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Gradient (other)
Gradient in vector calculus is a vector field representing the maximum rate of increase of a scalar field or a multivariate function and the direction of this maximal rate. Gradient may also refer to: *Gradient sro, a Czech aircraft manufacturer * Image gradient, a gradual change or blending of color ** Color gradient, a range of position-dependent colors, usually used to fill a region ** Texture gradient, the distortion in size which closer objects have compared to objects farther away * Spatial gradient, a gradient whose components are spatial derivatives * Grade (slope), the inclination of a road or other geographic feature Mathematics * Gradient descent, a first-order iterative optimization algorithm for finding the minimum of a function * Gradient theorem, theorem that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve * Gradient method, an algorithm to solve problems with search directions defined b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradient of a function is non-zero at a point , the direction of the gradient is the direction in which the function increases most quickly from , and the magnitude of the gradient is the rate of increase in that direction, the greatest absolute directional derivative. Further, a point where the gradient is the zero vector is known as a stationary point. The gradient thus plays a fundamental role in optimization theory, where it is used to maximize a function by gradient ascent. In coordinate-free terms, the gradient of a function f(\bf) may be defined by: :df=\nabla f \cdot d\bf where ''df'' is the total infinitesimal change in ''f'' for an infinitesimal displacement d\bf, and is seen to be maximal when d\bf is in the direction of the gradi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjugate Gradient Method
In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is positive-definite. The conjugate gradient method is often implemented as an iterative algorithm, applicable to sparse systems that are too large to be handled by a direct implementation or other direct methods such as the Cholesky decomposition. Large sparse systems often arise when numerically solving partial differential equations or optimization problems. The conjugate gradient method can also be used to solve unconstrained optimization problems such as energy minimization. It is commonly attributed to Magnus Hestenes and Eduard Stiefel, who programmed it on the Z4, and extensively researched it. The biconjugate gradient method provides a generalization to non-symmetric matrices. Various nonlinear conjugate gradient methods seek minima of nonlinear optimization problems. Description of the problem addressed by co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grade (other)
Grade most commonly refers to: * Grade (education), a measurement of a student's performance * Grade, the number of the year a student has reached in a given educational stage * Grade (slope), the steepness of a slope Grade or grading may also refer to: Music * Grade (music), a formally assessed level of profiency in a musical instrument * Grade (band), punk rock band * Grades (producer), British electronic dance music producer and DJ Science and technology Biology and medicine * Grading (tumors), a measure of the aggressiveness of a tumor in medicine * The Grading of Recommendations Assessment, