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Least-squares adjustmentis a model for the solution of anoverdetermined systemof equations based on the principle ofleast squaresofobservation residuals. It is used extensively in the disciplines ofsurveying,geodesy, andphotogrammetry—the field ofgeomatics, collectively.
There are three forms of least squares adjustme... | https://en.wikipedia.org/wiki/Least-squares_adjustment |
Instatistics,best linear unbiased prediction(BLUP) is used in linearmixed modelsfor the estimation ofrandom effects. BLUP was derived byCharles Roy Hendersonin 1950 but the term "best linear unbiased predictor" (or "prediction") seems not to have been used until 1962.[1]"Best linear unbiased predictions" (BLUPs) of ran... | https://en.wikipedia.org/wiki/Best_linear_unbiased_prediction |
Inmathematics, anormis afunctionfrom a real or complexvector spaceto the non-negative real numbers that behaves in certain ways like the distance from theorigin: itcommuteswith scaling, obeys a form of thetriangle inequality, and zero is only at the origin. In particular, theEuclidean distancein aEuclidean spaceis defi... | https://en.wikipedia.org/wiki/L2_norm |
Least-squares spectral analysis(LSSA) is a method of estimating afrequency spectrumbased on aleast-squaresfit ofsinusoidsto data samples, similar toFourier analysis.[1][2]Fourier analysis, the most used spectral method in science, generally boosts long-periodic noise in the long and gapped records; LSSA mitigates such ... | https://en.wikipedia.org/wiki/Least-squares_spectral_analysis |
Inlinear algebraandfunctional analysis, aprojectionis alinear transformationP{\displaystyle P}from avector spaceto itself (anendomorphism) such thatP∘P=P{\displaystyle P\circ P=P}. That is, wheneverP{\displaystyle P}is applied twice to any vector, it gives the same result as if it were applied once (i.e.P{\displaystyle... | https://en.wikipedia.org/wiki/Orthogonal_projection |
Proximal gradient(forward backward splitting)methods for learningis an area of research inoptimizationandstatistical learning theorywhich studies algorithms for a general class ofconvexregularizationproblems where the regularization penalty may not bedifferentiable. One such example isℓ1{\displaystyle \ell _{1}}regular... | https://en.wikipedia.org/wiki/Proximal_gradient_methods_for_learning |
Inmathematical optimizationanddecision theory, aloss functionorcost function(sometimes also called an error function)[1]is a function that maps aneventor values of one or more variables onto areal numberintuitively representing some "cost" associated with the event. Anoptimization problemseeks to minimize a loss functi... | https://en.wikipedia.org/wiki/Quadratic_loss_function |
Inmathematics, theroot mean square(abbrev.RMS,RMSorrms) of asetof values is thesquare rootof the set'smean square.[1]Given a setxi{\displaystyle x_{i}}, its RMS is denoted as eitherxRMS{\displaystyle x_{\mathrm {RMS} }}orRMSx{\displaystyle \mathrm {RMS} _{x}}. The RMS is also known as thequadratic mean(denotedM2{\displ... | https://en.wikipedia.org/wiki/Root_mean_square |
Squared deviations from the mean(SDM) result fromsquaringdeviations. Inprobability theoryandstatistics, the definition ofvarianceis either theexpected valueof the SDM (when considering a theoreticaldistribution) or its average value (for actual experimental data). Computations foranalysis of varianceinvolve the partiti... | https://en.wikipedia.org/wiki/Squared_deviations_from_the_mean |
Aneural circuitis a population ofneuronsinterconnected bysynapsesto carry out a specific function when activated.[1]Multiple neural circuits interconnect with one another to formlarge scale brain networks.[2]
Neural circuits have inspired the design ofartificial neural networks, though there are significant difference... | https://en.wikipedia.org/wiki/Neural_circuit |
Neural backpropagationis the phenomenon in which, after theaction potentialof aneuroncreates a voltage spike down theaxon(normal propagation), another impulse is generated from thesomaand propagates towards theapicalportions of the dendritic arbor ordendrites(from which much of the original input current originated). ... | https://en.wikipedia.org/wiki/Neural_backpropagation |
Backpropagation through time(BPTT) is agradient-based techniquefor training certain types ofrecurrent neural networks, such asElman networks. The algorithm was independently derived by numerous researchers.[1][2][3]
The training data for a recurrent neural network is an ordered sequence ofk{\displaystyle k}input-outpu... | https://en.wikipedia.org/wiki/Backpropagation_through_time |
Backpropagation through structure(BPTS) is agradient-based techniquefor trainingrecursive neural networks, proposed in a 1996 paper written by Christoph Goller and Andreas Küchler.[1]
Thisartificial intelligence-related article is astub. You can help Wikipedia byexpanding it. | https://en.wikipedia.org/wiki/Backpropagation_through_structure |
In neuroscience and machine learning,three-factor learningis the combinaison ofHebbian plasticitywith a third modulatory factor to stabilise and enhance synaptic learning.[1]This third factor can represent various signals such as reward, punishment, error, surprise, or novelty, often implemented throughneuromodulators.... | https://en.wikipedia.org/wiki/Three-factor_learning |
This is alist ofnumerical analysistopics.
Error analysis (mathematics)
Numerical linear algebra— study of numerical algorithms for linear algebra problems
Eigenvalue algorithm— a numerical algorithm for locating the eigenvalues of a matrix
Interpolation— construct a function going through some given data points
Po... | https://en.wikipedia.org/wiki/List_of_numerical_analysis_topics#Eigenvalue_algorithms |
Cluster analysisorclusteringis the data analyzing technique in which task of grouping a set of objects in such a way that objects in the same group (called acluster) are moresimilar(in some specific sense defined by the analyst) to each other than to those in other groups (clusters). It is a main task ofexploratory dat... | https://en.wikipedia.org/wiki/Clustering_(statistics) |
Ingeometry, thehyperplane separation theoremis a theorem aboutdisjointconvex setsinn-dimensionalEuclidean space. There are several rather similar versions. In one version of the theorem, if both these sets areclosedand at least one of them iscompact, then there is ahyperplanein between them and even two parallel hyperp... | https://en.wikipedia.org/wiki/Hyperplane_separation_theorem |
Kirchberger's theoremis a theorem indiscrete geometry, onlinear separability. The two-dimensional version of the theorem states that, if a finite set of red and blue points in theEuclidean planehas the property that, for every four points, there exists a line separating the red and blue points within those four, then t... | https://en.wikipedia.org/wiki/Kirchberger%27s_theorem |
Inmachine learning, theperceptronis an algorithm forsupervised learningofbinary classifiers. A binary classifier is a function that can decide whether or not an input, represented by a vector of numbers, belongs to some specific class.[1]It is a type oflinear classifier, i.e. a classification algorithm that makes its ... | https://en.wikipedia.org/wiki/Perceptron |
Theaverage absolute deviation(AAD) of a data set is theaverageof theabsolutedeviationsfrom acentral point. It is asummary statisticofstatistical dispersionor variability. In the general form, the central point can be amean,median,mode, or the result of any other measure of central tendency or any reference value relate... | https://en.wikipedia.org/wiki/Average_absolute_deviation |
Instatistics, themean signed difference(MSD),[1]also known asmean signed deviation,mean signed error, ormean bias error[2]is a samplestatisticthat summarizes how well a set of estimatesθ^i{\displaystyle {\hat {\theta }}_{i}}match the quantitiesθi{\displaystyle \theta _{i}}that they are supposed to estimate. It is one o... | https://en.wikipedia.org/wiki/Mean_signed_deviation |
Instatisticsandoptimization,errorsandresidualsare two closely related and easily confused measures of thedeviationof anobserved valueof anelementof astatistical samplefrom its "true value" (not necessarily observable). Theerrorof anobservationis the deviation of the observed value from the true value of a quantity of i... | https://en.wikipedia.org/wiki/Errors_and_residuals |
Convex analysisis the branch ofmathematicsdevoted to the study of properties ofconvex functionsandconvex sets, often with applications inconvex minimization, a subdomain ofoptimization theory.
A subsetC⊆X{\displaystyle C\subseteq X}of somevector spaceX{\displaystyle X}isconvexif it satisfies any of the following equiv... | https://en.wikipedia.org/wiki/Convex_analysis |
Inmathematicsandmathematical optimization, theconvex conjugateof a function is a generalization of theLegendre transformationwhich applies to non-convex functions. It is also known asLegendre–Fenchel transformation,Fenchel transformation, orFenchel conjugate(afterAdrien-Marie LegendreandWerner Fenchel). The convex con... | https://en.wikipedia.org/wiki/Convex_conjugate |
Ingeometry, aconvex curveis aplane curvethat has asupporting linethrough each of its points. There are many other equivalent definitions of these curves, going back toArchimedes. Examples of convex curves include theconvex polygons, theboundariesofconvex sets, and thegraphsofconvex functions. Important subclasses of co... | https://en.wikipedia.org/wiki/Convex_curve |
Inmathematics— specifically, inRiemannian geometry—geodesic convexityis a natural generalization ofconvexity for setsandfunctionstoRiemannian manifolds. It is common to drop the prefix "geodesic" and refer simply to "convexity" of a set or function.
