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  Quantum image processing

  Quantum image processing (QIMP) is using quantum computing or quantum information processing to create and work with quantum images. Due to some of the properties inherent to quantum computation, notably entanglement and parallelism, it is hoped that QIMP technologies will offer capabilities and performances that surpass their traditional equivalents, in terms of computing speed, security, and minimum storage requirements. == Background == A. Y. Vlasov's work in 1997 focused on using a quantum system to recognize orthogonal images. This was followed by efforts using quantum algorithms to search specific patterns in binary images and detect the posture of certain targets. Notably, more optics-based interpretations for quantum imaging were initially experimentally demonstrated in and formalized in after seven years. In 2003, Salvador Venegas-Andraca and S. Bose presented Qubit Lattice, the first published general model for storing, processing and retrieving images using quantum systems. Later on, in 2005, Latorre proposed another kind of representation, called the Real Ket, whose purpose was to encode quantum images as a basis for further applications in QIMP. Furthermore, in 2010 Venegas-Andraca and Ball presented a method for storing and retrieving binary geometrical shapes in quantum mechanical systems in which it is shown that maximally entangled qubits can be used to reconstruct images without using any additional information. Technically, these pioneering efforts with the subsequent studies related to them can be classified into three main groups: Quantum-assisted digital image processing (QDIP): These applications aim at improving digital or classical image processing tasks and applications. Optics-based quantum imaging (OQI) Classically inspired quantum image processing (QIMP) A survey of quantum image representation has been published in. Furthermore, the recently published book Quantum Image Processing provides a comprehensive introduction to quantum image processing, which focuses on extending conventional image processing tasks to the quantum computing frameworks. It summarizes the available quantum image representations and their operations, reviews the possible quantum image applications and their implementation, and discusses the open questions and future development trends. == Quantum image representations == There are various approaches for quantum image representation, that are usually based on the encoding of color information. A common representation is FRQI (Flexible Representation for Quantum Images), that captures the color and position at every pixel of the image, and defined as: | I ⟩ = 1 2 n ∑ i = 0 2 2 n − 1 | c i ⟩ ⊗ | i ⟩ {\displaystyle \vert I\rangle ={\frac {1}{2^{n}}}\sum _{i=0}^{2^{2n-1}}\vert c_{i}\rangle \otimes \vert i\rangle } where | i ⟩ {\textstyle |i\rangle } is the position and | c i ⟩ = c o s θ i | 0 ⟩ + s i n θ i | 1 ⟩ {\textstyle \vert c_{i}\rangle =cos\theta _{i}\vert 0\rangle +sin\theta _{i}\vert 1\rangle } the color with a vector of angles θ i ∈ [ 0 , π / 2 ] {\textstyle \theta _{i}\in \left[0,\pi /2\right]} . As it can be seen, | c i ⟩ {\textstyle \vert c_{i}\rangle } is a regular qubit state of the form | ψ ⟩ = α | 0 ⟩ + β | 1 ⟩ {\displaystyle \vert \psi \rangle =\alpha \vert 0\rangle +\beta \vert 1\rangle } , with basis states | 0 ⟩ = ( 1 0 ) {\textstyle \vert 0\rangle ={\begin{pmatrix}1\\0\end{pmatrix}}} and | 1 ⟩ = ( 0 1 ) {\textstyle \vert 1\rangle ={\begin{pmatrix}0\\1\end{pmatrix}}} , as well as amplitudes α {\textstyle \alpha } and β {\textstyle \beta } that satisfy | α | 2 + | β | 2 = 1 {\textstyle \left|\alpha \right|^{2}+\left|\beta \right|^{2}=1} . Another common representation is MCQI (Multi-Channel Representation for Quantum Images), that uses the RGB channels with quantum states and following FRQI definition: | I ⟩ = 1 2 n + 1 ∑ i = 0 2 2 n − 1 | C R G B i ⟩ ⊗ | i ⟩ {\displaystyle \vert I\rangle ={\frac {1}{2^{n+1}}}\sum _{i=0}^{2^{2n-1}}\vert C_{RGB}^{i}\rangle \otimes \vert i\rangle } | C R G B i ⟩ = cos ⁡ θ R i | 000 ⟩ + cos ⁡ θ G i | 001 ⟩ + cos ⁡ θ B i | 010 ⟩ + sin ⁡ θ R i | 100 ⟩ + sin ⁡ θ G i | 101 ⟩ + sin ⁡ θ B i | 110 ⟩ + cos ⁡ θ α | 011 ⟩ + sin ⁡ θ α | 111 ⟩ {\displaystyle {\begin{aligned}{\begin{aligned}\vert C_{RGB}^{i}\rangle &={\cos \theta _{R}^{i}\vert 000\rangle }+{\cos \theta _{G}^{i}\vert 001\rangle }+{\cos \theta _{B}^{i}\vert 010\rangle }\\&\quad +{\sin \theta _{R}^{i}\vert 100\rangle }+{\sin \theta _{G}^{i}\vert 101\rangle }+{\sin \theta _{B}^{i}\vert 110\rangle }\\&\quad +{\cos {\theta _{\alpha }}\vert 011\rangle }+{\sin \theta _{\alpha }\vert 111\rangle }\end{aligned}}\end{aligned}}} Departing from the angle-based approach of FRQI and MCQI, and using a qubit sequence, NEQR (Novel Enhanced Representation for Quantum Images) is another representation approach, that uses a function f ( y , x ) = C y x q − 1 C y x q − 2 … C y x 1 C y x 0 {\textstyle f\left(y,x\right)=C_{yx}^{q-1}C_{yx}^{q-2}\ldots C_{yx}^{1}C_{yx}^{0}} to encode color values for a 2 n × 2 n {\displaystyle 2^{n}\times 2^{n}} image: | I ⟩ = 1 2 n ∑ y = 0 2 n − 1 ∑ x = 0 2 n − 1 | f ( y , x ) ⟩ | y x ⟩ {\displaystyle \vert I\rangle ={\frac {1}{2^{n}}}\sum _{y=0}^{2^{n}-1}\sum _{x=0}^{2^{n}-1}\vert f\left(y,x\right)\rangle \vert yx\rangle } == Quantum image manipulations == A lot of the effort in QIMP has been focused on designing algorithms to manipulate the position and color information encoded using flexible representation of quantum images (FRQI) and its many variants. For instance, FRQI-based fast geometric transformations including (two-point) swapping, flip, (orthogonal) rotations and restricted geometric transformations to constrain these operations to a specified area of an image were initially proposed. Recently, NEQR-based quantum image translation to map the position of each picture element in an input image into a new position in an output image and quantum image scaling to resize a quantum image were discussed. While FRQI-based general form of color transformations were first proposed by means of the single qubit gates such as X, Z, and H gates. Later, Multi-Channel Quantum Image-based channel of interest (CoI) operator to entail shifting the grayscale value of the preselected color channel and the channel swapping (CS) operator to swap the grayscale values between two channels have been fully discussed. To illustrate the feasibility and capability of QIMP algorithms and application, researchers always prefer to simulate the digital image processing tasks on the basis of the QIRs that we already have. By using the basic quantum gates and the aforementioned operations, so far, researchers have contributed to quantum image feature extraction, quantum image segmentation, quantum image morphology, quantum image comparison, quantum image filtering, quantum image classification, quantum image stabilization, among others. In particular, QIMP-based security technologies have attracted extensive interest of researchers as presented in the ensuing discussions. Similarly, these advancements have led to many applications in the areas of watermarking, encryption, and steganography etc., which form the core security technologies highlighted in this area. In general, the work pursued by the researchers in this area are focused on expanding the applicability of QIMP to realize more classical-like digital image processing algorithms; propose technologies to physically realize the QIMP hardware; or simply to note the likely challenges that could impede the realization of some QIMP protocols. == Quantum image transform == By encoding and processing the image information in quantum-mechanical systems, a framework of quantum image processing is presented, where a pure quantum state encodes the image information: to encode the pixel values in the probability amplitudes and the pixel positions in the computational basis states. Given an image F = ( F i , j ) M × L {\displaystyle F=(F_{i,j})_{M\times L}} , where F i , j {\displaystyle F_{i,j}} represents the pixel value at position ( i , j ) {\displaystyle (i,j)} with i = 1 , … , M {\displaystyle i=1,\dots ,M} and j = 1 , … , L {\displaystyle j=1,\dots ,L} , a vector f → {\displaystyle {\vec {f}}} with M L {\displaystyle ML} elements can be formed by letting the first M {\displaystyle M} elements of f → {\displaystyle {\vec {f}}} be the first column of F {\displaystyle F} , the next M {\displaystyle M} elements the second column, etc. A large class of image operations is linear, e.g., unitary transformations, convolutions, and linear filtering. In the quantum computing, the linear transformation can be represented as | g ⟩ = U ^ | f ⟩ {\displaystyle |g\rangle ={\hat {U}}|f\rangle } with the input image state | f ⟩ {\displaystyle |f\rangle } and the output image state | g ⟩ {\displaystyle |g\rangle } . A unitary transformation can be implemented as a unitary evolution. Some basic and commonly used image transforms (e.g., the Fourier, Hadamard, an

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  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). == Motivation and informal explanation == 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 {\displaystyle i} -th training example ( x i , y i ) {\displaystyle (\mathbf {x} _{i},y_{i})} and learn for it a corresponding weight w i {\displaystyle w_{i}} . Prediction for unlabeled inputs, i.e., those not in the training set, are treated by the application of a similarity function k {\displaystyle k} , called a kernel, between the unlabeled input x ′ {\displaystyle \mathbf {x'} } and each of the training inputs x i {\displaystyle \mathbf {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 ′ ) , {\displaystyle {\hat {y}}=\operatorname {sgn} \sum _{i=1}^{n}w_{i}y_{i}k(\mathbf {x} _{i},\mathbf {x'} ),} where y ^ ∈ { − 1 , + 1 } {\displaystyle {\hat {y}}\in \{-1,+1\}} is the kernelized binary classifier's predicted label for the unlabeled input x ′ {\displaystyle \mathbf {x'} } whose hidden true label y {\displaystyle y} is of interest; k : X × X → R {\displaystyle k\colon {\mathcal {X}}\times {\mathcal {X}}\to \mathbb {R} } is the kernel function that measures similarity between any pair of inputs x , x ′ ∈ X {\displaystyle \mathbf {x} ,\mathbf {x'} \in {\mathcal {X}}} ; the sum ranges over the n labeled examples { ( x i , y i ) } i = 1 n {\displaystyle \{(\mathbf {x} _{i},y_{i})\}_{i=1}^{n}} in the classifier's training set, with y i ∈ { − 1 , + 1 } {\displaystyle y_{i}\in \{-1,+1\}} ; the w i ∈ R {\displaystyle w_{i}\in \mathbb {R} } are the weights for the training examples, as determined by the learning algorithm; the sign function sgn {\displaystyle \operatorname {sgn} } determines whether the predicted classification y ^ {\displaystyle {\hat {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. == Mathematics: the kernel trick == 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 {\displaystyle \mathbf {x} } and x ′ {\displaystyle \mathbf {x'} } in the input space X {\displaystyle {\mathcal {X}}} , certain functions k ( x , x ′ ) {\displaystyle k(\mathbf {x} ,\mathbf {x'} )} can be expressed as an inner product in another space V {\displaystyle {\mathcal {V}}} . The function k : X × X → R {\displaystyle k\colon {\mathcal {X}}\times {\mathcal {X}}\to \mathbb {R} } 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 {\displaystyle k} . The computation is made much simpler if the kernel can be written in the form of a "feature map" φ : X → V {\displaystyle \varphi \colon {\mathcal {X}}\to {\mathcal {V}}} which satisfies k ( x , x ′ ) = ⟨ φ ( x ) , φ ( x ′ ) ⟩ V . {\displaystyle k(\mathbf {x} ,\mathbf {x'} )=\langle \varphi (\mathbf {x} ),\varphi (\mathbf {x'} )\rangle _{\mathcal {V}}.} The key restriction is that ⟨ ⋅ , ⋅ ⟩ V {\displaystyle \langle \cdot ,\cdot \rangle _{\mathcal {V}}} must be a proper inner product. On the other hand, an explicit representation for φ {\displaystyle \varphi } is not necessary, as long as V {\displaystyle {\mathcal {V}}} is an inner product space. The alternative follows from Mercer's theorem: an implicitly defined function φ {\displaystyle \varphi } exists whenever the space X {\displaystyle {\mathcal {X}}} can be equipped with a suitable measure ensuring the function k {\displaystyle 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 | {\displaystyle \mu (T)=|T|} for all T ⊂ X {\displaystyle T\subset X} , which counts the number of points inside the set T {\displaystyle 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. {\displaystyle \sum _{i=1}^{n}\sum _{j=1}^{n}k(\mathbf {x} _{i},\mathbf {x} _{j})c_{i}c_{j}\geq 0.} If this summation holds for all finite sequences of points ( x 1 , … , x n ) {\displaystyle (\mathbf {x} _{1},\dotsc ,\mathbf {x} _{n})} in X {\displaystyle {\mathcal {X}}} and all choices of n {\displaystyle n} real-valued coefficients ( c 1 , … , c n ) {\displaystyle (c_{1},\dots ,c_{n})} (cf. positive definite kernel), then the function k {\displaystyle k} satisfies Mercer's condition. Some algorithms that depend on arbitrary relationships in the native space X {\displaystyle {\mathcal {X}}} would, in fact, have a linear interpretation in a different setting: the range space of φ {\displaystyle \varphi } . The linear interpretation gives us insight about the algorithm. Furthermore, there is often no need to compute φ {\displaystyle \varphi } 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 {\displaystyle \mathbf {K} \in \mathbb {R} ^{n\times n}} with respect to { x 1 , … , x n } {\displaystyle \{\mathbf {x} _{1},\dotsc ,\mathbf {x} _{n}\}} (sometimes also called a "kernel matrix"), where K i j = k ( x i , x j ) {\displaystyle K_{ij}=k(\mathbf {x} _{i},\mathbf {x} _{j})} , must be positive semi-definite (PSD). Empirically, for machine learning heuristics, choices of a function k {\displaystyle k} that do not satisfy Mercer's condition may still perform reasonably if k {\displaystyle k} at least approximates the intuitive idea of similarity. Regardless of whether k {\displaystyle k} is a Mercer kernel, k {\displaystyle k} may still be referred to as a "kernel". If the kernel function k {\displaystyle k} is also a covariance function as used in Gaussian processes, then the Gram matrix K {\displaystyle \mathbf {K} } can also be called a covariance matrix. == Applications == Application areas of kernel methods are diverse and include geostatistics, kriging, inverse distance weighting, 3D reconstruction, bioinformatics, cheminformatics, information extraction and handwriting recognition. == Popular kernels == Fisher kernel Graph kernels Kernel smoother Polynomial kernel Radial basis function kern

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  Ensemble learning

  In statistics and machine learning, ensemble methods use multiple learning algorithms to obtain better predictive performance than could be obtained from any of the constituent learning algorithms alone. Unlike a statistical ensemble in statistical mechanics, which is usually infinite, a machine learning ensemble consists of only a concrete finite set of alternative models, but typically allows for much more flexible structure to exist among those alternatives. == Overview == Supervised learning algorithms search through a hypothesis space to find a suitable hypothesis that will make good predictions with a particular problem. Even if this space contains hypotheses that are very well-suited for a particular problem, it may be very difficult to find a good one. Ensembles combine multiple hypotheses to form one which should be theoretically better. Ensemble learning trains two or more machine learning algorithms on a specific classification or regression task. The algorithms within the ensemble model are generally referred as "base models", "base learners", or "weak learners" in literature. These base models can be constructed using a single modelling algorithm, or several different algorithms. The idea is to train a diverse set of weak models on the same modelling task, such that the outputs of each weak learner have poor predictive ability (i.e., high bias), and among all weak learners, the outcome and error values exhibit high variance. Fundamentally, an ensemble learning model trains at least two high-bias (weak) and high-variance (diverse) models to be combined into a better-performing model. The set of weak models — which would not produce satisfactory predictive results individually — are combined or averaged to produce a single, high performing, accurate, and low-variance model to fit the task as required. Ensemble learning typically refers to bagging (bootstrap aggregating), boosting or stacking/blending techniques to induce high variance among the base models. Bagging creates diversity by generating random samples from the training observations and fitting the same model to each different sample — also known as homogeneous parallel ensembles. Boosting follows an iterative process by sequentially training each base model on the up-weighted errors of the previous base model, producing an additive model to reduce the final model errors — also known as sequential ensemble learning. Stacking or blending consists of different base models, each trained independently (i.e. diverse/high variance) to be combined into the ensemble model — producing a heterogeneous parallel ensemble. Common applications of ensemble learning include random forests (an extension of bagging), Boosted Tree models, and Gradient Boosted Tree Models. Models in applications of stacking are generally more task-specific — such as combining clustering techniques with other parametric and/or non-parametric techniques. Evaluating the prediction of an ensemble typically requires more computation than evaluating the prediction of a single model. In one sense, ensemble learning may be thought of as a way to compensate for poor learning algorithms by performing a lot of extra computation. On the other hand, the alternative is to do a lot more learning with one non-ensemble model. An ensemble may be more efficient at improving overall accuracy for the same increase in compute, storage, or communication resources by using that increase on two or more methods, than would have been improved by increasing resource use for a single method. Fast algorithms such as decision trees are commonly used in ensemble methods (e.g., random forests), although slower algorithms can benefit from ensemble techniques as well. By analogy, ensemble techniques have been used also in unsupervised learning scenarios, for example in consensus clustering or in anomaly detection. == Ensemble theory == Empirically, ensembles tend to yield better results when there is a significant diversity among the models. Many ensemble methods, therefore, seek to promote diversity among the models they combine. Although perhaps non-intuitive, more random algorithms (like random decision trees) can be used to produce a stronger ensemble than very deliberate algorithms (like entropy-reducing decision trees). Using a variety of strong learning algorithms, however, has been shown to be more effective than using techniques that attempt to dumb-down the models in order to promote diversity. It is possible to increase diversity in the training stage of the model using correlation for regression tasks or using information measures such as cross entropy for classification tasks. Theoretically, one can justify the diversity concept because the lower bound of the error rate of an ensemble system can be decomposed into accuracy, diversity, and the other term. === The geometric framework === Ensemble learning, including both regression and classification tasks, can be explained using a geometric framework. Within this framework, the output of each individual classifier or regressor for the entire dataset can be viewed as a point in a multi-dimensional space. Additionally, the target result is also represented as a point in this space, referred to as the "ideal point." The Euclidean distance is used as the metric to measure both the performance of a single classifier or regressor (the distance between its point and the ideal point) and the dissimilarity between two classifiers or regressors (the distance between their respective points). This perspective transforms ensemble learning into a deterministic problem. For example, within this geometric framework, it can be proved that the averaging of the outputs (scores) of all base classifiers or regressors can lead to equal or better results than the average of all the individual models. It can also be proved that if the optimal weighting scheme is used, then a weighted averaging approach can outperform any of the individual classifiers or regressors that make up the ensemble or as good as the best performer at least. == Ensemble size == While the number of component classifiers of an ensemble has a great impact on the accuracy of prediction, there is a limited number of studies addressing this problem. A priori determining of ensemble size and the volume and velocity of big data streams make this even more crucial for online ensemble classifiers. Mostly statistical tests were used for determining the proper number of components. More recently, a theoretical framework suggested that there is an ideal number of component classifiers for an ensemble such that having more or less than this number of classifiers would deteriorate the accuracy. It is called "the law of diminishing returns in ensemble construction." Their theoretical framework shows that using the same number of independent component classifiers as class labels gives the highest accuracy. == Common types of ensembles == === Bayes optimal classifier === The Bayes optimal classifier is a classification technique. It is an ensemble of all the hypotheses in the hypothesis space. On average, no other ensemble can outperform it. The Naive Bayes classifier is a version of this that assumes that the data is conditionally independent on the class and makes the computation more feasible. Each hypothesis is given a vote proportional to the likelihood that the training dataset would be sampled from a system if that hypothesis were true. To facilitate training data of finite size, the vote of each hypothesis is also multiplied by the prior probability of that hypothesis. The Bayes optimal classifier can be expressed with the following equation: y = a r g m a x c j ∈ C ∑ h i ∈ H P ( c j | h i ) P ( T | h i ) P ( h i ) {\displaystyle y={\underset {c_{j}\in C}{\mathrm {argmax} }}\sum _{h_{i}\in H}{P(c_{j}|h_{i})P(T|h_{i})P(h_{i})}} where y {\displaystyle y} is the predicted class, C {\displaystyle C} is the set of all possible classes, H {\displaystyle H} is the hypothesis space, P {\displaystyle P} refers to a probability, and T {\displaystyle T} is the training data. As an ensemble, the Bayes optimal classifier represents a hypothesis that is not necessarily in H {\displaystyle H} . The hypothesis represented by the Bayes optimal classifier, however, is the optimal hypothesis in ensemble space (the space of all possible ensembles consisting only of hypotheses in H {\displaystyle H} ). This formula can be restated using Bayes' theorem, which says that the posterior is proportional to the likelihood times the prior: P ( h i | T ) ∝ P ( T | h i ) P ( h i ) {\displaystyle P(h_{i}|T)\propto P(T|h_{i})P(h_{i})} hence, y = a r g m a x c j ∈ C ∑ h i ∈ H P ( c j | h i ) P ( h i | T ) {\displaystyle y={\underset {c_{j}\in C}{\mathrm {argmax} }}\sum _{h_{i}\in H}{P(c_{j}|h_{i})P(h_{i}|T)}} === Bootstrap aggregating (bagging) === Bootstrap aggregation (bagging) involves training an ensemble on bootstrapped data sets. A bootstrapped set is cr

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  Radial basis function

  In mathematics a radial basis function (RBF) is a real-valued function φ {\textstyle \varphi } whose value depends only on the distance between the input and some fixed point, either the origin, so that φ ( x ) = φ ^ ( ‖ x ‖ ) {\textstyle \varphi (\mathbf {x} )={\hat {\varphi }}(\left\|\mathbf {x} \right\|)} , or some other fixed point c {\textstyle \mathbf {c} } , called a center, so that φ ( x ) = φ ^ ( ‖ x − c ‖ ) {\textstyle \varphi (\mathbf {x} )={\hat {\varphi }}(\left\|\mathbf {x} -\mathbf {c} \right\|)} . Any function φ {\textstyle \varphi } that satisfies the property φ ( x ) = φ ^ ( ‖ x ‖ ) {\textstyle \varphi (\mathbf {x} )={\hat {\varphi }}(\left\|\mathbf {x} \right\|)} is a radial function. The distance is usually Euclidean distance, although other metrics are sometimes used. They are often used as a collection { φ k } k {\displaystyle \{\varphi _{k}\}_{k}} which forms a basis for some function space of interest, hence the name. Sums of radial basis functions are typically used to approximate given functions. This approximation process can also be interpreted as a simple kind of neural network; this was the context in which they were originally applied to machine learning, in work by David Broomhead and David Lowe in 1988, which stemmed from Michael J. D. Powell's seminal research from 1977. RBFs are also used as a kernel in support vector classification. The technique has proven effective and flexible enough that radial basis functions are now applied in a variety of engineering applications. == Definition == A radial function is a function φ : [ 0 , ∞ ) → R {\textstyle \varphi :[0,\infty )\to \mathbb {R} } . When paired with a norm ‖ ⋅ ‖ : V → [ 0 , ∞ ) {\textstyle \|\cdot \|:V\to [0,\infty )} on a vector space, a function of the form φ c = φ ( ‖ x − c ‖ ) {\textstyle \varphi _{\mathbf {c} }=\varphi (\|\mathbf {x} -\mathbf {c} \|)} is said to be a radial kernel centered at c ∈ V {\textstyle \mathbf {c} \in V} . A radial function and the associated radial kernels are said to be radial basis functions if, for any finite set of nodes { x k } k = 1 n ⊆ V {\displaystyle \{\mathbf {x} _{k}\}_{k=1}^{n}\subseteq V} , all of the following conditions are true: === Examples === Commonly used types of radial basis functions include (writing r = ‖ x − x i ‖ {\textstyle r=\left\|\mathbf {x} -\mathbf {x} _{i}\right\|} and using ε {\textstyle \varepsilon } to indicate a shape parameter that can be used to scale the input of the radial kernel): == Approximation == Radial basis functions are typically used to build up function approximations of the form where the approximating function y ( x ) {\textstyle y(\mathbf {x} )} is represented as a sum of N {\displaystyle N} radial basis functions, each associated with a different center x i {\textstyle \mathbf {x} _{i}} , and weighted by an appropriate coefficient w i . {\textstyle w_{i}.} The weights w i {\textstyle w_{i}} can be estimated using the matrix methods of linear least squares, because the approximating function is linear in the weights w i {\textstyle w_{i}} . Approximation schemes of this kind have been particularly used in time series prediction and control of nonlinear systems exhibiting sufficiently simple chaotic behaviour and 3D reconstruction in computer graphics (for example, hierarchical RBF and Pose Space Deformation). == RBF Network == The sum can also be interpreted as a rather simple single-layer type of artificial neural network called a radial basis function network, with the radial basis functions taking on the role of the activation functions of the network. It can be shown that any continuous function on a compact interval can in principle be interpolated with arbitrary accuracy by a sum of this form, if a sufficiently large number N {\textstyle N} of radial basis functions is used. The approximant y ( x ) {\textstyle y(\mathbf {x} )} is differentiable with respect to the weights w i {\textstyle w_{i}} . The weights could thus be learned using any of the standard iterative methods for neural networks. Using radial basis functions in this manner yields a reasonable interpolation approach provided that the fitting set has been chosen such that it covers the entire range systematically (equidistant data points are ideal). However, without a polynomial term that is orthogonal to the radial basis functions, estimates outside the fitting set tend to perform poorly. == RBFs for PDEs == Radial basis functions are used to approximate functions and so can be used to discretize and numerically solve Partial Differential Equations (PDEs). This was first done in 1990 by E. J. Kansa who developed the first RBF based numerical method. It is called the Kansa method and was used to solve the elliptic Poisson equation and the linear advection-diffusion equation. The function values at points x {\displaystyle \mathbf {x} } in the domain are approximated by the linear combination of RBFs: The derivatives are approximated as such: where N {\displaystyle N} are the number of points in the discretized domain, d {\displaystyle d} the dimension of the domain and λ {\displaystyle \lambda } the scalar coefficients that are unchanged by the differential operator. Different numerical methods based on Radial Basis Functions were developed thereafter. Some methods are the RBF-FD method, the RBF-QR method and the RBF-PUM method.

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  LakeFS

  lakeFS is an open-source data version control system for managing data stored in object storage. It provides Git-like operations such as branching, committing, merging, and reverting for large-scale data stored in systems including Amazon S3, Azure Blob Storage, and Google Cloud Storage, as well as other S3-compatible object storage platforms. lakeFS is used in data engineering and machine learning workflows to manage changes to data, support reproducibility, and enable data governance across data lakes. The software is available as an open-source project, as well as in enterprise and managed service offerings, including lakeFS Cloud. == History == lakeFS was created in 2020 by Einat Orr and Oz Katz at Treeverse. Its first public release, version 0.8.1, appeared in August 2020 and introduced Git-style operations with support for Amazon S3. In 2021, Treeverse raised $23 million in a Series A funding round led by Dell Technologies Capital, Norwest Venture Partners, and Zeev Ventures. The same year, lakeFS was included in InfoWorld’s Best of Open Source Software (Bossie) awards. In June 2022, Treeverse introduced lakeFS Cloud, a managed service providing hosted lakeFS deployments for cloud-based data lakes. Version 1.0 was released in October 2023, adding integrations with platforms such as Databricks and Apache Iceberg, as well as support for orchestration tools including Apache Airflow. Public case studies and conference materials have described usage of lakeFS by organizations such as Microsoft, Volvo, and NASA. In July 2025, Treeverse announced an additional $20 million in growth funding to support further development of lakeFS. In November 2025, Treeverse announced the acquisition of the open-source data version control project DVC. == Software == === Overview === lakeFS provides Git-like operations such as branching, committing, merging, and reverting for datasets stored in object storage. These operations are used to manage changes to data, test modifications in isolation, reproduce specific data states, and recover from errors or unintended updates. === Architecture === lakeFS operates as a metadata layer on top of object storage systems such as Amazon S3, Azure Blob Storage, and Google Cloud Storage. It stores repository metadata describing commits, branches, and tags, enabling versioned views of data without copying underlying objects. The system provides access through multiple interfaces, including a web user interface, command-line tools, a REST API, and software development kits. It is designed to integrate with existing data engineering and machine learning workflows, and can be deployed either in self-hosted environments or as a managed service. === Functions === lakeFS provides version control functionality for data stored in object storage–based data lakes. Core features include: Atomic commits and version tracking for datasets, supporting reproducibility and auditability. Branching and merging mechanisms that allow isolated development and testing without duplicating data. Configurable hooks that can validate data or trigger external processes during commit and merge operations. The ability to revert repositories to earlier states to recover from data errors or failed changes. Recording of commit history and associated metadata for lineage tracking. Support for managing data across multiple object storage systems, including Amazon S3, Azure Blob Storage, Google Cloud Storage, and MinIO. Use of fixed data versions to reproduce experiments and machine learning model training. === Integrations === Coverage of lakeFS has described integrations with platforms such as Databricks and Apache Iceberg, as well as support for environments including Red Hat OpenShift. Additional materials describe its use with Trino, including validation of data changes prior to merging in versioned data workflows, as well as compatibility with orchestration tools such as Apache Airflow.

