0054769. ISBN 978-3-540-64613-6. Fergus, R.; Perona, P.; Zisserman, A. (2003). "Object Class Recognition by Unsupervised Scale-Invariant Learning" (PDF). Proc. Computer Vision and Pattern Recognition: 264–271. Weber, M.; Welling, M.; Perona, P. (2000). "Unsupervised Learning of Models for Recognition" (PDF). Proc. European Conf. Computer Vision. Lecture Notes in Computer Science. 1842: 101–108. doi:10.1007/3-540-45054-8_2. ISBN 978-3-540-67685-0. Kadir, T.; Brady, M. (2001). "Scale, Saliency, and Image Description". International Journal of Computer Vision. 45 (2): 83–105. doi:10.1023/A:1012460413855. S2CID 825395.
Scale-invariant feature operator : In the fields of computer vision and image analysis, the scale-invariant feature operator (or SFOP) is an algorithm to detect local features in images. The algorithm was published by Förstner et al. in 2009.
Scale-invariant feature operator : The scale-invariant feature operator (SFOP) is based on two theoretical concepts: spiral model feature operator Desired properties of keypoint detectors: Invariance and repeatability for object recognition Accuracy to support camera calibration Interpretability: Especially corners and circles, should be part of the detected keypoints (see figure). As few control parameters as possible with clear semantics Complementarity to known detectors scale-invariant corner/circle detector.
Scale-invariant feature operator : Corner detection Feature detection (computer vision)
Scale-invariant feature operator : Project website at University of Bonn
Fisher kernel : In statistical classification, the Fisher kernel, named after Ronald Fisher, is a function that measures the similarity of two objects on the basis of sets of measurements for each object and a statistical model. In a classification procedure, the class for a new object (whose real class is unknown) can be estimated by minimising, across classes, an average of the Fisher kernel distance from the new object to each known member of the given class. The Fisher kernel was introduced in 1998. It combines the advantages of generative statistical models (like the hidden Markov model) and those of discriminative methods (like support vector machines): generative models can process data of variable length (adding or removing data is well-supported) discriminative methods can have flexible criteria and yield better results.
Fisher kernel : Fisher information metric
Fisher kernel : Nello Cristianini and John Shawe-Taylor. An Introduction to Support Vector Machines and other kernel-based learning methods. Cambridge University Press, 2000. ISBN 0-521-78019-5 ([1] SVM Book)
Gaussian process : In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution. The distribution of a Gaussian process is the joint distribution of all those (infinitely many) random variables, and as such, it is a distribution over functions with a continuous domain, e.g. time or space. The concept of Gaussian processes is named after Carl Friedrich Gauss because it is based on the notion of the Gaussian distribution (normal distribution). Gaussian processes can be seen as an infinite-dimensional generalization of multivariate normal distributions. Gaussian processes are useful in statistical modelling, benefiting from properties inherited from the normal distribution. For example, if a random process is modelled as a Gaussian process, the distributions of various derived quantities can be obtained explicitly. Such quantities include the average value of the process over a range of times and the error in estimating the average using sample values at a small set of times. While exact models often scale poorly as the amount of data increases, multiple approximation methods have been developed which often retain good accuracy while drastically reducing computation time.
Gaussian process : A time continuous stochastic process ;t\in T\right\ is Gaussian if and only if for every finite set of indices t 1 , … , t k ,\ldots ,t_ in the index set T X t 1 , … , t k = ( X t 1 , … , X t k ) _,\ldots ,t_=(X_,\ldots ,X_) is a multivariate Gaussian random variable. As the sum of independent and Gaussian distributed random variables is again Gaussian distributed, that is the same as saying every linear combination of ( X t 1 , … , X t k ) ,\ldots ,X_) has a univariate Gaussian (or normal) distribution. Using characteristic functions of random variables with i denoting the imaginary unit such that i 2 = − 1 =-1 , the Gaussian property can be formulated as follows: ;t\in T\right\ is Gaussian if and only if, for every finite set of indices t 1 , … , t k ,\ldots ,t_ , there are real-valued σ ℓ j , μ ℓ with σ j j > 0 >0 such that the following equality holds for all s 1 , s 2 , … , s k ∈ R ,s_,\ldots ,s_\in \mathbb , E [ exp ⁡ ( i ∑ ℓ = 1 k s ℓ X t ℓ ) ] = exp ⁡ ( − 1 2 ∑ ℓ , j σ ℓ j s ℓ s j + i ∑ ℓ μ ℓ s ℓ ) , \left[\exp \left(i\sum _^s_\,\mathbf _\right)\right]=\exp \left(-\sum _\sigma _s_s_+i\sum _\mu _s_\right), or E [ e i s ( X t − μ ) ] = e − s σ s / 2 \left[ ^ \,(\mathbf _-\mathbf )\right]= ^ \,\sigma \,\mathbf /2 . The numbers σ ℓ j and μ ℓ can be shown to be the covariances and means of the variables in the process.
Gaussian process : The variance of a Gaussian process is finite at any time t , formally: p. 515 var ⁡ [ X ( t ) ] = E [ \| X ( t ) − E ⁡ [ X ( t ) ] \| 2 ] < ∞ for all t ∈ T . [X(t)]= \left[\left\|X(t)-\operatorname [X(t)]\right\|^\right]<\infty \quad t\in T.
Gaussian process : For general stochastic processes strict-sense stationarity implies wide-sense stationarity but not every wide-sense stationary stochastic process is strict-sense stationary. However, for a Gaussian stochastic process the two concepts are equivalent.: p. 518 A Gaussian stochastic process is strict-sense stationary if and only if it is wide-sense stationary.
Gaussian process : There is an explicit representation for stationary Gaussian processes. A simple example of this representation is X t = cos ⁡ ( a t ) ξ 1 + sin ⁡ ( a t ) ξ 2 =\cos(at)\,\xi _+\sin(at)\,\xi _ where ξ 1 and ξ 2 are independent random variables with the standard normal distribution.
Gaussian process : A key fact of Gaussian processes is that they can be completely defined by their second-order statistics. Thus, if a Gaussian process is assumed to have mean zero, defining the covariance function completely defines the process' behaviour. Importantly the non-negative definiteness of this function enables its spectral decomposition using the Karhunen–Loève expansion. Basic aspects that can be defined through the covariance function are the process' stationarity, isotropy, smoothness and periodicity. Stationarity refers to the process' behaviour regarding the separation of any two points x and x ′ . If the process is stationary, the covariance function depends only on x − x ′ . For example, the Ornstein–Uhlenbeck process is stationary. If the process depends only on \| x − x ′ \| , the Euclidean distance (not the direction) between x and x ′ , then the process is considered isotropic. A process that is concurrently stationary and isotropic is considered to be homogeneous; in practice these properties reflect the differences (or rather the lack of them) in the behaviour of the process given the location of the observer. Ultimately Gaussian processes translate as taking priors on functions and the smoothness of these priors can be induced by the covariance function. If we expect that for "near-by" input points x and x ′ their corresponding output points y and y ′ to be "near-by" also, then the assumption of continuity is present. If we wish to allow for significant displacement then we might choose a rougher covariance function. Extreme examples of the behaviour is the Ornstein–Uhlenbeck covariance function and the squared exponential where the former is never differentiable and the latter infinitely differentiable. Periodicity refers to inducing periodic patterns within the behaviour of the process. Formally, this is achieved by mapping the input x to a two dimensional vector u ( x ) = ( cos ⁡ ( x ) , sin ⁡ ( x ) ) .
Gaussian process : For a Gaussian process, continuity in probability is equivalent to mean-square continuity: 145 : 91 "Gaussian processes are discontinuous at fixed points." and continuity with probability one is equivalent to sample continuity. The latter implies, but is not implied by, continuity in probability. Continuity in probability holds if and only if the mean and autocovariance are continuous functions. In contrast, sample continuity was challenging even for stationary Gaussian processes (as probably noted first by Andrey Kolmogorov), and more challenging for more general processes.: Sect. 2.8 : 69, 81 : 80 As usual, by a sample continuous process one means a process that admits a sample continuous modification. : 292 : 424
Gaussian process : A Wiener process (also known as Brownian motion) is the integral of a white noise generalized Gaussian process. It is not stationary, but it has stationary increments. The Ornstein–Uhlenbeck process is a stationary Gaussian process. The Brownian bridge is (like the Ornstein–Uhlenbeck process) an example of a Gaussian process whose increments are not independent. The fractional Brownian motion is a Gaussian process whose covariance function is a generalisation of that of the Wiener process.