Let (M,g) be a Riemannian manifold. | https://en.wikipedia.org/wiki/Geodesic_convexity |
Infunctional analysis, theHahn–Banach theoremis a central result that allows the extension ofbounded linear functionalsdefined on avector subspaceof somevector spaceto the whole space. The theorem also shows that there are sufficientcontinuouslinear functionals defined on everynormed vector spacein order to study the... | https://en.wikipedia.org/wiki/Hahn%E2%80%93Banach_theorem |
Inmathematics, theHermite–Hadamard inequality, named afterCharles HermiteandJacques Hadamardand sometimes also calledHadamard's inequality, states that if a functionf: [a,b] →Risconvex, then the following chain of inequalities hold:
The inequality has been generalized to higher dimensions: ifΩ⊂Rn{\displaystyle \Omega ... | https://en.wikipedia.org/wiki/Hermite%E2%80%93Hadamard_inequality |
Invector calculus, aninvex functionis adifferentiable functionf{\displaystyle f}fromRn{\displaystyle \mathbb {R} ^{n}}toR{\displaystyle \mathbb {R} }for which there exists a vector valued functionη{\displaystyle \eta }such that
for allxandu.
Invex functions were introduced by Hanson as a generalization ofconvex funct... | https://en.wikipedia.org/wiki/Invex_function |
Inmathematics,Jensen's inequality, named after the Danish mathematicianJohan Jensen, relates the value of aconvex functionof anintegralto the integral of the convex function. It wasprovedby Jensen in 1906,[1]building on an earlier proof of the same inequality for doubly-differentiable functions byOtto Hölderin 1889.[2]... | https://en.wikipedia.org/wiki/Jensen%27s_inequality |
K-convex functions, first introduced byScarf,[1]are a special weakening of the concept ofconvex functionwhich is crucial in the proof of theoptimalityof the(s,S){\displaystyle (s,S)}policy ininventory control theory. The policy is characterized by two numberssandS,S≥s{\displaystyle S\geq s}, such that when the inventor... | https://en.wikipedia.org/wiki/K-convex_function |
Inmathematics,Kachurovskii's theoremis a theorem relating theconvexityof a function on aBanach spaceto themonotonicityof itsFréchet derivative.
LetKbe aconvex subsetof a Banach spaceVand letf:K→R∪ {+∞} be anextended real-valued functionthat is Fréchet differentiable with derivative df(x) :V→Rat each pointxinK. (In fac... | https://en.wikipedia.org/wiki/Kachurovskii%27s_theorem |
Inmathematics, amonotonic function(ormonotone function) is afunctionbetweenordered setsthat preserves or reverses the givenorder.[1][2][3]This concept first arose incalculus, and was later generalized to the more abstract setting oforder theory.
Incalculus, a functionf{\displaystyle f}defined on asubsetof thereal numb... | https://en.wikipedia.org/wiki/Monotone_operator |
Inmathematics,Karamata's inequality,[1]named afterJovan Karamata,[2]also known as themajorization inequality, is a theorem inelementary algebrafor convex and concave real-valued functions, defined on an interval of the real line. It generalizes the discrete form ofJensen's inequality, and generalizes in turn to the con... | https://en.wikipedia.org/wiki/Karamata%27s_inequality |
Inmathematics, afunctionfislogarithmically convexorsuperconvex[1]iflog∘f{\displaystyle {\log }\circ f}, thecompositionof thelogarithmwithf, is itself aconvex function.
LetXbe aconvex subsetof arealvector space, and letf:X→Rbe a function takingnon-negativevalues. Thenfis:
Here we interpretlog0{\displaystyle \log 0}a... | https://en.wikipedia.org/wiki/Logarithmically_convex_function |
Inconvex analysisand thecalculus of variations, both branches ofmathematics, apseudoconvex functionis afunctionthat behaves like aconvex functionwith respect to finding itslocal minima, but need not actually be convex. Informally, a differentiable function is pseudoconvex if it is increasing in any direction where it ... | https://en.wikipedia.org/wiki/Pseudoconvex_function |
Inmathematics, aquasiconvex functionis areal-valuedfunctiondefined on anintervalor on aconvex subsetof a realvector spacesuch that theinverse imageof any set of the form(−∞,a){\displaystyle (-\infty ,a)}is aconvex set. For a function of a single variable, along any stretch of the curve the highest point is one of the e... | https://en.wikipedia.org/wiki/Quasiconvex_function |
Inmathematics,subderivatives(orsubgradient) generalizes thederivativeto convex functions which are not necessarilydifferentiable. The set of subderivatives at a point is called thesubdifferentialat that point.[1]Subderivatives arise inconvex analysis, the study ofconvex functions, often in connection toconvex optimizat... | https://en.wikipedia.org/wiki/Subderivative |
Acobweb plot, known also asLémeray DiagramorVerhulst diagramis a visual tool used indynamical systems, a field ofmathematicsto investigate the qualitative behaviour of one-dimensionaliterated functions, such as thelogistic map. The technique was introduced in the 1890s by E.-M. Lémeray.[1]Using a cobweb plot, it is po... | https://en.wikipedia.org/wiki/Cobweb_plot |
Combinatory logicis a notation to eliminate the need forquantifiedvariables inmathematical logic. It was introduced byMoses Schönfinkel[1]andHaskell Curry,[2]and has more recently been used incomputer scienceas a theoreticalmodel of computationand also as a basis for the design offunctional programming languages. It is... | https://en.wikipedia.org/wiki/Combinatory_logic |
Ring homomorphisms
Algebraic structures
Related structures
Algebraic number theory
Noncommutative algebraic geometry
Free algebra
Clifford algebra
Inmathematics, acomposition ring, introduced in (Adler 1962), is acommutative ring(R, 0, +, −, ·), possibly without anidentity 1, together with an operation
such th... | https://en.wikipedia.org/wiki/Composition_ring |
Inmathematics, aflowformalizes the idea of the motion of particles in a fluid. Flows are ubiquitous in science, includingengineeringandphysics. The notion of flow is basic to the study ofordinary differential equations. Informally, a flow may be viewed as a continuous motion of points over time. More formally, a flow ... | https://en.wikipedia.org/wiki/Flow_(mathematics) |
Incomputer science,function compositionis an act or mechanism to combine simplefunctionsto build more complicated ones. Like the usualcomposition of functionsinmathematics, the result of each function is passed as the argument of the next, and the result of the last one is the result of the whole.
Programmers frequent... | https://en.wikipedia.org/wiki/Function_composition_(computer_science) |
Arandom variable(also calledrandom quantity,aleatory variable, orstochastic variable) is amathematicalformalization of a quantity or object which depends onrandomevents.[1]The term 'random variable' in its mathematical definition refers to neither randomness nor variability[2]but instead is a mathematicalfunctionin whi... | https://en.wikipedia.org/wiki/Random_variable#Functions_of_random_variables |
Inengineering,functional decompositionis the process of resolving afunctionalrelationship into its constituent parts in such a way that the original function can be reconstructed (i.e., recomposed) from those parts.
This process of decomposition may be undertaken to gain insight into the identity of the constituent co... | https://en.wikipedia.org/wiki/Functional_decomposition |
Inmathematics, afunctional equation[1][2][irrelevant citation]is, in the broadest meaning, anequationin which one or several functions appear asunknowns. So,differential equationsandintegral equationsare functional equations. However, a more restricted meaning is often used, where afunctional equationis an equation tha... | https://en.wikipedia.org/wiki/Functional_equation |
Explainable AI(XAI), often overlapping withinterpretable AI, orexplainable machine learning(XML), is a field of research withinartificial intelligence(AI) that explores methods that provide humans with the ability ofintellectual oversightover AI algorithms.[1][2]The main focus is on the reasoning behind the decisions o... | https://en.wikipedia.org/wiki/Explainable_artificial_intelligence |
Inprobabilityandstatistics, amixture distributionis theprobability distributionof arandom variablethat is derived from a collection of other random variables as follows: first, a random variable is selected by chance from the collection according to given probabilities of selection, and then the value of the selected r... | https://en.wikipedia.org/wiki/Mixture_density |
Inprobability theoryandstatistics, amixtureis a probabilistic combination of two or more probability distributions.[1]The concept arises mostly in two contexts:
Thisprobability-related article is astub. You can help Wikipedia byexpanding it. | https://en.wikipedia.org/wiki/Mixture_(probability) |
Subspace Gaussian mixture model(SGMM) is an acoustic modeling approach in which all phonetic states share a common Gaussianmixture modelstructure, and the means and mixture weights vary in a subspace of the total parameter space.[1]
Thiscomputer sciencearticle is astub. You can help Wikipedia byexpanding it. | https://en.wikipedia.org/wiki/Subspace_Gaussian_mixture_model |
Inmathematics, theGiry monadis a construction that assigns to ameasurable spacea space ofprobability measuresover it, equipped with a canonicalsigma-algebra.[1][2][3][4][5]It is one of the main examples of aprobability monad.
It is implicitly used inprobability theorywhenever one considersprobability measureswhich dep... | https://en.wikipedia.org/wiki/Giry_monad |
Alpha–beta pruningis asearch algorithmthat seeks to decrease the number of nodes that are evaluated by theminimax algorithmin itssearch tree. It is an adversarial search algorithm used commonly for machine playing of two-playercombinatorial games(Tic-tac-toe,Chess,Connect 4, etc.). It stops evaluating a move when at le... | https://en.wikipedia.org/wiki/Alpha%E2%80%93beta_pruning |
Theexpectiminimaxalgorithm is a variation of theminimaxalgorithm, for use inartificial intelligencesystems that play two-playerzero-sum games, such asbackgammon, in which the outcome depends on a combination of the player's skill andchance elementssuch as dice rolls. In addition to "min" and "max" nodes of the traditio... | https://en.wikipedia.org/wiki/Expectiminimax |
Ingame theory, ann-player gameis a game which is well defined for any number of players. This is usually used in contrast to standard2-player gamesthat are only specified for two players. In definingn-player games,game theoristsusually provide a definition that allow for any (finite) number of players.[1]The limiting c... | https://en.wikipedia.org/wiki/Maxn_algorithm |
Computer chessincludes both hardware (dedicated computers) andsoftwarecapable of playingchess. Computer chess provides opportunities for players to practice even in the absence of human opponents, and also provides opportunities for analysis, entertainment and training. Computer chess applications that play at the leve... | https://en.wikipedia.org/wiki/Computer_chess |
Thehorizon effect, also known as thehorizon problem, is a problem inartificial intelligencewhereby, in many games, the number of possible states or positions is immense and computers can only feasibly search a small portion of them, typically a fewpliesdown thegame tree. Thus, for a computer searching only a fixed numb... | https://en.wikipedia.org/wiki/Horizon_effect |
Thelesser of two evils principle, also referred to as thelesser evil principleandlesser-evilism, is the principle that when faced with selecting from two immoral options, the least immoral one should be chosen. The principle is most often invoked in reference to binary political choices under systems that make it impos... | https://en.wikipedia.org/wiki/Lesser_of_two_evils_principle |
Condorcet methods
Positional voting
Cardinal voting
Quota-remainder methods
Approval-based committees
Fractional social choice
Semi-proportional representation
By ballot type
Pathological response
Strategic voting
Paradoxes ofmajority rule
Positive results
Invoting systems, theMinimax Condorcet methodis asi... | https://en.wikipedia.org/wiki/Minimax_Condorcet |
Indecision theory,regret aversion(oranticipated regret) describes how the human emotional response ofregretcan influence decision-making underuncertainty. When individuals make choices without complete information, they often experience regret if they later discover that a different choice would have produced a better ... | https://en.wikipedia.org/wiki/Regret_(decision_theory)#Minimax_regret |
Incomputer science,Monte Carlo tree search(MCTS) is aheuristicsearch algorithmfor some kinds ofdecision processes, most notably those employed insoftwarethat playsboard games. In that context MCTS is used to solve thegame tree.
MCTS was combined with neural networks in 2016[1]and has been used in multiple board games ... | https://en.wikipedia.org/wiki/Monte_Carlo_tree_search |
Negamaxsearch is a variant form ofminimaxsearch that relies on thezero-sumproperty of atwo-player game.