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  Low-rank approximation

  In mathematics, low-rank approximation refers to the process of approximating a given matrix by a matrix of lower rank. More precisely, it is a minimization problem, in which the cost function measures the fit between a given matrix (the data) and an approximating matrix (the optimization variable), subject to a constraint that the approximating matrix has reduced rank. The problem is used for mathematical modeling and data compression. The rank constraint is related to a constraint on the complexity of a model that fits the data. In applications, often there are other constraints on the approximating matrix apart from the rank constraint, e.g., non-negativity and Hankel structure. Low-rank approximation is closely related to numerous other techniques, including principal component analysis, factor analysis, total least squares, latent semantic analysis, orthogonal regression, and dynamic mode decomposition. == Definition == Given structure specification S : R n p → R m × n {\displaystyle {\mathcal {S}}:\mathbb {R} ^{n_{p}}\to \mathbb {R} ^{m\times n}} , vector of structure parameters p ∈ R n p {\displaystyle p\in \mathbb {R} ^{n_{p}}} , norm ‖ ⋅ ‖ {\displaystyle \|\cdot \|} , and desired rank r {\displaystyle r} , minimize over p ^ ‖ p − p ^ ‖ subject to rank ⁡ ( S ( p ^ ) ) ≤ r . {\displaystyle {\text{minimize}}\quad {\text{over }}{\widehat {p}}\quad \|p-{\widehat {p}}\|\quad {\text{subject to}}\quad \operatorname {rank} {\big (}{\mathcal {S}}({\widehat {p}}){\big )}\leq r.} == Applications == Linear system identification, in which case the approximating matrix is Hankel structured. Machine learning, in which case the approximating matrix is nonlinearly structured. Recommender systems, in which cases the data matrix has missing values and the approximation is categorical. Distance matrix completion, in which case there is a positive definiteness constraint. Natural language processing, in which case the approximation is nonnegative. Computer algebra, in which case the approximation is Sylvester structured. Matrix product states, in which case the approximation is usually rescaled to have fixed Frobenius norm. == Basic low-rank approximation problem == The unstructured problem with fit measured by the Frobenius norm, i.e., minimize over D ^ ‖ D − D ^ ‖ F subject to rank ⁡ ( D ^ ) ≤ r {\displaystyle {\text{minimize}}\quad {\text{over }}{\widehat {D}}\quad \|D-{\widehat {D}}\|_{\text{F}}\quad {\text{subject to}}\quad \operatorname {rank} {\big (}{\widehat {D}}{\big )}\leq r} has an analytic solution in terms of the singular value decomposition of the data matrix. The result is referred to as the matrix approximation lemma or Eckart–Young–Mirsky theorem. This problem was originally solved by Erhard Schmidt in the infinite dimensional context of integral operators (although his methods easily generalize to arbitrary compact operators on Hilbert spaces) and later rediscovered by C. Eckart and G. Young. L. Mirsky generalized the result to arbitrary unitarily invariant norms. Let D = U Σ V ⊤ ∈ R m × n , m ≥ n {\displaystyle D=U\Sigma V^{\top }\in \mathbb {R} ^{m\times n},\quad m\geq n} be the singular value decomposition of D {\displaystyle D} , where Σ =: diag ⁡ ( σ 1 , … , σ r ) {\displaystyle \Sigma =:\operatorname {diag} (\sigma _{1},\ldots ,\sigma _{r})} , where r ≤ min { m , n } = n {\displaystyle r\leq \min\{m,n\}=n} , is the m × n {\displaystyle m\times n} rectangular diagonal matrix with r {\displaystyle r} non-zero singular values σ 1 ≥ … ≥ σ r > σ r + 1 = … = σ n = 0 {\displaystyle \sigma _{1}\geq \ldots \geq \sigma _{r}>\sigma _{r+1}=\ldots =\sigma _{n}=0} . For a given k ∈ { 1 , … , r } {\displaystyle k\in \{1,\dots ,r\}} , partition U {\displaystyle U} , Σ {\displaystyle \Sigma } , and V {\displaystyle V} as follows: U =: [ U 1 U 2 ] , Σ =: [ Σ 1 0 0 Σ 2 ] , and V =: [ V 1 V 2 ] , {\displaystyle U=:{\begin{bmatrix}U_{1}&U_{2}\end{bmatrix}},\quad \Sigma =:{\begin{bmatrix}\Sigma _{1}&0\\0&\Sigma _{2}\end{bmatrix}},\quad {\text{and}}\quad V=:{\begin{bmatrix}V_{1}&V_{2}\end{bmatrix}},} where U 1 {\displaystyle U_{1}} is m × k {\displaystyle m\times k} , Σ 1 {\displaystyle \Sigma _{1}} is k × k {\displaystyle k\times k} , and V 1 {\displaystyle V_{1}} is n × k {\displaystyle n\times k} . Then the rank k {\displaystyle k} matrix D ^ ∗ := U 1 Σ 1 V 1 ⊤ , {\displaystyle {\widehat {D}}^{}:=U_{1}\Sigma _{1}V_{1}^{\top },} obtained from the truncated singular value decomposition is such that ‖ D − D ^ ∗ ‖ F = min rank ⁡ ( D ^ ) ≤ k ‖ D − D ^ ‖ F = σ k + 1 2 + ⋯ + σ r 2 . {\displaystyle \|D-{\widehat {D}}^{}\|_{\text{F}}=\min _{\operatorname {rank} ({\widehat {D}})\leq k}\|D-{\widehat {D}}\|_{\text{F}}={\sqrt {\sigma _{k+1}^{2}+\cdots +\sigma _{r}^{2}}}.} The minimizer D ^ ∗ {\displaystyle {\widehat {D}}^{}} is unique if and only if σ k > σ k + 1 {\displaystyle \sigma _{k}>\sigma _{k+1}} . == Proof of Eckart–Young–Mirsky theorem (for spectral norm) == Let A ∈ R m × n {\displaystyle A\in \mathbb {R} ^{m\times n}} be a real (possibly rectangular) matrix with m ≤ n {\displaystyle m\leq n} . Suppose that A = U Σ V ⊤ {\displaystyle A=U\Sigma V^{\top }} is the singular value decomposition of A {\displaystyle A} . Recall that U {\displaystyle U} and V {\displaystyle V} are orthogonal matrices, and Σ {\displaystyle \Sigma } is an m × n {\displaystyle m\times n} diagonal matrix with entries ( σ 1 , σ 2 , ⋯ , σ m ) {\displaystyle (\sigma _{1},\sigma _{2},\cdots ,\sigma _{m})} such that σ 1 ≥ σ 2 ≥ ⋯ ≥ σ m ≥ 0 {\displaystyle \sigma _{1}\geq \sigma _{2}\geq \cdots \geq \sigma _{m}\geq 0} . We claim that the best rank- k {\displaystyle k} approximation to A {\displaystyle A} in the spectral norm, denoted by ‖ ⋅ ‖ 2 {\displaystyle \|\cdot \|_{2}} , is given by A k := ∑ i = 1 k σ i u i v i ⊤ {\displaystyle A_{k}:=\sum _{i=1}^{k}\sigma _{i}u_{i}v_{i}^{\top }} where u i {\displaystyle u_{i}} and v i {\displaystyle v_{i}} denote the i {\displaystyle i} th column of U {\displaystyle U} and V {\displaystyle V} , respectively. First, note that we have ‖ A − A k ‖ 2 = ‖ ∑ i = 1 n σ i u i v i ⊤ − ∑ i = 1 k σ i u i v i ⊤ ‖ 2 = ‖ ∑ i = k + 1 n σ i u i v i ⊤ ‖ 2 = σ k + 1 {\displaystyle \|A-A_{k}\|_{2}=\left\|\sum _{i=1}^{\color {red}{n}}\sigma _{i}u_{i}v_{i}^{\top }-\sum _{i=1}^{\color {red}{k}}\sigma _{i}u_{i}v_{i}^{\top }\right\|_{2}=\left\|\sum _{i=\color {red}{k+1}}^{n}\sigma _{i}u_{i}v_{i}^{\top }\right\|_{2}=\sigma _{k+1}} Therefore, we need to show that if B k = X Y ⊤ {\displaystyle B_{k}=XY^{\top }} where X {\displaystyle X} and Y {\displaystyle Y} have k {\displaystyle k} columns then ‖ A − A k ‖ 2 = σ k + 1 ≤ ‖ A − B k ‖ 2 {\displaystyle \|A-A_{k}\|_{2}=\sigma _{k+1}\leq \|A-B_{k}\|_{2}} . Since Y {\displaystyle Y} has k {\displaystyle k} columns, then there must be a nontrivial linear combination of the first k + 1 {\displaystyle k+1} columns of V {\displaystyle V} , i.e., w = γ 1 v 1 + ⋯ + γ k + 1 v k + 1 , {\displaystyle w=\gamma _{1}v_{1}+\cdots +\gamma _{k+1}v_{k+1},} such that Y ⊤ w = 0 {\displaystyle Y^{\top }w=0} . Without loss of generality, we can scale w {\displaystyle w} so that ‖ w ‖ 2 = 1 {\displaystyle \|w\|_{2}=1} or (equivalently) γ 1 2 + ⋯ + γ k + 1 2 = 1 {\displaystyle \gamma _{1}^{2}+\cdots +\gamma _{k+1}^{2}=1} . Therefore, ‖ A − B k ‖ 2 2 ≥ ‖ ( A − B k ) w ‖ 2 2 = ‖ A w ‖ 2 2 = γ 1 2 σ 1 2 + ⋯ + γ k + 1 2 σ k + 1 2 ≥ σ k + 1 2 . {\displaystyle \|A-B_{k}\|_{2}^{2}\geq \|(A-B_{k})w\|_{2}^{2}=\|Aw\|_{2}^{2}=\gamma _{1}^{2}\sigma _{1}^{2}+\cdots +\gamma _{k+1}^{2}\sigma _{k+1}^{2}\geq \sigma _{k+1}^{2}.} The result follows by taking the square root of both sides of the above inequality. == Proof of Eckart–Young–Mirsky theorem (for Frobenius norm) == Let A ∈ R m × n {\displaystyle A\in \mathbb {R} ^{m\times n}} be a real (possibly rectangular) matrix with m ≤ n {\displaystyle m\leq n} . Suppose that A = U Σ V ⊤ {\displaystyle A=U\Sigma V^{\top }} is the singular value decomposition of A {\displaystyle A} . We claim that the best rank k {\displaystyle k} approximation to A {\displaystyle A} in the Frobenius norm, denoted by ‖ ⋅ ‖ F {\displaystyle \|\cdot \|_{F}} , is given by A k = ∑ i = 1 k σ i u i v i ⊤ {\displaystyle A_{k}=\sum _{i=1}^{k}\sigma _{i}u_{i}v_{i}^{\top }} where u i {\displaystyle u_{i}} and v i {\displaystyle v_{i}} denote the i {\displaystyle i} th column of U {\displaystyle U} and V {\displaystyle V} , respectively. First, note that we have ‖ A − A k ‖ F 2 = ‖ ∑ i = k + 1 n σ i u i v i ⊤ ‖ F 2 = ∑ i = k + 1 n σ i 2 {\displaystyle \|A-A_{k}\|_{F}^{2}=\left\|\sum _{i=k+1}^{n}\sigma _{i}u_{i}v_{i}^{\top }\right\|_{F}^{2}=\sum _{i=k+1}^{n}\sigma _{i}^{2}} Therefore, we need to show that if B k = X Y ⊤ {\displaystyle B_{k}=XY^{\top }} where X {\displaystyle X} and Y {\displaystyle Y} have k {\displaystyle k} columns then ‖ A − A k ‖ F 2 = ∑ i = k + 1 n σ i 2 ≤ ‖ A − B k ‖ F 2 . {\displaystyle \|A-A_{k}\|_{F}^{2}=\sum _{i=k+1}^{n}\sigma _{i}^{2}\leq \|A-B_{k}\|_{F}^{2}.} By the triangle inequality with the spectral norm

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  Tanagra (machine learning)

  Tanagra is a free suite of machine learning software for research and academic purposes developed by Ricco Rakotomalala at the Lumière University Lyon 2, France. Tanagra supports several standard data mining tasks such as: Visualization, Descriptive statistics, Instance selection, feature selection, feature construction, regression, factor analysis, clustering, classification and association rule learning. Tanagra is an academic project. It is widely used in French-speaking universities. Tanagra is frequently used in real studies and in software comparison papers. == History == The development of Tanagra was started in June 2003. The first version was distributed in December 2003. Tanagra is the successor of Sipina, another free data mining tool which is intended only for supervised learning tasks (classification), especially the interactive and visual construction of decision trees. Sipina is still available online and is maintained. Tanagra is an "open source project" as every researcher can access the source code and add their own algorithms, as long as they agree and conform to the software distribution license. The main purpose of the Tanagra project is to give researchers and students a user-friendly data mining software, conforming to the present norms of the software development in this domain (especially in the design of its GUI and the way to use it), and allowing the analyzation of either real or synthetic data. From 2006, Ricco Rakotomalala made an important documentation effort. A large number of tutorials are published on a dedicated website. They describe the statistical and machine learning methods and their implementation with Tanagra on real case studies. The use of other free data mining tools on the same problems is also widely described. The comparison of the tools enables readers to understand the possible differences in the presentation of results. == Description == Tanagra works similarly to current data mining tools. The user can design visually a data mining process in a diagram. Each node is a statistical or machine learning technique, the connection between two nodes represents the data transfer. But unlike the majority of tools which are based on the workflow paradigm, Tanagra is very simplified. The treatments are represented in a tree diagram. The results are displayed in an HTML format. This makes it is easy to export the outputs in order to visualize the results in a browser. It is also possible to copy the result tables to a spreadsheet. Tanagra makes a good compromise between statistical approaches (e.g. parametric and nonparametric statistical tests), multivariate analysis methods (e.g. factor analysis, correspondence analysis, cluster analysis, regression) and machine learning techniques (e.g. neural network, support vector machine, decision trees, random forest).

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  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.

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  GNU toolchain

  The GNU toolchain is a broad collection of programming tools produced by the GNU Project. These tools form a toolchain (a suite of tools used in a serial manner) used for developing software applications and operating systems. The GNU toolchain plays a vital role in development of Linux, some BSD systems, and software for embedded systems. Parts of the GNU toolchain are also directly used with or ported to other platforms such as Solaris, macOS, Microsoft Windows (via Cygwin and MinGW/MSYS/WSL2), Sony PlayStation Portable (used by PSP modding scene) and Sony PlayStation 3. == Components == Projects in the GNU toolchain are: GNU Autotools (build system) – Software build toolset from GNU GNU Binutils – GNU software development tools for executable code GNU Bison – Yacc-compatible parser generator program GNU C Library – GNU implementation of the standard C libraryPages displaying short descriptions of redirect targets GNU Compiler Collection – Free and open-source compiler for various programming languages GNU Debugger – Source-level debugger GNU m4 – General-purpose macro processor GNU make – Software build automation tool

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  Moral graph

  In graph theory, a moral graph is used to find the equivalent undirected form of a directed acyclic graph. It is a key step of the junction tree algorithm, used in belief propagation on graphical models. The moralized counterpart of a directed acyclic graph is formed by adding edges between all pairs of non-adjacent nodes that have a common child, and then making all edges in the graph undirected. Equivalently, a moral graph of a directed acyclic graph G is an undirected graph in which each node of the original G is now connected to its Markov blanket. The name stems from the fact that, in a moral graph, two nodes that have a common child are required to be married by sharing an edge. Moralization may also be applied to mixed graphs, called in this context "chain graphs". In a chain graph, a connected component of the undirected subgraph is called a chain. Moralization adds an undirected edge between any two vertices that both have outgoing edges to the same chain, and then forgets the orientation of the directed edges of the graph. == Weakly recursively simplicial == A graph is weakly recursively simplicial if it has a simplicial vertex and the subgraph after removing a simplicial vertex and some edges (possibly none) between its neighbours is weakly recursively simplicial. A graph is moral if and only if it is weakly recursively simplicial. A chordal graph (a.k.a., recursive simplicial) is a special case of weakly recursively simplicial when no edge is removed during the elimination process. Therefore, a chordal graph is also moral. But a moral graph is not necessarily chordal. == Recognising moral graphs == Unlike chordal graphs that can be recognised in polynomial time, Verma & Pearl (1993) proved that deciding whether or not a graph is moral is NP-complete.

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  Nearest neighbor search

  Nearest neighbor search (NNS), as a form of proximity search, is the optimization problem of finding the point in a given set that is closest (or most similar) to a given point. Closeness is typically expressed in terms of a dissimilarity function: the less similar the objects, the larger the function values. Formally, the nearest neighbor (NN) search problem is defined as follows: given a set S of points in a space M and a query point q ∈ M {\displaystyle q\in M} , find the closest point in S to q. Donald Knuth in volume 3 of The Art of Computer Programming (1973) called it the post-office problem, referring to an application of assigning to a residence the nearest post office. A direct generalization of this problem is a k-NN search, where we need to find the k closest points. Most commonly M is a metric space and dissimilarity is expressed as a distance metric, which is symmetric and satisfies the triangle inequality. Even more common, M is taken to be the d-dimensional vector space where dissimilarity is measured using the Euclidean distance, Manhattan distance or other distance metric. However, the dissimilarity function can be arbitrary. One example is asymmetric Bregman divergence, for which the triangle inequality does not hold. == Applications == The nearest neighbor search problem arises in numerous fields of application, including: Pattern recognition – in particular for optical character recognition Statistical classification – see k-nearest neighbor algorithm Computer vision – for point cloud registration Computational geometry – see Closest pair of points problem Cryptanalysis – for lattice problem Databases – e.g. content-based image retrieval Coding theory – see maximum likelihood decoding Semantic search Vector databases, where nearest-neighbor lookup over embeddings is used to retrieve semantically similar records Retrieval-augmented generation systems, where nearest-neighbor retrieval over embeddings is used to fetch candidate passages or documents before generation Data compression – see MPEG-2 standard Robotic sensing Recommendation systems, e.g. see Collaborative filtering Internet marketing – see contextual advertising and behavioral targeting DNA sequencing Spell checking – suggesting correct spelling Plagiarism detection Similarity scores for predicting career paths of professional athletes. Cluster analysis – assignment of a set of observations into subsets (called clusters) so that observations in the same cluster are similar in some sense, usually based on Euclidean distance Chemical similarity Sampling-based motion planning == Methods == Various solutions to the NNS problem have been proposed. The quality and usefulness of the algorithms are determined by the time complexity of queries as well as the space complexity of any search data structures that must be maintained. The informal observation usually referred to as the curse of dimensionality states that there is no general-purpose exact solution for NNS in high-dimensional Euclidean space using polynomial preprocessing and polylogarithmic search time. === Exact methods === ==== Linear search ==== The simplest solution to the NNS problem is to compute the distance from the query point to every other point in the database, keeping track of the "best so far". This algorithm, sometimes referred to as the naive approach, has a running time of O(dN), where N is the cardinality of S and d is the dimensionality of S. There are no search data structures to maintain, so the linear search has no space complexity beyond the storage of the database. Naive search can, on average, outperform space partitioning approaches on higher dimensional spaces. The absolute distance is not required for distance comparison, only the relative distance. In geometric coordinate systems the distance calculation can be sped up considerably by omitting the square root calculation from the distance calculation between two coordinates. The distance comparison will still yield identical results. ==== Space partitioning ==== Since the 1970s, the branch and bound methodology has been applied to the problem. In the case of Euclidean space, this approach encompasses spatial index or spatial access methods. Several space-partitioning methods have been developed for solving the NNS problem. Perhaps the simplest is the k-d tree, which iteratively bisects the search space into two regions containing half of the points of the parent region. Queries are performed via traversal of the tree from the root to a leaf by evaluating the query point at each split. Depending on the distance specified in the query, neighboring branches that might contain hits may also need to be evaluated. For constant dimension query time, average complexity is O(log N) in the case of randomly distributed points, worst case complexity is O(kN^(1-1/k)) Alternatively the R-tree data structure was designed to support nearest neighbor search in dynamic context, as it has efficient algorithms for insertions and deletions such as the R tree. R-trees can yield nearest neighbors not only for Euclidean distance, but can also be used with other distances. In the case of general metric space, the branch-and-bound approach is known as the metric tree approach. Particular examples include vp-tree and BK-tree methods. Using a set of points taken from a 3-dimensional space and put into a BSP tree, and given a query point taken from the same space, a possible solution to the problem of finding the nearest point-cloud point to the query point is given in the following description of an algorithm. (Strictly speaking, no such point may exist, because it may not be unique. But in practice, usually we only care about finding any one of the subset of all point-cloud points that exist at the shortest distance to a given query point.) The idea is, for each branching of the tree, guess that the closest point in the cloud resides in the half-space containing the query point. This may not be the case, but it is a good heuristic. After having recursively gone through all the trouble of solving the problem for the guessed half-space, now compare the distance returned by this result with the shortest distance from the query point to the partitioning plane. This latter distance is that between the query point and the closest possible point that could exist in the half-space not searched. If this distance is greater than that returned in the earlier result, then clearly there is no need to search the other half-space. If there is such a need, then you must go through the trouble of solving the problem for the other half space, and then compare its result to the former result, and then return the proper result. The performance of this algorithm is nearer to logarithmic time than linear time when the query point is near the cloud, because as the distance between the query point and the closest point-cloud point nears zero, the algorithm needs only perform a look-up using the query point as a key to get the correct result. === Approximation methods === An approximate nearest neighbor search algorithm is allowed to return points whose distance from the query is at most c {\displaystyle c} times the distance from the query to its nearest points. The appeal of this approach is that, in many cases, an approximate nearest neighbor is almost as good as the exact one. In particular, if the distance measure accurately captures the notion of user quality, then small differences in the distance should not matter. ==== Greedy search in proximity neighborhood graphs ==== Proximity graph methods (such as navigable small world graphs and HNSW) are considered the current state-of-the-art for the approximate nearest neighbors search. The methods are based on greedy traversing in proximity neighborhood graphs G ( V , E ) {\displaystyle G(V,E)} in which every point x i ∈ S {\displaystyle x_{i}\in S} is uniquely associated with vertex v i ∈ V {\displaystyle v_{i}\in V} . The search for the nearest neighbors to a query q in the set S takes the form of searching for the vertex in the graph G ( V , E ) {\displaystyle G(V,E)} . The basic algorithm – greedy search – works as follows: search starts from an enter-point vertex v i ∈ V {\displaystyle v_{i}\in V} by computing the distances from the query q to each vertex of its neighborhood { v j : ( v i , v j ) ∈ E } {\displaystyle \{v_{j}:(v_{i},v_{j})\in E\}} , and then finds a vertex with the minimal distance value. If the distance value between the query and the selected vertex is smaller than the one between the query and the current element, then the algorithm moves to the selected vertex, and it becomes new enter-point. The algorithm stops when it reaches a local minimum: a vertex whose neighborhood does not contain a vertex that is closer to the query than the vertex itself. The idea of proximity neighborhood graphs was exploited in multiple publications, including the seminal paper by Arya and Mount, in the VoroNet syst

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  Information Harvesting

  Information Harvesting (IH) was an early data mining product from the 1990s. It was invented by Ralphe Wiggins and produced by the Ryan Corp, later Information Harvesting Inc., of Cambridge, Massachusetts. Wiggins had a background in genetic algorithms and fuzzy logic. IH sought to infer rules from sets of data. It did this first by classifying various input variables into one of a number of bins, thereby putting some structure on the continuous variables in the input. IH then proceeds to generate rules, trading off generalization against memorization, that will infer the value of the prediction variable, possibly creating many levels of rules in the process. It included strategies for checking if overfitting took place and, if so, correcting for it. Because of its strategies for correcting for overfitting by considering more data, and refining the rules based on that data, IH might also be considered to be a form of machine learning. The advantage of IH, as compared with other data mining products of its time and even later, was that it provided a mechanism for finding multiple rules that would classify the data and determining, according to set criteria, the best rules to use.

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  Comparison of vector graphics editors

  A number of vector graphics editors exist for various platforms. Potential users of these editors will make comparisons based on factors such as the availability for the user's platform, the software license, the feature set, the merits of the user interface (UI) and the focus of the program. Some programs are more suitable for artistic work while others are better for technical drawings. Another important factor is the application's support of various vector and bitmap image formats for import and export. The tables in this article compare general and technical information for a number of vector graphics editors. See the article on each editor for further information. This article is neither all-inclusive nor necessarily up-to-date. == Some editors in detail == Adobe Fireworks (formerly Macromedia Fireworks) is a vector editor with bitmap editing capabilities with its main purpose being the creation of graphics for Web and screen. Fireworks supports RGB color scheme and has no CMYK support. This means it is mostly used for screen design. The native Fireworks file format is editable PNG (FWPNG or PNG). Adobe Fireworks has a competitive price, but its features can seem limited in comparison with other products. It is easier to learn than other products and can produce complex vector artwork. The Fireworks editable PNG file format is not supported by other Adobe products. Fireworks can manage the PSD and AI file formats which enables it to be integrated with other Adobe apps. Fireworks can also open FWPNG/PNG, PSD, AI, EPS, JPG, GIF, BMP, TIFF file formats, and save/export to FWPNG/PNG, PSD, AI (v.8), FXG (v.2.0), JPG, GIF, PDF, SWF and some others. Some support for exporting to SVG is available via a free Export extension. On May 6, 2013, Adobe announced that Fireworks would be phased out. Adobe Flash (formerly a Macromedia product) has straightforward vector editing tools that make it easier for designers and illustrators to use. The most important of these tools are vector lines and fills with bitmap-like selectable areas, simple modification of curves via the "selection" or the control points/handles through "direct selection" tools. Flash uses Actionscript for OOP, and has full XML functionality through E4X support. Adobe FreeHand (formerly Macromedia Freehand and Aldus Freehand) is mainly used by professional graphic designers. The functionality of FreeHand includes the flexibility of the application in the wide design environment, catering to the output needs of both traditional image reproduction methods and to contemporary print and digital media with its page-layout capabilities and text attribute controls. Specific functions of FreeHand include a superior image-tracing operation for vector editing, page layout features within multiple-page documents, and embedding custom print-settings (such as variable halftone-screen specifications within a single graphic, etc.) to each document independent of auxiliary printer-drivers. User-operation is considered to be more suited for designers with an artistic background compared to designers with a technical background. When being marketed, FreeHand lacked the promotional backing, development and PR support in comparison to other similar products. FreeHand was transferred to the classic print group after Macromedia was purchased by Adobe in 2005. On May 16, 2007, Adobe announced that no further updates to Freehand would be developed but continues to sell FreeHand MX as a Macromedia product. FreeHand continues to run on Mac OS X Snow Leopard (using an Adobe fix) and on Windows 7. For macOS, Affinity Designer is able to open version 10 & MX Freehand files. Adobe Illustrator is a commonly used editor because of Adobe's market dominance, but is more expensive than other similar products. It is primarily developed consistently in line with other Adobe products and is best integrated with Adobe's Creative Suite packages. The ai file format is proprietary, but some vector editors can open and save in that format. Illustrator imports over two dozen formats, including PSD, PDF and SVG, and exports AI, PDF, SVG, SVGZ, GIF, JPG, PNG, WBMP, and SWF. However, the user must be aware of unchecking the "Preserve Illustrator Editing Capabilities" option if generating interoperable SVG files is desired. Affinity Designer by Serif Europe (the successor to their previous product, DrawPlus) is non-subscription-based software that is often described as an alternative to Adobe Illustrator. The application can open Portable Document Format (PDF), Adobe Photoshop, and Adobe Illustrator files, as well as export to those formats and to the Scalable Vector Graphics (SVG) and Encapsulated PostScript (EPS) formats. It also supports import from some Adobe Freehand files (specifically versions 10 & MX). Apache OpenOffice Draw is the vector graphics editor of the Apache OpenOffice open source office suite. It supports many import and export file formats and is available for multiple desktop operating systems. Boxy SVG is a chromium-based vector graphics editor for creating illustrations, as well as logos, icons, and other elements of graphic design. It is primarily focused on editing drawings in the SVG file format. The program is available as both a web app and a desktop application for Windows, macOS, ChromeOS, and Linux-based operating systems. Collabora Online Draw is the vector graphics editor of the Collabora Online open source office suite. It supports many import and export file formats and is accessible via any modern web browser, it also supports desktop editing features, Collabora Office is available for desktop and mobile operating systems, it is the enterprise ready version of LibreOffice. ConceptDraw PRO is a business diagramming tool and vector graphics editor available for both Windows and macOS. It supports multi-page documents, and includes an integrated presentation mode. ConceptDraw PRO supports imports and exports several formats, including Microsoft Visio and Microsoft PowerPoint. Corel Designer (originally Micrografx Designer) is one of the earliest vector-based graphics editors for the Microsoft Windows platform. The product is mainly used for the creation of engineering drawings and is shipped with extensive libraries for the needs of engineers. It is also flexible enough for most vector graphics design applications. CorelDRAW is an editor used in the graphic design, sign making and fashion design industries. CorelDRAW is capable of limited interoperation by reading file formats from Adobe Illustrator. CorelDRAW has over 50 import and export filters, on-screen and dialog box editing and the ability to create multi-page documents. It can also generate TrueType and Type 1 fonts, although refined typographic control is better suited to a more specific application. Some other features of CorelDRAW include the creation and execution of VBA macros, viewing of colour separations in print preview mode and integrated professional imposing options. Dia is a free and open-source diagramming and vector graphics editor available for Windows, Linux and other Unix-based computer operating systems. Dia has a modular design and several shape packages for flowcharting, network diagrams and circuit diagrams. Its design was inspired by Microsoft Visio, although it uses a Single Document Interface similar to other GNOME software (such as GIMP). DrawPlus, first built for the Windows platform in 1993, has matured into a full featured vector graphics editor for home and professional users. Also available as a feature-limited free 'starter edition': DrawPlus SE. DrawPlus developers, Serif Europe, have now ceased its development in order to focus on its successor, Affinity Designer. Edraw Max is a cross-platform diagram software and vector graphics editor available for Windows, Mac and Linux. It supports kinds of diagram types. It supports imports and exports SVG, PDF, HTML, Multiple page TIFF, Microsoft Visio and Microsoft PowerPoint. Embroidermodder is a free machine embroidery software tool that supports a variety of formats and allows the user to add custom modifications to their embroidery designs. Fatpaint is a free, light-weight, browser-based graphic design application with built-in vector drawing tools. It can be accessed through any browser with Flash 9 installed. Its integration with Zazzle makes it particularly suitable for people who want to create graphics for custom printed products such as T-shirts, mugs, iPhone cases, flyers and other promotional products. Figma is a collaborative web-based online vector graphics editor, used primarily for UX design and prototyping. GIMP, which works mainly with raster images, offers a limited set of features to create and record SVG files. It can also load and handle SVG files created with other software like Inkscape. Inkscape is a free and open-source vector editor with the primary native format being SVG. Inkscape is available for Linux, Windows, Mac OS X, and

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  Vladimir Batagelj

  Vladimir Batagelj (born June 14, 1948 in Idrija, Yugoslavia) is a Slovenian mathematician and an emeritus professor of mathematics at the University of Ljubljana. He is known for his work in discrete mathematics and combinatorial optimization, particularly analysis of social networks and other large networks (blockmodeling). == Education and career == Vladimir Batagelj completed his Ph.D. at the University of Ljubljana in 1986 under the direction of Tomaž Pisanski. He stayed at the University of Ljubljana as a professor until his retirement, where he was a professor of sociology and statistics, while also being a chair of the Department of Sociology of the Faculty of Social Sciences. As visiting professor, he was taught at the University of Pittsburgh (1990-91) and at the University of Konstanz (2002). He was also a member of editorial boards of two journals: Informatica and Journal of Social Structure. His work has been cited over 11000 times. His book Exploratory Social Network Analysis with Pajek on blockmodeling, coauthored with Wouter de Nooy and Andrej Mrvar, is Batagelj's most cited work and has over 3300 citations. The book was translated into Chinese and Japanese. The revised and expanded third edition has been published by Cambridge University Press. In 1975, 11 years before completing his PhD, Batagelj published a solo paper in Communications of the ACM. Batagelj authored more than 20 textbooks in Slovenian, covering topics like TeX, combinatorics and discrete mathematics. He has also written extensively in the Slovenian popular science journal Presek. Batagelj has advised 9 Ph.D. students. == Pajek == Batagelj is particularly known for his work on Pajek, a freely available software for analysis and visualization of large networks. He began work on Pajek in 1996 with Andrej Mrvar, who was then his PhD student. == Awards and honors == First prizes for contributions (with Andrej Mrvar) to Graph Drawing Contests in years: 1995, 1996, 1997, 1998, 1999, 2000 and 2005 / Graph Drawing Hall of Fame. In 2007 the book Generalized blockmodeling was awarded the Harrison White Outstanding Book Award by the Mathematical Sociology Section of American Sociological Association In 2007 he was awarded (together with Anuška Ferligoj) the Simmel Award by INSNA. In 2013, Vladimir Batagelj and Andrej Mrvar received the INSNA's William D. Richards Software award for their work on Pajek. == Selected bibliography == Vladimir Batagelj, Social Network Analysis, Large-Scale [1]. in R.A. Meyers, ed., Encyclopedia of Complexity and Systems Science, Springer 2009: 8245–8265. Vladimir Batagelj, Complex Networks, Visualization of [2]. in R.A. Meyers, ed., Encyclopedia of Complexity and Systems Science, Springer 2009: 1253–1268. Wouter de Nooy, Andrej Mrvar, Vladimir Batagelj, Mark Granovetter (Series Editor), Exploratory Social Network Analysis with Pajek (Structural Analysis in the Social Sciences), Cambridge University Press 2005 (ISBN 0-521-60262-9). ESNA in Japanese, TDU, 2010. Patrick Doreian, Vladimir Batagelj, Anuška Ferligoj, Mark Granovetter (Series Editor), Generalized Blockmodeling (Structural Analysis in the Social Sciences), Cambridge University Press 2004 (ISBN 0-521-84085-6)

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  Quadratic unconstrained binary optimization

  Quadratic unconstrained binary optimization (QUBO), also known as unconstrained binary quadratic programming (UBQP), is a combinatorial optimization problem with a wide range of applications from finance and economics to machine learning. QUBO is an NP hard problem, and for many classical problems from theoretical computer science, like maximum cut, graph coloring and the partition problem, embeddings into QUBO have been formulated. Embeddings for machine learning models include support-vector machines, clustering and probabilistic graphical models. Moreover, due to its close connection to Ising models, QUBO constitutes a central problem class for adiabatic quantum computation, where it is solved through a physical process called quantum annealing. == Definition == Let B = { 0 , 1 } {\displaystyle \mathbb {B} =\lbrace 0,1\rbrace } the set of binary digits (or bits), then B n {\displaystyle \mathbb {B} ^{n}} is the set of binary vectors of fixed length n ∈ N {\displaystyle n\in \mathbb {N} } . Given a symmetric or upper triangular matrix Q ∈ R n × n {\displaystyle {\boldsymbol {Q}}\in \mathbb {R} ^{n\times n}} , whose entries Q i j {\displaystyle Q_{ij}} define a weight for each pair of indices i , j ∈ { 1 , … , n } {\displaystyle i,j\in \lbrace 1,\dots ,n\rbrace } , we can define the function f Q : B n → R {\displaystyle f_{\boldsymbol {Q}}:\mathbb {B} ^{n}\rightarrow \mathbb {R} } that assigns a value to each binary vector x {\displaystyle {\boldsymbol {x}}} through f Q ( x ) = x ⊺ Q x = ∑ i = 1 n ∑ j = 1 n Q i j x i x j . {\displaystyle f_{\boldsymbol {Q}}({\boldsymbol {x}})={\boldsymbol {x}}^{\intercal }{\boldsymbol {Qx}}=\sum _{i=1}^{n}\sum _{j=1}^{n}Q_{ij}x_{i}x_{j}.} Alternatively, the linear and quadratic parts can be separated as f Q ′ , q ( x ) = x ⊺ Q ′ x + q ⊺ x , {\displaystyle f_{{\boldsymbol {Q}}',{\boldsymbol {q}}}({\boldsymbol {x}})={\boldsymbol {x}}^{\intercal }{\boldsymbol {Q}}'{\boldsymbol {x}}+{\boldsymbol {q}}^{\intercal }{\boldsymbol {x}},} where Q ′ ∈ R n × n {\displaystyle {\boldsymbol {Q}}'\in \mathbb {R} ^{n\times n}} and q ∈ R n {\displaystyle {\boldsymbol {q}}\in \mathbb {R} ^{n}} . This is equivalent to the previous definition through Q = Q ′ + diag ⁡ [ q ] {\displaystyle {\boldsymbol {Q}}={\boldsymbol {Q}}'+\operatorname {diag} [{\boldsymbol {q}}]} using the diag operator, exploiting that x = x ⋅ x {\displaystyle x=x\cdot x} for all binary values x {\displaystyle x} . Intuitively, the weight Q i j {\displaystyle Q_{ij}} is added if both x i = 1 {\displaystyle x_{i}=1} and x j = 1 {\displaystyle x_{j}=1} . The QUBO problem consists of finding a binary vector x ∗ {\displaystyle {\boldsymbol {x}}^{}} that minimizes f Q {\displaystyle f_{\boldsymbol {Q}}} , i.e., ∀ x ∈ B n : f Q ( x ∗ ) ≤ f Q ( x ) {\displaystyle \forall {\boldsymbol {x}}\in \mathbb {B} ^{n}:~f_{\boldsymbol {Q}}({\boldsymbol {x}}^{})\leq f_{\boldsymbol {Q}}({\boldsymbol {x}})} . In general, x ∗ {\displaystyle {\boldsymbol {x}}^{}} is not unique, meaning there may be a set of minimizing vectors with equal value w.r.t. f Q {\displaystyle f_{\boldsymbol {Q}}} . The complexity of QUBO arises from the number of candidate binary vectors to be evaluated, as | B n | = 2 n {\displaystyle \left|\mathbb {B} ^{n}\right|=2^{n}} grows exponentially in n {\displaystyle n} . Sometimes, QUBO is defined as the problem of maximizing f Q {\displaystyle f_{\boldsymbol {Q}}} , which is equivalent to minimizing f − Q = − f Q {\displaystyle f_{-{\boldsymbol {Q}}}=-f_{\boldsymbol {Q}}} . == Properties == QUBO is scale invariant for positive factors α > 0 {\displaystyle \alpha >0} , which leave the optimum x ∗ {\displaystyle {\boldsymbol {x}}^{}} unchanged: f α Q ( x ) = x ⊺ ( α Q ) x = α ( x ⊺ Q x ) = α f Q ( x ) {\displaystyle f_{\alpha {\boldsymbol {Q}}}({\boldsymbol {x}})={\boldsymbol {x}}^{\intercal }(\alpha {\boldsymbol {Q}}){\boldsymbol {x}}=\alpha ({\boldsymbol {x}}^{\intercal }{\boldsymbol {Qx}})=\alpha f_{\boldsymbol {Q}}({\boldsymbol {x}})} . In its general form, QUBO is NP-hard and cannot be solved efficiently by any known polynomial-time algorithm. However, there are polynomially-solvable special cases, where Q {\displaystyle {\boldsymbol {Q}}} has certain properties, for example: If all coefficients are positive, the optimum is trivially x ∗ = ( 0 , … , 0 ) ⊺ {\displaystyle {\boldsymbol {x}}^{}=(0,\dots ,0)^{\intercal }} . Similarly, if all coefficients are negative, the optimum is x ∗ = ( 1 , … , 1 ) ⊺ {\displaystyle {\boldsymbol {x}}^{}=(1,\dots ,1)^{\intercal }} . If Q {\displaystyle {\boldsymbol {Q}}} is diagonal, the bits can be optimized independently, and the problem is solvable in O ( n ) {\displaystyle {\mathcal {O}}(n)} . The optimal variable assignments are simply x i ∗ = 1 {\displaystyle x_{i}^{}=1} if Q i i < 0 {\displaystyle Q_{ii}<0} , and x i ∗ = 0 {\displaystyle x_{i}^{}=0} otherwise. If all off-diagonal elements of Q {\displaystyle {\boldsymbol {Q}}} are non-positive, the corresponding QUBO problem is solvable in polynomial time. QUBO can be solved using integer linear programming solvers like CPLEX or Gurobi Optimizer. This is possible since QUBO can be reformulated as a linear constrained binary optimization problem. To achieve this, substitute the product x i x j {\displaystyle x_{i}x_{j}} by an additional binary variable z i j ∈ B {\displaystyle z_{ij}\in \mathbb {B} } and add the constraints x i ≥ z i j {\displaystyle x_{i}\geq z_{ij}} , x j ≥ z i j {\displaystyle x_{j}\geq z_{ij}} and x i + x j − 1 ≤ z i j {\displaystyle x_{i}+x_{j}-1\leq z_{ij}} . Note that z i j {\displaystyle z_{ij}} can also be relaxed to continuous variables within the bounds zero and one. == Applications == QUBO is a structurally simple, yet computationally hard optimization problem. It can be used to encode a wide range of optimization problems from various scientific areas. === Maximum Cut === Given a graph G = ( V , E ) {\displaystyle G=(V,E)} with vertex set V = { 1 , … , n } {\displaystyle V=\lbrace 1,\dots ,n\rbrace } and edges E ⊆ V × V {\displaystyle E\subseteq V\times V} , the maximum cut (max-cut) problem consists of finding two subsets S , T ⊆ V {\displaystyle S,T\subseteq V} with T = V ∖ S {\displaystyle T=V\setminus S} , such that the number of edges between S {\displaystyle S} and T {\displaystyle T} is maximized. The more general weighted max-cut problem assumes edge weights w i j ≥ 0 ∀ i , j ∈ V {\displaystyle w_{ij}\geq 0~\forall i,j\in V} , with ( i , j ) ∉ E ⇒ w i j = 0 {\displaystyle (i,j)\notin E\Rightarrow w_{ij}=0} , and asks for a partition S , T ⊆ V {\displaystyle S,T\subseteq V} that maximizes the sum of edge weights between S {\displaystyle S} and T {\displaystyle T} , i.e., max S ⊆ V ∑ i ∈ S , j ∉ S w i j . {\displaystyle \max _{S\subseteq V}\sum _{i\in S,j\notin S}w_{ij}.} By setting w i j = 1 {\displaystyle w_{ij}=1} for all ( i , j ) ∈ E {\displaystyle (i,j)\in E} this becomes equivalent to the original max-cut problem above, which is why we focus on this more general form in the following. For every vertex in i ∈ V {\displaystyle i\in V} we introduce a binary variable x i {\displaystyle x_{i}} with the interpretation x i = 0 {\displaystyle x_{i}=0} if i ∈ S {\displaystyle i\in S} and x i = 1 {\displaystyle x_{i}=1} if i ∈ T {\displaystyle i\in T} . As T = V ∖ S {\displaystyle T=V\setminus S} , every i {\displaystyle i} is in exactly one set, meaning there is a 1:1 correspondence between binary vectors x ∈ B n {\displaystyle {\boldsymbol {x}}\in \mathbb {B} ^{n}} and partitions of V {\displaystyle V} into two subsets. We observe that, for any i , j ∈ V {\displaystyle i,j\in V} , the expression x i ( 1 − x j ) + ( 1 − x i ) x j {\displaystyle x_{i}(1-x_{j})+(1-x_{i})x_{j}} evaluates to 1 if and only if i {\displaystyle i} and j {\displaystyle j} are in different subsets, equivalent to logical XOR. Let W ∈ R + n × n {\displaystyle {\boldsymbol {W}}\in \mathbb {R} _{+}^{n\times n}} with W i j = w i j ∀ i , j ∈ V {\displaystyle W_{ij}=w_{ij}~\forall i,j\in V} . By extending above expression to matrix-vector form we find that x ⊺ W ( 1 − x ) + ( 1 − x ) ⊺ W x = − 2 x ⊺ W x + ( W 1 + W ⊺ 1 ) ⊺ x {\displaystyle {\boldsymbol {x}}^{\intercal }{\boldsymbol {W}}({\boldsymbol {1}}-{\boldsymbol {x}})+({\boldsymbol {1}}-{\boldsymbol {x}})^{\intercal }{\boldsymbol {Wx}}=-2{\boldsymbol {x}}^{\intercal }{\boldsymbol {Wx}}+({\boldsymbol {W1}}+{\boldsymbol {W}}^{\intercal }{\boldsymbol {1}})^{\intercal }{\boldsymbol {x}}} is the sum of weights of all edges between S {\displaystyle S} and T {\displaystyle T} , where 1 = ( 1 , 1 , … , 1 ) ⊺ ∈ R n {\displaystyle {\boldsymbol {1}}=(1,1,\dots ,1)^{\intercal }\in \mathbb {R} ^{n}} . As this is a quadratic function over x {\displaystyle {\boldsymbol {x}}} , it is a QUBO problem whose parameter matrix we can read from above expression as Q = 2 W − diag ⁡ [ W 1 + W ⊺ 1 ] , {\displaystyle {\boldsymbol {Q}}=2{\boldsymbol {W}}-\operatorname {diag} [{\boldsymbol {W1}}+{\boldsymbol {W}}^{\intercal }{\bol

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