Gaussian process : Let f be a mean-zero Gaussian process ;t\in T\right\ with a non-negative definite covariance function K and let R be a symmetric and positive semidefinite function. Then, there exists a Gaussian process X which has the covariance R . Moreover, the reproducing kernel Hilbert space (RKHS) associated to R coincides with the Cameron–Martin theorem associated space R ( H ) of X , and all the spaces R ( H ) , H X , and H ( K ) (K) are isometric. From now on, let H ( R ) (R) be a reproducing kernel Hilbert space with positive definite kernel R . Driscoll's zero-one law is a result characterizing the sample functions generated by a Gaussian process: lim n → ∞ tr ⁡ [ K n R n − 1 ] < ∞ , \operatorname [K_R_^]<\infty , where K n and R n are the covariance matrices of all possible pairs of n points, implies Pr [ f ∈ H ( R ) ] = 1. (R)]=1. Moreover, lim n → ∞ tr ⁡ [ K n R n − 1 ] = ∞ \operatorname [K_R_^]=\infty implies Pr [ f ∈ H ( R ) ] = 0. (R)]=0. This has significant implications when K = R , as lim n → ∞ tr ⁡ [ R n R n − 1 ] = lim n → ∞ tr ⁡ [ I ] = lim n → ∞ n = ∞ . \operatorname [R_R_^]=\lim _\operatorname [I]=\lim _n=\infty . As such, almost all sample paths of a mean-zero Gaussian process with positive definite kernel K will lie outside of the Hilbert space H ( K ) (K) .
Gaussian process : For many applications of interest some pre-existing knowledge about the system at hand is already given. Consider e.g. the case where the output of the Gaussian process corresponds to a magnetic field; here, the real magnetic field is bound by Maxwell's equations and a way to incorporate this constraint into the Gaussian process formalism would be desirable as this would likely improve the accuracy of the algorithm. A method on how to incorporate linear constraints into Gaussian processes already exists: Consider the (vector valued) output function f ( x ) which is known to obey the linear constraint (i.e. F X _ is a linear operator) F X ( f ( x ) ) = 0. _(f(x))=0. Then the constraint F X _ can be fulfilled by choosing f ( x ) = G X ( g ( x ) ) _(g(x)) , where g ( x ) ∼ G P ( μ g , K g ) (\mu _,K_) is modelled as a Gaussian process, and finding G X _ such that F X ( G X ( g ) ) = 0 ∀ g . _(_(g))=0\qquad \forall g. Given G X _ and using the fact that Gaussian processes are closed under linear transformations, the Gaussian process for f obeying constraint F X _ becomes f ( x ) = G X g ∼ G P ( G X μ g , G X K g G X ′ T ) . _g\sim (_\mu _,_K__^). Hence, linear constraints can be encoded into the mean and covariance function of a Gaussian process.
Gaussian process : A Gaussian process can be used as a prior probability distribution over functions in Bayesian inference. Given any set of N points in the desired domain of your functions, take a multivariate Gaussian whose covariance matrix parameter is the Gram matrix of your N points with some desired kernel, and sample from that Gaussian. For solution of the multi-output prediction problem, Gaussian process regression for vector-valued function was developed. In this method, a 'big' covariance is constructed, which describes the correlations between all the input and output variables taken in N points in the desired domain. This approach was elaborated in detail for the matrix-valued Gaussian processes and generalised to processes with 'heavier tails' like Student-t processes. Inference of continuous values with a Gaussian process prior is known as Gaussian process regression, or kriging; extending Gaussian process regression to multiple target variables is known as cokriging. Gaussian processes are thus useful as a powerful non-linear multivariate interpolation tool. Kriging is also used to extend Gaussian process in the case of mixed integer inputs. Gaussian processes are also commonly used to tackle numerical analysis problems such as numerical integration, solving differential equations, or optimisation in the field of probabilistic numerics. Gaussian processes can also be used in the context of mixture of experts models, for example. The underlying rationale of such a learning framework consists in the assumption that a given mapping cannot be well captured by a single Gaussian process model. Instead, the observation space is divided into subsets, each of which is characterized by a different mapping function; each of these is learned via a different Gaussian process component in the postulated mixture. In the natural sciences, Gaussian processes have found use as probabilistic models of astronomical time series and as predictors of molecular properties. They are also being increasingly used as surrogate models for force field optimization.
Gaussian process : In practical applications, Gaussian process models are often evaluated on a grid leading to multivariate normal distributions. Using these models for prediction or parameter estimation using maximum likelihood requires evaluating a multivariate Gaussian density, which involves calculating the determinant and the inverse of the covariance matrix. Both of these operations have cubic computational complexity which means that even for grids of modest sizes, both operations can have a prohibitive computational cost. This drawback led to the development of multiple approximation methods.
Gaussian process : Bayes linear statistics Bayesian interpretation of regularization Kriging Gaussian free field Gauss–Markov process Gradient-enhanced kriging (GEK) Student's t-process
Gram matrix : In linear algebra, the Gram matrix (or Gramian matrix, Gramian) of a set of vectors v 1 , … , v n ,\dots ,v_ in an inner product space is the Hermitian matrix of inner products, whose entries are given by the inner product G i j = ⟨ v i , v j ⟩ =\left\langle v_,v_\right\rangle . If the vectors v 1 , … , v n ,\dots ,v_ are the columns of matrix X then the Gram matrix is X † X X in the general case that the vector coordinates are complex numbers, which simplifies to X ⊤ X X for the case that the vector coordinates are real numbers. An important application is to compute linear independence: a set of vectors are linearly independent if and only if the Gram determinant (the determinant of the Gram matrix) is non-zero. It is named after Jørgen Pedersen Gram.
Gram matrix : For finite-dimensional real vectors in R n ^ with the usual Euclidean dot product, the Gram matrix is G = V ⊤ V V , where V is a matrix whose columns are the vectors v k and V ⊤ is its transpose whose rows are the vectors v k ⊤ ^ . For complex vectors in C n ^ , G = V † V V , where V † is the conjugate transpose of V . Given square-integrable functions (\cdot ),\,i=1,\dots ,n\ on the interval [ t 0 , t f ] ,t_\right] , the Gram matrix G = [ G i j ] \right] is: G i j = ∫ t 0 t f ℓ i ∗ ( τ ) ℓ j ( τ ) d τ . =\int _^\ell _^(\tau )\ell _(\tau )\,d\tau . where ℓ i ∗ ( τ ) ^(\tau ) is the complex conjugate of ℓ i ( τ ) (\tau ) . For any bilinear form B on a finite-dimensional vector space over any field we can define a Gram matrix G attached to a set of vectors v 1 , … , v n ,\dots ,v_ by G i j = B ( v i , v j ) =B\left(v_,v_\right) . The matrix will be symmetric if the bilinear form B is symmetric.
Gram matrix : The Gram determinant or Gramian is the determinant of the Gram matrix: \| G ( v 1 , … , v n ) \| = \| ⟨ v 1 , v 1 ⟩ ⟨ v 1 , v 2 ⟩ … ⟨ v 1 , v n ⟩ ⟨ v 2 , v 1 ⟩ ⟨ v 2 , v 2 ⟩ … ⟨ v 2 , v n ⟩ ⋮ ⋮ ⋱ ⋮ ⟨ v n , v 1 ⟩ ⟨ v n , v 2 ⟩ … ⟨ v n , v n ⟩ \| . G(v_,\dots ,v_)=\langle v_,v_\rangle &\langle v_,v_\rangle &\dots &\langle v_,v_\rangle \\\langle v_,v_\rangle &\langle v_,v_\rangle &\dots &\langle v_,v_\rangle \\\vdots &\vdots &\ddots &\vdots \\\langle v_,v_\rangle &\langle v_,v_\rangle &\dots &\langle v_,v_\rangle \end. If v 1 , … , v n ,\dots ,v_ are vectors in R m ^ then it is the square of the n-dimensional volume of the parallelotope formed by the vectors. In particular, the vectors are linearly independent if and only if the parallelotope has nonzero n-dimensional volume, if and only if Gram determinant is nonzero, if and only if the Gram matrix is nonsingular. When n > m the determinant and volume are zero. When n = m, this reduces to the standard theorem that the absolute value of the determinant of n n-dimensional vectors is the n-dimensional volume. The Gram determinant is also useful for computing the volume of the simplex formed by the vectors; its volume is Volume(parallelotope) / n!. The Gram determinant can also be expressed in terms of the exterior product of vectors by \| G ( v 1 , … , v n ) \| = ‖ v 1 ∧ ⋯ ∧ v n ‖ 2 . G(v_,\dots ,v_)=\\|v_\wedge \cdots \wedge v_\\|^. The Gram determinant therefore supplies an inner product for the space ⁠ ⋀ n ( V ) ^(V) ⁠. If an orthonormal basis ei, i = 1, 2, ..., n on ⁠ V ⁠ is given, the vectors e i 1 ∧ ⋯ ∧ e i n , i 1 < ⋯ < i n , \wedge \cdots \wedge e_,\quad i_<\cdots <i_, will constitute an orthonormal basis of n-dimensional volumes on the space ⁠ ⋀ n ( V ) ^(V) ⁠. Then the Gram determinant \| G ( v 1 , … , v n ) \| G(v_,\dots ,v_) amounts to an n-dimensional Pythagorean Theorem for the volume of the parallelotope formed by the vectors v 1 ∧ ⋯ ∧ v n \wedge \cdots \wedge v_ in terms of its projections onto the basis volumes e i 1 ∧ ⋯ ∧ e i n \wedge \cdots \wedge e_ . When the vectors v 1 , … , v n ∈ R m ,\ldots ,v_\in \mathbb ^ are defined from the positions of points p 1 , … , p n ,\ldots ,p_ relative to some reference point p n + 1 , ( v 1 , v 2 , … , v n ) = ( p 1 − p n + 1 , p 2 − p n + 1 , … , p n − p n + 1 ) , ,v_,\ldots ,v_)=(p_-p_,p_-p_,\ldots ,p_-p_)\,, then the Gram determinant can be written as the difference of two Gram determinants, \| G ( v 1 , … , v n ) \| = \| G ( ( p 1 , 1 ) , … , ( p n + 1 , 1 ) ) \| − \| G ( p 1 , … , p n + 1 ) \| , G(v_,\dots ,v_)=G((p_,1),\dots ,(p_,1))-G(p_,\dots ,p_)\,, where each ( p j , 1 ) ,1) is the corresponding point p j supplemented with the coordinate value of 1 for an ( m + 1 ) -st dimension. Note that in the common case that n = m, the second term on the right-hand side will be zero.
Gram matrix : Given a set of linearly independent vectors \ with Gram matrix G defined by G i j := ⟨ v i , v j ⟩ :=\langle v_,v_\rangle , one can construct an orthonormal basis u i := ∑ j ( G − 1 / 2 ) j i v j . :=\sum _G^_v_. In matrix notation, U = V G − 1 / 2 , where U has orthonormal basis vectors \ and the matrix V is composed of the given column vectors \ . The matrix G − 1 / 2 is guaranteed to exist. Indeed, G is Hermitian, and so can be decomposed as G = U D U † with U a unitary matrix and D a real diagonal matrix. Additionally, the v i are linearly independent if and only if G is positive definite, which implies that the diagonal entries of D are positive. G − 1 / 2 is therefore uniquely defined by G − 1 / 2 := U D − 1 / 2 U † :=UD^U^ . One can check that these new vectors are orthonormal: ⟨ u i , u j ⟩ = ∑ i ′ ∑ j ′ ⟨ ( G − 1 / 2 ) i ′ i v i ′ , ( G − 1 / 2 ) j ′ j v j ′ ⟩ = ∑ i ′ ∑ j ′ ( G − 1 / 2 ) i i ′ G i ′ j ′ ( G − 1 / 2 ) j ′ j = ( G − 1 / 2 G G − 1 / 2 ) i j = δ i j \langle u_,u_\rangle &=\sum _\sum _G^_v_,G^_v_\\[10mu]&=\sum _\sum _G^_G_G^_\\[8mu]&=G^GG^_=\delta _\end where we used ( G − 1 / 2 ) † = G − 1 / 2 G^^=G^ .
Gram matrix : Controllability Gramian Observability Gramian
Gram matrix : Horn, Roger A.; Johnson, Charles R. (2013). Matrix Analysis (2nd ed.). Cambridge University Press. ISBN 978-0-521-54823-6.
Gram matrix : "Gram matrix", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Volumes of parallelograms by Frank Jones
Graph kernel : In structure mining, a graph kernel is a kernel function that computes an inner product on graphs. Graph kernels can be intuitively understood as functions measuring the similarity of pairs of graphs. They allow kernelized learning algorithms such as support vector machines to work directly on graphs, without having to do feature extraction to transform them to fixed-length, real-valued feature vectors. They find applications in bioinformatics, in chemoinformatics (as a type of molecule kernels), and in social network analysis. Concepts of graph kernels have been around since the 1999, when D. Haussler introduced convolutional kernels on discrete structures. The term graph kernels was more officially coined in 2002 by R. I. Kondor and J. Lafferty as kernels on graphs, i.e. similarity functions between the nodes of a single graph, with the World Wide Web hyperlink graph as a suggested application. In 2003, Gärtner et al. and Kashima et al. defined kernels between graphs. In 2010, Vishwanathan et al. gave their unified framework. In 2018, Ghosh et al. described the history of graph kernels and their evolution over two decades.
Graph kernel : The marginalized graph kernel has been shown to allow accurate predictions of the atomization energy of small organic molecules.
Graph kernel : An example of a kernel between graphs is the random walk kernel, which conceptually performs random walks on two graphs simultaneously, then counts the number of paths that were produced by both walks. This is equivalent to doing random walks on the direct product of the pair of graphs, and from this, a kernel can be derived that can be efficiently computed. Another examples is the Weisfeiler-Leman graph kernel which computes multiple rounds of the Weisfeiler-Leman algorithm and then computes the similarity of two graphs as the inner product of the histogram vectors of both graphs. In those histogram vectors the kernel collects the number of times a color occurs in the graph in every iteration. Note that the Weisfeiler-Leman kernel in theory has an infinite dimension as the number of possible colors assigned by the Weisfeiler-Leman algorithm is infinite. By restricting to the colors that occur in both graphs, the computation is still feasible.
Graph kernel : Tree kernel, as special case of non-cyclic graphs Molecule mining, as special case of small multi-label graphs == References ==
Kernel adaptive filter : In signal processing, a kernel adaptive filter is a type of nonlinear adaptive filter. An adaptive filter is a filter that adapts its transfer function to changes in signal properties over time by minimizing an error or loss function that characterizes how far the filter deviates from ideal behavior. The adaptation process is based on learning from a sequence of signal samples and is thus an online algorithm. A nonlinear adaptive filter is one in which the transfer function is nonlinear. Kernel adaptive filters implement a nonlinear transfer function using kernel methods. In these methods, the signal is mapped to a high-dimensional linear feature space and a nonlinear function is approximated as a sum over kernels, whose domain is the feature space. If this is done in a reproducing kernel Hilbert space, a kernel method can be a universal approximator for a nonlinear function. Kernel methods have the advantage of having convex loss functions, with no local minima, and of being only moderately complex to implement. Because high-dimensional feature space is linear, kernel adaptive filters can be thought of as a generalization of linear adaptive filters. As with linear adaptive filters, there are two general approaches to adapting a filter: the least mean squares filter (LMS) and the recursive least squares filter (RLS). Self organising kernel adaptive filters that use iteration to achieve convex LMS error minimisation address some of the statistical and practical issues of non-linear models that do not arise in the linear case. Regularisation is particularly important feature for non-linear models and also often used in linear adaptive filters to reduce statistical uncertainties. However because nonlinear filters typically have a much higher potential structural complexity (or higher dimensional feature space) compared to the subspace actually required, regularisation of some kind must deal with the under-determined model. Though some specific forms of parameter regularisation such as prescribed by Vapink's SRM & SVM address the dimensionality problem statistically to some extent, there remain further statistical and practical issues for truly adaptive non-linear filters. Adaptive filters are often used for tracking the behaviour of a time-varying system or systems which cannot be fully modelled from the data and structure available, hence the models may not only need to adapt parameters, but structure too. Where structural parameters of kernels are derived directly from data being processed (as in the above "Support Vector" approach) there are convenient opportunities for analytically robust methods of self organisation of the kernels available to the filter. The linearised feature space induced by kernels allows linear projection of new samples on to the current structure of the model where novelty in new data can be easily differentiated from noise-born errors which should not result in a change to model structure. Analytical metrics for structure analysis can be used to parsimoniously grow model complexity when required or optimally prune the existing structure when processor resource limits are reached. Structure updates are also relevant when system variation is detected and the long-term memory of the model should be updated as for the Kalman Filter case in linear filters. Iterative gradient descent that is typically used in adaptive filters has also gained popularity in offline batch-mode support vector based machine learning because of its computational efficiency for large data set processing. Both time series and batch data processing performance is reported to be able to easily handle over 100,000 training examples using as little as 10kB RAM. Data sizes this large are challenging to the original formulations of support vector machines and other kernel methods, which for example relied on constrained optimisation using linear or quadratic programming techniques. == References ==
Kernel eigenvoice : Speaker adaptation is an important technology to fine-tune either features or speech models for mis-match due to inter-speaker variation. In the last decade, eigenvoice (EV) speaker adaptation has been developed. It makes use of the prior knowledge of training speakers to provide a fast adaptation algorithm (in other words, only a small amount of adaptation data is needed). Inspired by the kernel eigenface idea in face recognition, kernel eigenvoice (KEV) is proposed. KEV is a non-linear generalization to EV. This incorporates Kernel principal component analysis, a non-linear version of Principal Component Analysis, to capture higher order correlations in order to further explore the speaker space and enhance recognition performance.
Kernel eigenvoice : Kernel Eigenvoice Speaker Adaptation, ScientificCommons Mak, B.; Ho, S. (2005). "Various Reference Speakers Determination Methods for Embedded Kernel Eigenvoice Speaker Adaptation". IEEE International Conference on Acoustics, Speech, and Signal Processing, 2005. Proceedings. ICASSP '05. Vol. 1. pp. 981–984. doi:10.1109/ICASSP.2005.1415280. Mak, B.; Kwok, J. T.; Ho, S. (September 2005). "Kernel Eigenvoice Speaker Adaptation". IEEE Transactions on Speech and Audio Processing. 13 (5): 984–992. doi:10.1109/TSA.2005.851971. ISSN 1063-6676. S2CID 7361772. Retrieved 2017-11-15. Speedup of Kernel Eigenvoice Speaker Adaptation by Embedded Kernel PCA, ICSLP 2004. Speaker Adaptation via Composite Kernel PCA, NIPS 2003. Mak, Brian Kan-Wing; Hsiao, Roger Wend-Huu; Ho, Simon Ka-Lung; Kwok, J. T. (July 2006). "Embedded kernel eigenvoice speaker adaptation and its implication to reference speaker weighting". IEEE Transactions on Audio, Speech, and Language Processing. 14 (4): 1267–1280. CiteSeerX 10.1.1.206.4596. doi:10.1109/TSA.2005.860836. S2CID 7527119.
Kernel method : In machine learning, kernel machines are a class of algorithms for pattern analysis, whose best known member is the support-vector machine (SVM). These methods involve using linear classifiers to solve nonlinear problems. The general task of pattern analysis is to find and study general types of relations (for example clusters, rankings, principal components, correlations, classifications) in datasets. For many algorithms that solve these tasks, the data in raw representation have to be explicitly transformed into feature vector representations via a user-specified feature map: in contrast, kernel methods require only a user-specified kernel, i.e., a similarity function over all pairs of data points computed using inner products. The feature map in kernel machines is infinite dimensional but only requires a finite dimensional matrix from user-input according to the representer theorem. Kernel machines are slow to compute for datasets larger than a couple of thousand examples without parallel processing. Kernel methods owe their name to the use of kernel functions, which enable them to operate in a high-dimensional, implicit feature space without ever computing the coordinates of the data in that space, but rather by simply computing the inner products between the images of all pairs of data in the feature space. This operation is often computationally cheaper than the explicit computation of the coordinates. This approach is called the "kernel trick". Kernel functions have been introduced for sequence data, graphs, text, images, as well as vectors. Algorithms capable of operating with kernels include the kernel perceptron, support-vector machines (SVM), Gaussian processes, principal components analysis (PCA), canonical correlation analysis, ridge regression, spectral clustering, linear adaptive filters and many others. Most kernel algorithms are based on convex optimization or eigenproblems and are statistically well-founded. Typically, their statistical properties are analyzed using statistical learning theory (for example, using Rademacher complexity).
Kernel method : Kernel methods can be thought of as instance-based learners: rather than learning some fixed set of parameters corresponding to the features of their inputs, they instead "remember" the i -th training example ( x i , y i ) _,y_) and learn for it a corresponding weight w i . Prediction for unlabeled inputs, i.e., those not in the training set, is treated by the application of a similarity function k , called a kernel, between the unlabeled input x ′ and each of the training inputs x i _ . For instance, a kernelized binary classifier typically computes a weighted sum of similarities y ^ = sgn ⁡ ∑ i = 1 n w i y i k ( x i , x ′ ) , =\operatorname \sum _^w_y_k(\mathbf _,\mathbf ), where y ^ ∈ \in \ is the kernelized binary classifier's predicted label for the unlabeled input x ′ whose hidden true label y is of interest; k : X × X → R \times \to \mathbb is the kernel function that measures similarity between any pair of inputs x , x ′ ∈ X ,\mathbf \in ; the sum ranges over the n labeled examples i = 1 n _,y_)\_^ in the classifier's training set, with y i ∈ \in \ ; the w i ∈ R \in \mathbb are the weights for the training examples, as determined by the learning algorithm; the sign function sgn determines whether the predicted classification y ^ comes out positive or negative. Kernel classifiers were described as early as the 1960s, with the invention of the kernel perceptron. They rose to great prominence with the popularity of the support-vector machine (SVM) in the 1990s, when the SVM was found to be competitive with neural networks on tasks such as handwriting recognition.
Kernel method : The kernel trick avoids the explicit mapping that is needed to get linear learning algorithms to learn a nonlinear function or decision boundary. For all x and x ′ in the input space X , certain functions k ( x , x ′ ) ,\mathbf ) can be expressed as an inner product in another space V . The function k : X × X → R \times \to \mathbb is often referred to as a kernel or a kernel function. The word "kernel" is used in mathematics to denote a weighting function for a weighted sum or integral. Certain problems in machine learning have more structure than an arbitrary weighting function k . The computation is made much simpler if the kernel can be written in the form of a "feature map" φ : X → V \to which satisfies k ( x , x ′ ) = ⟨ φ ( x ) , φ ( x ′ ) ⟩ V . ,\mathbf )=\langle \varphi (\mathbf ),\varphi (\mathbf )\rangle _. The key restriction is that ⟨ ⋅ , ⋅ ⟩ V must be a proper inner product. On the other hand, an explicit representation for φ is not necessary, as long as V is an inner product space. The alternative follows from Mercer's theorem: an implicitly defined function φ exists whenever the space X can be equipped with a suitable measure ensuring the function k satisfies Mercer's condition. Mercer's theorem is similar to a generalization of the result from linear algebra that associates an inner product to any positive-definite matrix. In fact, Mercer's condition can be reduced to this simpler case. If we choose as our measure the counting measure μ ( T ) = \| T \| for all T ⊂ X , which counts the number of points inside the set T , then the integral in Mercer's theorem reduces to a summation ∑ i = 1 n ∑ j = 1 n k ( x i , x j ) c i c j ≥ 0. ^\sum _^k(\mathbf _,\mathbf _)c_c_\geq 0. If this summation holds for all finite sequences of points ( x 1 , … , x n ) _,\dotsc ,\mathbf _) in X and all choices of n real-valued coefficients ( c 1 , … , c n ) ,\dots ,c_) (cf. positive definite kernel), then the function k satisfies Mercer's condition. Some algorithms that depend on arbitrary relationships in the native space X would, in fact, have a linear interpretation in a different setting: the range space of φ . The linear interpretation gives us insight about the algorithm. Furthermore, there is often no need to compute φ directly during computation, as is the case with support-vector machines. Some cite this running time shortcut as the primary benefit. Researchers also use it to justify the meanings and properties of existing algorithms. Theoretically, a Gram matrix K ∈ R n × n \in \mathbb ^ with respect to _,\dotsc ,\mathbf _\ (sometimes also called a "kernel matrix"), where K i j = k ( x i , x j ) =k(\mathbf _,\mathbf _) , must be positive semi-definite (PSD). Empirically, for machine learning heuristics, choices of a function k that do not satisfy Mercer's condition may still perform reasonably if k at least approximates the intuitive idea of similarity. Regardless of whether k is a Mercer kernel, k may still be referred to as a "kernel". If the kernel function k is also a covariance function as used in Gaussian processes, then the Gram matrix K can also be called a covariance matrix.
Kernel method : Application areas of kernel methods are diverse and include geostatistics, kriging, inverse distance weighting, 3D reconstruction, bioinformatics, cheminformatics, information extraction and handwriting recognition.
Kernel method : Fisher kernel Graph kernels Kernel smoother Polynomial kernel Radial basis function kernel (RBF) String kernels Neural tangent kernel Neural network Gaussian process (NNGP) kernel
Kernel method : Kernel methods for vector output Kernel density estimation Representer theorem Similarity learning Cover's theorem
Kernel method : Shawe-Taylor, J.; Cristianini, N. (2004). Kernel Methods for Pattern Analysis. Cambridge University Press. ISBN 9780511809682. Liu, W.; Principe, J.; Haykin, S. (2010). Kernel Adaptive Filtering: A Comprehensive Introduction. Wiley. ISBN 9781118211212. Schölkopf, B.; Smola, A. J.; Bach, F. (2018). Learning with Kernels : Support Vector Machines, Regularization, Optimization, and Beyond. MIT Press. ISBN 978-0-262-53657-8.
Kernel method : Kernel-Machines Org—community website onlineprediction.net Kernel Methods Article
Kernel perceptron : In machine learning, the kernel perceptron is a variant of the popular perceptron learning algorithm that can learn kernel machines, i.e. non-linear classifiers that employ a kernel function to compute the similarity of unseen samples to training samples. The algorithm was invented in 1964, making it the first kernel classification learner.
Kernel perceptron : To derive a kernelized version of the perceptron algorithm, we must first formulate it in dual form, starting from the observation that the weight vector w can be expressed as a linear combination of the n training samples. The equation for the weight vector is w = ∑ i n α i y i x i =\sum _^\alpha _y_\mathbf _ where αi is the number of times xi was misclassified, forcing an update w ← w + yi xi. Using this result, we can formulate the dual perceptron algorithm, which loops through the samples as before, making predictions, but instead of storing and updating a weight vector w, it updates a "mistake counter" vector α. We must also rewrite the prediction formula to get rid of w: y ^ = sgn ⁡ ( w T x ) = sgn ⁡ ( ∑ i n α i y i x i ) T x = sgn ⁡ ∑ i n α i y i ( x i ⋅ x ) &=\operatorname (\mathbf ^\mathbf )\\&=\operatorname \left(\sum _^\alpha _y_\mathbf _\right)^\mathbf \\&=\operatorname \sum _^\alpha _y_(\mathbf _\cdot \mathbf )\end Plugging these two equations into the training loop turn it into the dual perceptron algorithm. Finally, we can replace the dot product in the dual perceptron by an arbitrary kernel function, to get the effect of a feature map Φ without computing Φ(x) explicitly for any samples. Doing this yields the kernel perceptron algorithm: Initialize α to an all-zeros vector of length n, the number of training samples. For some fixed number of iterations, or until some stopping criterion is met: For each training example xj, yj: Let y ^ = sgn ⁡ ∑ i n α i y i K ( x i , x j ) =\operatorname \sum _^\alpha _y_K(\mathbf _,\mathbf _) If ŷ ≠ yj, perform an update by incrementing the mistake counter: αj ← αj + 1
Kernel perceptron : One problem with the kernel perceptron, as presented above, is that it does not learn sparse kernel machines. Initially, all the αi are zero so that evaluating the decision function to get ŷ requires no kernel evaluations at all, but each update increments a single αi, making the evaluation increasingly more costly. Moreover, when the kernel perceptron is used in an online setting, the number of non-zero αi and thus the evaluation cost grow linearly in the number of examples presented to the algorithm. The forgetron variant of the kernel perceptron was suggested to deal with this problem. It maintains an active set of examples with non-zero αi, removing ("forgetting") examples from the active set when it exceeds a pre-determined budget and "shrinking" (lowering the weight of) old examples as new ones are promoted to non-zero αi. Another problem with the kernel perceptron is that it does not regularize, making it vulnerable to overfitting. The NORMA online kernel learning algorithm can be regarded as a generalization of the kernel perceptron algorithm with regularization. The sequential minimal optimization (SMO) algorithm used to learn support vector machines can also be regarded as a generalization of the kernel perceptron. The voted perceptron algorithm of Freund and Schapire also extends to the kernelized case, giving generalization bounds comparable to the kernel SVM. == References ==
Low-rank matrix approximations : Low-rank matrix approximations are essential tools in the application of kernel methods to large-scale learning problems. Kernel methods (for instance, support vector machines or Gaussian processes) project data points into a high-dimensional or infinite-dimensional feature space and find the optimal splitting hyperplane. In the kernel method the data is represented in a kernel matrix (or, Gram matrix). Many algorithms can solve machine learning problems using the kernel matrix. The main problem of kernel method is its high computational cost associated with kernel matrices. The cost is at least quadratic in the number of training data points, but most kernel methods include computation of matrix inversion or eigenvalue decomposition and the cost becomes cubic in the number of training data. Large training sets cause large storage and computational costs. While low rank decomposition methods (Cholesky decomposition) reduce this cost, they still require computing the kernel matrix. One of the approaches to deal with this problem is low-rank matrix approximations. The most popular examples of them are the Nyström approximation and randomized feature maps approximation methods. Both of them have been successfully applied to efficient kernel learning.
Low-rank matrix approximations : Kernel methods become computationally unfeasible when the number of points D is so large such that the kernel matrix K cannot be stored in memory. If D is the number of training examples, the storage and computational cost required to find the solution of the problem using general kernel method is O ( D 2 ) ) and O ( D 3 ) ) respectively. The Nyström approximation can allow a significant speed-up of the computations. This speed-up is achieved by using, instead of the kernel matrix, its approximation K ~ of rank d . An advantage of the method is that it is not necessary to compute or store the whole kernel matrix, but only a submatrix of size d × D . It reduces the storage and complexity requirements to O ( D d ) and O ( D d 2 ) ) respectively. The method is named "Nyström approximation" because it can be interpreted as a case of the Nyström method from integral equation theory.
Low-rank matrix approximations : Let x , x ′ ∈ R d ,\mathbf \in \mathbb ^ – samples of data, z : R d → R D ^\to \mathbb ^ – a randomized feature map (maps a single vector to a vector of higher dimensionality) so that the inner product between a pair of transformed points approximates their kernel evaluation: K ( x , x ′ ) = ⟨ Φ ( x ) , Φ ( x ′ ) ⟩ ≈ z ( x ) T z ( x ′ ) , ,\mathbf )=\langle \Phi (\mathbf ),\Phi (\mathbf )\rangle \approx z(\mathbf )^z(\mathbf ), where Φ is the mapping embedded in the RBF kernel. Since z is low-dimensional, the input can be easily transformed with z , after that different linear learning methods to approximate the answer of the corresponding nonlinear kernel can be applied. There are different randomized feature maps to compute the approximations to the RBF kernels. For instance, random Fourier features and random binning features.
Low-rank matrix approximations : The approaches for large-scale kernel learning (Nyström method and random features) differs in the fact that the Nyström method uses data dependent basis functions while in random features approach the basis functions are sampled from a distribution independent from the training data. This difference leads to an improved analysis for kernel learning approaches based on the Nyström method. When there is a large gap in the eigen-spectrum of the kernel matrix, approaches based on the Nyström method can achieve better results than the random features based approach.
Low-rank matrix approximations : Nyström method Support vector machine Radial basis function kernel Regularized least squares
Low-rank matrix approximations : Andreas Müller (2012). Kernel Approximations for Efficient SVMs (and other feature extraction methods). == References ==
Neural tangent kernel : In the study of artificial neural networks (ANNs), the neural tangent kernel (NTK) is a kernel that describes the evolution of deep artificial neural networks during their training by gradient descent. It allows ANNs to be studied using theoretical tools from kernel methods. In general, a kernel is a positive-semidefinite symmetric function of two inputs which represents some notion of similarity between the two inputs. The NTK is a specific kernel derived from a given neural network; in general, when the neural network parameters change during training, the NTK evolves as well. However, in the limit of large layer width the NTK becomes constant, revealing a duality between training the wide neural network and kernel methods: gradient descent in the infinite-width limit is fully equivalent to kernel gradient descent with the NTK. As a result, using gradient descent to minimize least-square loss for neural networks yields the same mean estimator as ridgeless kernel regression with the NTK. This duality enables simple closed form equations describing the training dynamics, generalization, and predictions of wide neural networks. The NTK was introduced in 2018 by Arthur Jacot, Franck Gabriel and Clément Hongler, who used it to study the convergence and generalization properties of fully connected neural networks. Later works extended the NTK results to other neural network architectures. In fact, the phenomenon behind NTK is not specific to neural networks and can be observed in generic nonlinear models, usually by a suitable scaling.
Neural tangent kernel : Let f ( x ; θ ) denote the scalar function computed by a given neural network with parameters θ on input x . Then the neural tangent kernel is defined as Θ ( x , x ′ ; θ ) = ∇ θ f ( x ; θ ) ⋅ ∇ θ f ( x ′ ; θ ) . f(x;\theta )\cdot \nabla _f(x';\theta ). Since it is written as a dot product between mapped inputs (with the gradient of the neural network function serving as the feature map), we are guaranteed that the NTK is symmetric and positive semi-definite. The NTK is thus a valid kernel function. Consider a fully connected neural network whose parameters are chosen i.i.d. according to any mean-zero distribution. This random initialization of θ induces a distribution over f ( x ; θ ) whose statistics we will analyze, both at initialization and throughout training (gradient descent on a specified dataset). We can visualize this distribution via a neural network ensemble which is constructed by drawing many times from the initial distribution over f ( x ; θ ) and training each draw according to the same training procedure. The number of neurons in each layer is called the layer’s width. Consider taking the width of every hidden layer to infinity and training the neural network with gradient descent (with a suitably small learning rate). In this infinite-width limit, several nice properties emerge: At initialization (before training), the neural network ensemble is a zero-mean Gaussian process (GP). This means that distribution of functions is the maximum-entropy distribution with mean E θ [ f ( x ; θ ) ] = 0 _[f(x;\theta )]=0 and covariance E θ [ f ( x ; θ ) f ( x ′ ; θ ) ] = Σ ( x , x ′ ) _[f(x;\theta )f(x';\theta )]=\Sigma (x,x') , where the GP covariance Σ ( x , x ′ ) can be computed from the network architecture. In other words, the distribution of neural network functions at initialization has no structure other than its first and second moments (mean and covariance). This follows from the central limit theorem. The NTK is deterministic. In other words, the NTK is independent of the random parameter initialization. The NTK does not change during training. Each parameter changes negligibly throughout training. As Lee et al. note, "although individual parameters move by a vanishingly small amount, they collectively conspire to provide a finite change in the final output of the network, as is necessary for training." During training, the neural network is linearized, i.e., its parameter dependence can be captured by its first-order Taylor expansion: f ( x ; θ 0 + Δ θ ) = f ( x ; θ 0 ) + Δ θ ⋅ ∇ θ f ( x ; θ 0 ) +\Delta \theta )=f(x;\theta _)+\Delta \theta \cdot \nabla _f(x;\theta _) , where θ 0 are the initial parameters. This follows from the fact that each parameter changes negligibly during training. (The neural network remains nonlinear with respect to the inputs.) The training dynamics are equivalent to kernel gradient descent using the NTK as the kernel. If the loss function is mean-squared error, the final distribution over f ( x ; θ ) is still a Gaussian process, but with a new mean and covariance. In particular, the mean converges to the same estimator yielded by kernel regression with the NTK as kernel and zero ridge regularization, and the covariance is expressible in terms of the NTK and the initial GP covariance. It can be shown that the ensemble variance vanishes at the training points (in other words, the neural network always interpolates the training data, regardless of initialization). From a physics point of view, the NTK can be understood as a type of Hamiltonian, since it generates the time-evolution of observables when the neural network is trained by gradient descent with infinitesimally small steps (the continuum limit).
Neural tangent kernel : The NTK can be studied for various ANN architectures, in particular convolutional neural networks (CNNs), recurrent neural networks (RNNs) and transformers. In such settings, the large-width limit corresponds to letting the number of parameters grow, while keeping the number of layers fixed: for CNNs, this involves letting the number of channels grow. Individual parameters of a wide neural network in the kernel regime change negligibly during training. However, this implies that infinite-width neural networks cannot exhibit feature learning, which is widely considered to be an important property of realistic deep neural networks. This is not a generic feature of infinite-width neural networks and is largely due to a specific choice of the scaling by which the width is taken to the infinite limit; indeed several works have found alternate infinite-width scaling limits of neural networks in which there is no duality with kernel regression and feature learning occurs during training. Others introduce a "neural tangent hierarchy" to describe finite-width effects, which may drive feature learning. Neural Tangents is a free and open-source Python library used for computing and doing inference with the infinite width NTK and neural network Gaussian process (NNGP) corresponding to various common ANN architectures. In addition, there exists a scikit-learn compatible implementation of the infinite width NTK for Gaussian processes called scikit-ntk.
Neural tangent kernel : When optimizing the parameters θ ∈ R P ^ of an ANN to minimize an empirical loss through gradient descent, the NTK governs the dynamics of the ANN output function f θ throughout the training.
Neural tangent kernel : Large width limits of neural networks
Neural tangent kernel : Ananthaswamy, Anil (2021-10-11). "A New Link to an Old Model Could Crack the Mystery of Deep Learning". Quanta Magazine.
Polynomial kernel : In machine learning, the polynomial kernel is a kernel function commonly used with support vector machines (SVMs) and other kernelized models, that represents the similarity of vectors (training samples) in a feature space over polynomials of the original variables, allowing learning of non-linear models. Intuitively, the polynomial kernel looks not only at the given features of input samples to determine their similarity, but also combinations of these. In the context of regression analysis, such combinations are known as interaction features. The (implicit) feature space of a polynomial kernel is equivalent to that of polynomial regression, but without the combinatorial blowup in the number of parameters to be learned. When the input features are binary-valued (booleans), then the features correspond to logical conjunctions of input features.
Polynomial kernel : For degree-d polynomials, the polynomial kernel is defined as K ( x , y ) = ( x T y + c ) d ,\mathbf )=(\mathbf ^\mathbf +c)^ where x and y are vectors of size n in the input space, i.e. vectors of features computed from training or test samples and c ≥ 0 is a free parameter trading off the influence of higher-order versus lower-order terms in the polynomial. When c = 0, the kernel is called homogeneous. (A further generalized polykernel divides xTy by a user-specified scalar parameter a.) As a kernel, K corresponds to an inner product in a feature space based on some mapping φ: K ( x , y ) = ⟨ φ ( x ) , φ ( y ) ⟩ ,\mathbf )=\langle \varphi (\mathbf ),\varphi (\mathbf )\rangle The nature of φ can be seen from an example. Let d = 2, so we get the special case of the quadratic kernel. After using the multinomial theorem (twice—the outermost application is the binomial theorem) and regrouping, K ( x , y ) = ( ∑ i = 1 n x i y i + c ) 2 = ∑ i = 1 n ( x i 2 ) ( y i 2 ) + ∑ i = 2 n ∑ j = 1 i − 1 ( 2 x i x j ) ( 2 y i y j ) + ∑ i = 1 n ( 2 c x i ) ( 2 c y i ) + c 2 ,\mathbf )=\left(\sum _^x_y_+c\right)^=\sum _^\left(x_^\right)\left(y_^\right)+\sum _^\sum _^\left(x_x_\right)\left(y_y_\right)+\sum _^\left(x_\right)\left(y_\right)+c^ From this it follows that the feature map is given by: φ ( x ) = ( x n 2 , … , x 1 2 , 2 x n x n − 1 , … , 2 x n x 1 , 2 x n − 1 x n − 2 , … , 2 x n − 1 x 1 , … , 2 x 2 x 1 , 2 c x n , … , 2 c x 1 , c ) ^,\ldots ,x_^,x_x_,\ldots ,x_x_,x_x_,\ldots ,x_x_,\ldots ,x_x_,x_,\ldots ,x_,c\right) generalizing for ( x T y + c ) d ^\mathbf +c\right)^ , where x ∈ R n \in \mathbb ^ , y ∈ R n \in \mathbb ^ and applying the multinomial theorem: ( x T y + c ) d = ∑ j 1 + j 2 + ⋯ + j n + 1 = d d ! j 1 ! ⋯ j n ! j n + 1 ! x 1 j 1 ⋯ x n j n c j n + 1 d ! j 1 ! ⋯ j n ! j n + 1 ! y 1 j 1 ⋯ y n j n c j n + 1 = φ ( x ) T φ ( y ) \left(\mathbf ^\mathbf +c\right)^&=\sum _+j_+\dots +j_=d!\cdots j_!j_!x_^\cdots x_^^!\cdots j_!j_!y_^\cdots y_^^\\&=\varphi (\mathbf )^\varphi (\mathbf )\end The last summation has l d = ( n + d d ) = elements, so that: φ ( x ) = ( a 1 , … , a l , … , a l d ) )=\left(a_,\dots ,a_,\dots ,a_\right) where l = ( j 1 , j 2 , . . . , j n , j n + 1 ) ,j_,...,j_,j_) and a l = d ! j 1 ! ⋯ j n ! j n + 1 ! x 1 j 1 ⋯ x n j n c j n + 1 \| j 1 + j 2 + ⋯ + j n + j n + 1 = d =!\cdots j_!j_!x_^\cdots x_^^\quad \|\quad j_+j_+\dots +j_+j_=d
Polynomial kernel : Although the RBF kernel is more popular in SVM classification than the polynomial kernel, the latter is quite popular in natural language processing (NLP). The most common degree is d = 2 (quadratic), since larger degrees tend to overfit on NLP problems. Various ways of computing the polynomial kernel (both exact and approximate) have been devised as alternatives to the usual non-linear SVM training algorithms, including: full expansion of the kernel prior to training/testing with a linear SVM, i.e. full computation of the mapping φ as in polynomial regression; basket mining (using a variant of the apriori algorithm) for the most commonly occurring feature conjunctions in a training set to produce an approximate expansion; inverted indexing of support vectors. One problem with the polynomial kernel is that it may suffer from numerical instability: when xTy + c < 1, K(x, y) = (xTy + c)d tends to zero with increasing d, whereas when xTy + c > 1, K(x, y) tends to infinity. == References ==
Relevance vector machine : In mathematics, a Relevance Vector Machine (RVM) is a machine learning technique that uses Bayesian inference to obtain parsimonious solutions for regression and probabilistic classification. A greedy optimisation procedure and thus fast version were subsequently developed. The RVM has an identical functional form to the support vector machine, but provides probabilistic classification. It is actually equivalent to a Gaussian process model with covariance function: k ( x , x ′ ) = ∑ j = 1 N 1 α j φ ( x , x j ) φ ( x ′ , x j ) ,\mathbf )=\sum _^\varphi (\mathbf ,\mathbf _)\varphi (\mathbf ',\mathbf _) where φ is the kernel function (usually Gaussian), α j are the variances of the prior on the weight vector w ∼ N ( 0 , α − 1 I ) I) , and x 1 , … , x N _,\ldots ,\mathbf _ are the input vectors of the training set. Compared to that of support vector machines (SVM), the Bayesian formulation of the RVM avoids the set of free parameters of the SVM (that usually require cross-validation-based post-optimizations). However RVMs use an expectation maximization (EM)-like learning method and are therefore at risk of local minima. This is unlike the standard sequential minimal optimization (SMO)-based algorithms employed by SVMs, which are guaranteed to find a global optimum (of the convex problem). The relevance vector machine was patented in the United States by Microsoft (patent expired September 4, 2019).
Relevance vector machine : Kernel trick Platt scaling: turns an SVM into a probability model
Relevance vector machine : dlib C++ Library The Kernel-Machine Library rvmbinary: R package for binary classification scikit-rvm fast-scikit-rvm, rvm tutorial
Relevance vector machine : Tipping's webpage on Sparse Bayesian Models and the RVM A Tutorial on RVM by Tristan Fletcher Applied tutorial on RVM Comparison of RVM and SVM
Inductive logic programming : Inductive logic programming (ILP) is a subfield of symbolic artificial intelligence which uses logic programming as a uniform representation for examples, background knowledge and hypotheses. The term "inductive" here refers to philosophical (i.e. suggesting a theory to explain observed facts) rather than mathematical (i.e. proving a property for all members of a well-ordered set) induction. Given an encoding of the known background knowledge and a set of examples represented as a logical database of facts, an ILP system will derive a hypothesised logic program which entails all the positive and none of the negative examples. Schema: positive examples + negative examples + background knowledge ⇒ hypothesis. Inductive logic programming is particularly useful in bioinformatics and natural language processing.
Inductive logic programming : Building on earlier work on Inductive inference, Gordon Plotkin was the first to formalise induction in a clausal setting around 1970, adopting an approach of generalising from examples. In 1981, Ehud Shapiro introduced several ideas that would shape the field in his new approach of model inference, an algorithm employing refinement and backtracing to search for a complete axiomatisation of given examples. His first implementation was the Model Inference System in 1981: a Prolog program that inductively inferred Horn clause logic programs from positive and negative examples. The term Inductive Logic Programming was first introduced in a paper by Stephen Muggleton in 1990, defined as the intersection of machine learning and logic programming. Muggleton and Wray Buntine introduced predicate invention and inverse resolution in 1988. Several inductive logic programming systems that proved influential appeared in the early 1990s. FOIL, introduced by Ross Quinlan in 1990 was based on upgrading propositional learning algorithms AQ and ID3. Golem, introduced by Muggleton and Feng in 1990, went back to a restricted form of Plotkin's least generalisation algorithm. The Progol system, introduced by Muggleton in 1995, first implemented inverse entailment, and inspired many later systems. Aleph, a descendant of Progol introduced by Ashwin Srinivasan in 2001, is still one of the most widely used systems as of 2022. At around the same time, the first practical applications emerged, particularly in bioinformatics, where by 2000 inductive logic programming had been successfully applied to drug design, carcinogenicity and mutagenicity prediction, and elucidation of the structure and function of proteins. Unlike the focus on automatic programming inherent in the early work, these fields used inductive logic programming techniques from a viewpoint of relational data mining. The success of those initial applications and the lack of progress in recovering larger traditional logic programs shaped the focus of the field. Recently, classical tasks from automated programming have moved back into focus, as the introduction of meta-interpretative learning makes predicate invention and learning recursive programs more feasible. This technique was pioneered with the Metagol system introduced by Muggleton, Dianhuan Lin, Niels Pahlavi and Alireza Tamaddoni-Nezhad in 2014. This allows ILP systems to work with fewer examples, and brought successes in learning string transformation programs, answer set grammars and general algorithms.
Inductive logic programming : Inductive logic programming has adopted several different learning settings, the most common of which are learning from entailment and learning from interpretations. In both cases, the input is provided in the form of background knowledge B, a logical theory (commonly in the form of clauses used in logic programming), as well as positive and negative examples, denoted E + and E − respectively. The output is given as a hypothesis H, itself a logical theory that typically consists of one or more clauses. The two settings differ in the format of examples presented.
Inductive logic programming : An inductive logic programming system is a program that takes as an input logic theories B , E + , E − ,E^ and outputs a correct hypothesis H with respect to theories B , E + , E − ,E^ . A system is complete if and only if for any input logic theories B , E + , E − ,E^ any correct hypothesis H with respect to these input theories can be found with its hypothesis search procedure. Inductive logic programming systems can be roughly divided into two classes, search-based and meta-interpretative systems. Search-based systems exploit that the space of possible clauses forms a complete lattice under the subsumption relation, where one clause C 1 subsumes another clause C 2 if there is a substitution θ such that C 1 θ \theta , the result of applying θ to C 1 , is a subset of C 2 . This lattice can be traversed either bottom-up or top-down.
Inductive logic programming : 1BC and 1BC2: first-order naive Bayesian classifiers: ACE (A Combined Engine) Aleph Atom Archived 2014-03-26 at the Wayback Machine Claudien DL-Learner Archived 2019-08-15 at the Wayback Machine DMax FastLAS (Fast Learning from Answer Sets) FOIL (First Order Inductive Learner) Golem ILASP (Inductive Learning of Answer Set Programs) Imparo Inthelex (INcremental THEory Learner from EXamples) Archived 2011-11-28 at the Wayback Machine Lime Metagol Mio MIS (Model Inference System) by Ehud Shapiro Ontolearn Popper PROGOL RSD Warmr (now included in ACE) ProGolem
Inductive logic programming : Probabilistic inductive logic programming adapts the setting of inductive logic programming to learning probabilistic logic programs. It can be considered as a form of statistical relational learning within the formalism of probabilistic logic programming. Given background knowledge as a probabilistic logic program B, and a set of positive and negative examples E + and E − the goal of probabilistic inductive logic programming is to find a probabilistic logic program H such that the probability of positive examples according to H ∪ B is maximized and the probability of negative examples is minimized. This problem has two variants: parameter learning and structure learning. In the former, one is given the structure (the clauses) of H and the goal is to infer the probabilities annotations of the given clauses, while in the latter the goal is to infer both the structure and the probability parameters of H. Just as in classical inductive logic programming, the examples can be given as examples or as (partial) interpretations.
Inductive logic programming : Commonsense reasoning Formal concept analysis Inductive reasoning Inductive programming Inductive probability Statistical relational learning Version space learning
Inductive logic programming : This article incorporates text from a free content work. Licensed under CC-BY 4.0 (license statement/permission). Text taken from A History of Probabilistic Inductive Logic Programming, Fabrizio Riguzzi, Elena Bellodi and Riccardo Zese, Frontiers Media. == Further reading ==
Aleph (ILP) : Aleph (A Learning Engine for Proposing Hypotheses) is an inductive logic programming system introduced by Ashwin Srinivasan in 2001. As of 2022 it is still one of the most widely used inductive logic programming systems. It is based on the earlier system Progol.
Aleph (ILP) : The input to Aleph is background knowledge, specified as a logic program, a language bias in the form of mode declarations, as well as positive and negative examples specified as ground facts. As output it returns a logic program which, together with the background knowledge, entails all of the positive examples and none of the negative examples.
Aleph (ILP) : Starting with an empty hypothesis, Aleph proceeds as follows: It chooses a positive example to generalise; if none are left, it aborts and outputs the current hypothesis. Then it constructs the bottom clause, that is, the most specific clause that is allowed by the mode declarations and covers the example. It then searches for a generalisation of the bottom clause that scores better on the chosen metric. It then adds the new clause to the hypothesis program and removes all examples that are covered by the new clause.
Aleph (ILP) : Aleph searches for clauses in a top-down manner, using the bottom clause constructed in the preceding step to bound the search from below. It searches the refinement graph in a breadth-first manner, with tunable parameters to bound the maximal clause size and proof depth. It scores each clause using one of 13 different evaluation metrics, as chosen in advance by the user.
Aleph (ILP) : Burnside, Elizabeth S.; Davis, Jesse; Costa, Vítor Santos; de Castro Dutra, Inês; Kahn, Charles E.; Fine, Jason; Page, David (2005). "Knowledge Discovery from Structured Mammography Reports Using Inductive Logic Programming". AMIA Annual Symposium Proceedings. 2005: 96–100. ISSN 1942-597X. PMC 1560852. PMID 16779009. Cropper, Andrew; Dumančić, Sebastijan (2022-06-15). "Inductive Logic Programming At 30: A New Introduction". Journal of Artificial Intelligence Research. 74: 766–850. arXiv:2008.07912. doi:10.1613/jair.1.13507. ISSN 1076-9757.
Anti-unification : Anti-unification is the process of constructing a generalization common to two given symbolic expressions. As in unification, several frameworks are distinguished depending on which expressions (also called terms) are allowed, and which expressions are considered equal. If variables representing functions are allowed in an expression, the process is called "higher-order anti-unification", otherwise "first-order anti-unification". If the generalization is required to have an instance literally equal to each input expression, the process is called "syntactical anti-unification", otherwise "E-anti-unification", or "anti-unification modulo theory". An anti-unification algorithm should compute for given expressions a complete and minimal generalization set, that is, a set covering all generalizations and containing no redundant members, respectively. Depending on the framework, a complete and minimal generalization set may have one, finitely many, or possibly infinitely many members, or may not exist at all; it cannot be empty, since a trivial generalization exists in any case. For first-order syntactical anti-unification, Gordon Plotkin gave an algorithm that computes a complete and minimal singleton generalization set containing the so-called "least general generalization" (lgg). Anti-unification should not be confused with dis-unification. The latter means the process of solving systems of inequations, that is of finding values for the variables such that all given inequations are satisfied. This task is quite different from finding generalizations.
Anti-unification : Formally, an anti-unification approach presupposes An infinite set V of variables. For higher-order anti-unification, it is convenient to choose V disjoint from the set of lambda-term bound variables. A set T of terms such that V ⊆ T. For first-order and higher-order anti-unification, T is usually the set of first-order terms (terms built from variable and function symbols) and lambda terms (terms containing some higher-order variables), respectively. An equivalence relation ≡ on T , indicating which terms are considered equal. For higher-order anti-unification, usually t ≡ u if t and u are alpha equivalent. For first-order E-anti-unification, ≡ reflects the background knowledge about certain function symbols; for example, if ⊕ is considered commutative, t ≡ u if u results from t by swapping the arguments of ⊕ at some (possibly all) occurrences. If there is no background knowledge at all, then only literally, or syntactically, identical terms are considered equal.
Anti-unification : The framework of first-order syntactical anti-unification is based on T being the set of first-order terms (over some given set V of variables, C of constants and F n of n -ary function symbols) and on ≡ being syntactic equality. In this framework, each anti-unification problem ⟨ t 1 , t 2 ⟩ ,t_\rangle has a complete, and obviously minimal, singleton solution set . Its member t is called the least general generalization (lgg) of the problem, it has an instance syntactically equal to t 1 and another one syntactically equal to t 2 . Any common generalization of t 1 and t 2 subsumes t . The lgg is unique up to variants: if S 1 and S 2 are both complete and minimal solution sets of the same syntactical anti-unification problem, then S 1 = =\\ and S 2 = =\\ for some terms s 1 and s 2 , that are renamings of each other. Plotkin has given an algorithm to compute the lgg of two given terms. It presupposes an injective mapping ϕ : T × T ⟶ V , that is, a mapping assigning each pair s , t of terms an own variable ϕ ( s , t ) , such that no two pairs share the same variable. The algorithm consists of two rules: For example, ( 0 ∗ 0 ) ⊔ ( 4 ∗ 4 ) ⇝ ( 0 ⊔ 4 ) ∗ ( 0 ⊔ 4 ) ⇝ ϕ ( 0 , 4 ) ∗ ϕ ( 0 , 4 ) ⇝ x ∗ x ; this least general generalization reflects the common property of both inputs of being square numbers. Plotkin used his algorithm to compute the "relative least general generalization (rlgg)" of two clause sets in first-order logic, which was the basis of the Golem approach to inductive logic programming.

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