This algorithm relies on the fact thatmin(a,b)=−max(−b,−a){\displaystyle \min(a,b)=-\max(-b,-a)}to simplify the implementation of theminimaxalgorithm. More precisely, the value of a position to player A in such a ... | https://en.wikipedia.org/wiki/Negamax |
Principal variation search(sometimes equated with the practically identicalNegaScout) is anegamaxalgorithm that can be faster thanalpha–beta pruning. Like alpha–beta pruning, NegaScout is a directional search algorithm for computing theminimaxvalue of a node in atree. It dominates alpha–beta pruning in the sense that i... | https://en.wikipedia.org/wiki/Negascout |
In the mathematical area ofgame theoryand ofconvex optimization, aminimax theoremis a theorem that claims that
under certain conditions on the setsX{\displaystyle X}andY{\displaystyle Y}and on the functionf{\displaystyle f}.[1]It is always true that the left-hand side is at most the right-hand side (max–min inequality... | https://en.wikipedia.org/wiki/Sion%27s_minimax_theorem |
Tit for tatis an English saying meaning "equivalentretaliation". It is analterationoftipfortap"blow for blow",[1]first recorded in 1558.[2]
It is also a highly effectivestrategyingame theory. Anagentusing this strategy will first cooperate, then subsequently replicate an opponent's previous action. If the opponent pre... | https://en.wikipedia.org/wiki/Tit_for_Tat |
Indecision theoryandgame theory,Wald'smaximinmodelis a non-probabilistic decision-making model according to which decisions are ranked on the basis of their worst-case outcomes – the optimal decision is one with the least bad outcome. It is one of the most important models inrobust decision makingin general androbust o... | https://en.wikipedia.org/wiki/Wald%27s_maximin_model |
In the statisticaldecision theory, where one is faced with making decisions in the presence of statistical knowledge,Γ-minimax inferenceis aminimaxapproach used to deal with partial prior information. It works with applications of Γ-minimax to statistical estimation, and contains Γ-minimax theory, used to pick applicab... | https://en.wikipedia.org/wiki/Gamma-minimax_inference |
Reversi Championis a video game adaptation of theOthelloboard game. Playable in single-player or two-player modes, it was developed and published byLoricielsand released in1984for theOric 1,Oric Atmos, andSega SC-3000computers.[1]AnAmstrad CPCversion followed in February1986.
While the Oric and SC-3000 versions offer ... | https://en.wikipedia.org/wiki/Reversi_Champion |
Linear discriminant analysis(LDA),normal discriminant analysis(NDA),canonical variates analysis(CVA), ordiscriminant function analysisis a generalization ofFisher's linear discriminant, a method used instatisticsand other fields, to find alinear combinationof features that characterizes or separates two or more classes... | https://en.wikipedia.org/wiki/Discriminant_function |
Invector calculus, thecurl, also known asrotor, is avector operatorthat describes theinfinitesimalcirculationof avector fieldin three-dimensionalEuclidean space. The curl at a point in the field is represented by avectorwhose length and direction denote themagnitudeand axis of the maximum circulation.[1]The curl of a f... | https://en.wikipedia.org/wiki/Curl_(mathematics) |
Invector calculus,divergenceis avector operatorthat operates on avector field, producing ascalar fieldgiving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the outwardfluxof a vector field from an infinitesimal volume around a given point.
As ... | https://en.wikipedia.org/wiki/Divergence |
Indifferential geometry, thefour-gradient(or4-gradient)∂{\displaystyle {\boldsymbol {\partial }}}is thefour-vectoranalogue of thegradient∇→{\displaystyle {\vec {\boldsymbol {\nabla }}}}fromvector calculus.
Inspecial relativityand inquantum mechanics, the four-gradient is used to define the properties and relations bet... | https://en.wikipedia.org/wiki/Four-gradient |
Inmathematics, theHessian matrix,Hessianor (less commonly)Hesse matrixis asquare matrixof second-orderpartial derivativesof a scalar-valuedfunction, orscalar field. It describes the localcurvatureof a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematicianLudwig Otto... | https://en.wikipedia.org/wiki/Hessian_matrix |
Inmathematics, askew gradientof aharmonic functionover asimply connected domainwith two real dimensions is avector fieldthat is everywhereorthogonalto thegradientof the function and that has the samemagnitudeas the gradient.
The skew gradient can be defined using complex analysis and theCauchy–Riemann equations.
Letf... | https://en.wikipedia.org/wiki/Skew_gradient |
Aspatial gradientis agradientwhosecomponentsare spatialderivatives, i.e.,rate of changeof a givenscalarphysical quantitywith respect to theposition coordinatesinphysical space.
Homogeneous regions have spatial gradientvector normequal to zero.
When evaluated oververtical position(altitude or depth), it is calledvertic... | https://en.wikipedia.org/wiki/Spatial_gradient |
Inmathematical optimizationtheory,dualityor theduality principleis the principle thatoptimization problemsmay be viewed from either of two perspectives, theprimal problemor thedual problem. If the primal is a minimization problem then the dual is a maximization problem (and vice versa). Any feasible solution to the pri... | https://en.wikipedia.org/wiki/Duality_(optimization) |
Inmathematical optimization, theKarush–Kuhn–Tucker(KKT)conditions, also known as theKuhn–Tucker conditions, arefirst derivative tests(sometimes called first-ordernecessary conditions) for a solution innonlinear programmingto beoptimal, provided that someregularity conditionsare satisfied.
Allowing inequality constrain... | https://en.wikipedia.org/wiki/Karush%E2%80%93Kuhn%E2%80%93Tucker_conditions |
Inmathematics,engineering,computer scienceandeconomics, anoptimization problemis theproblemof finding thebestsolution from allfeasible solutions.
Optimization problems can be divided into two categories, depending on whether thevariablesarecontinuousordiscrete:
In the context of an optimization problem, thesearch spa... | https://en.wikipedia.org/wiki/Optimization_problem |
Proximal gradient methodsare a generalized form of projection used to solve non-differentiableconvex optimizationproblems.
Many interesting problems can be formulated as convex optimization problems of the form
minx∈RN∑i=1nfi(x){\displaystyle \operatorname {min} \limits _{x\in \mathbb {R} ^{N}}\sum _{i=1}^{n}f_{i}(x... | https://en.wikipedia.org/wiki/Proximal_gradient_method |
Many problems inmathematical programmingcan be formulated asproblems onconvex setsorconvex bodies. Six kinds of problems are particularly important:[1]: Sec.2optimization,violation,validity,separation,membershipandemptiness. Each of these problems has a strong (exact) variant, and a weak (approximate) variant.
In all ... | https://en.wikipedia.org/wiki/Algorithmic_problems_on_convex_sets |
InBayesian inference, theBernstein–von Mises theoremprovides the basis for using Bayesian credible sets for confidence statements inparametric models. It states that under some conditions, a posterior distribution converges intotal variation distanceto a multivariate normal distribution centered at the maximum likeliho... | https://en.wikipedia.org/wiki/Bernstein%E2%80%93von_Mises_theorem |
Theprobability ofsuccess(POS)is a statistics concept commonly used in thepharmaceutical industryincluding byhealth authoritiesto supportdecision making.
The probability of success is a concept closely related to conditional power andpredictive power. Conditional power is the probability of observing statistical signif... | https://en.wikipedia.org/wiki/Probability_of_success |
Bayesian epistemologyis a formal approach to various topics inepistemologythat has its roots inThomas Bayes' work in the field of probability theory.[1]One advantage of its formal method in contrast to traditional epistemology is that its concepts and theorems can be defined with a high degree of precision. It is based... | https://en.wikipedia.org/wiki/Bayesian_epistemology |
Instatisticsandstatistical physics, theMetropolis–Hastings algorithmis aMarkov chain Monte Carlo(MCMC) method for obtaining a sequence ofrandom samplesfrom aprobability distributionfrom which direct sampling is difficult. New samples are added to the sequence in two steps: first a new sample is proposed based on the pr... | https://en.wikipedia.org/wiki/Metropolis%E2%80%93Hastings_algorithm |
Fawkesis a facial image cloaking software created by the SAND (Security, Algorithms, Networking and Data) Laboratory of theUniversity of Chicago.[1]It is a free tool that is available as a standalone executable.[2]The software creates small alterations in images usingartificial intelligenceto protect the images from be... | https://en.wikipedia.org/wiki/Fawkes_(image_cloaking_software) |
Inmathematics, aweak derivativeis a generalization of the concept of thederivativeof afunction(strong derivative) for functions not assumeddifferentiable, but onlyintegrable, i.e., to lie in theLpspaceL1([a,b]){\displaystyle L^{1}([a,b])}.
The method ofintegration by partsholds that for smooth functionsu{\displaystyle... | https://en.wikipedia.org/wiki/Weak_derivative |
Subgradient methodsareconvex optimizationmethods which usesubderivatives. Originally developed byNaum Z. Shorand others in the 1960s and 1970s, subgradient methods are convergent when applied even to a non-differentiable objective function. When the objective function is differentiable, sub-gradient methods for unconst... | https://en.wikipedia.org/wiki/Subgradient_method |
In mathematics, theClarke generalized derivativesare types generalized ofderivativesthat allow for the differentiation of nonsmooth functions. The Clarke derivatives were introduced byFrancis Clarkein 1975.[1]
For alocally Lipschitz continuousfunctionf:Rn→R,{\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} ,}th... | https://en.wikipedia.org/wiki/Clarke_generalized_derivative |
Inprobability theoryandstatistics, thecovariance functiondescribes how much tworandom variableschange together (theircovariance) with varying spatial or temporal separation. For arandom fieldorstochastic processZ(x) on a domainD, a covariance functionC(x,y) gives the covariance of the values of the random field at the ... | https://en.wikipedia.org/wiki/Covariance_function |
Inmathematical analysis, integral equations are equations in which an unknownfunctionappears under anintegralsign.[1]In mathematical notation, integral equations may thus be expressed as being of the form:f(x1,x2,x3,…,xn;u(x1,x2,x3,…,xn);I1(u),I2(u),I3(u),…,Im(u))=0{\displaystyle f(x_{1},x_{2},x_{3},\ldots ,x_{n};u(x_{... | https://en.wikipedia.org/wiki/Integral_equation |
In mathematics, and specifically inoperator theory, apositive-definite function on a grouprelates the notions of positivity, in the context ofHilbert spaces, and algebraicgroups. It can be viewed as a particular type ofpositive-definite kernelwhere the underlying set has the additional group structure.
LetG{\displayst... | https://en.wikipedia.org/wiki/Positive-definite_function_on_a_group |
Infunctional analysis, areproducing kernel Hilbert space(RKHS) is aHilbert spaceof functions in which point evaluation is a continuouslinear functional. Specifically, a Hilbert spaceH{\displaystyle H}of functions from a setX{\displaystyle X}(toR{\displaystyle \mathbb {R} }orC{\displaystyle \mathbb {C} }) is an RKHS if ... | https://en.wikipedia.org/wiki/Reproducing_kernel_Hilbert_space |
Kernel methodsare a well-established tool to analyze the relationship between input data and the corresponding output of a function. Kernels encapsulate the properties of functions in acomputationally efficientway and allow algorithms to easily swap functions of varying complexity.
In typicalmachine learningalgorithms... | https://en.wikipedia.org/wiki/Kernel_methods_for_vector_output |
Instatistics,kernel density estimation(KDE) is the application ofkernel smoothingforprobability density estimation, i.e., anon-parametricmethod toestimatetheprobability density functionof arandom variablebased onkernelsasweights. KDE answers a fundamental data smoothing problem where inferences about thepopulationare m... | https://en.wikipedia.org/wiki/Kernel_density_estimation |
Forcomputer science, instatistical learning theory, arepresenter theoremis any of several related results stating that a minimizerf∗{\displaystyle f^{*}}of a regularizedempirical risk functionaldefined over areproducing kernel Hilbert spacecan be represented as a finite linear combination of kernel products evaluated o... | https://en.wikipedia.org/wiki/Representer_theorem |
Similarity learningis an area ofsupervised machine learninginartificial intelligence. It is closely related toregressionandclassification, but the goal is to learn asimilarity functionthat measures howsimilaror related two objects are. It has applications inranking, inrecommendation systems, visual identity tracking, f... | https://en.wikipedia.org/wiki/Similarity_learning |
Cover's theoremis a statement incomputational learning theoryand is one of the primary theoretical motivations for the use of non-linearkernel methodsinmachine learningapplications. It is so termed after the information theoristThomas M. Coverwho stated it in 1965, referring to it ascounting function theorem.
Let the ... | https://en.wikipedia.org/wiki/Cover%27s_theorem |
Instatistics, aquadratic classifieris astatistical classifierthat uses aquadraticdecision surfaceto separate measurements of two or more classes of objects or events. It is a more general version of thelinear classifier.
Statistical classificationconsiders a set ofvectorsof observationsxof an object or event, each of ... | https://en.wikipedia.org/wiki/Quadratic_classifier |
Inmachine learning,support vector machines(SVMs, alsosupport vector networks[1]) aresupervisedmax-marginmodels with associated learningalgorithmsthat analyze data forclassificationandregression analysis. Developed atAT&T Bell Laboratories,[1][2]SVMs are one of the most studied models, being based on statistical learnin... | https://en.wikipedia.org/wiki/Support_vector_machines |
InEuclidean geometry, theintersection of a line and a linecan be theempty set, apoint, or anotherline. Distinguishing these cases and finding theintersectionhave uses, for example, incomputer graphics,motion planning, andcollision detection.
Inthree-dimensionalEuclidean geometry, if two lines are not in the sameplane,... | https://en.wikipedia.org/wiki/Line%E2%80%93line_intersection |
Line fittingis the process of constructing astraight linethat has the best fit to a series of data points.
Several methods exist, considering: | https://en.wikipedia.org/wiki/Line_fitting |
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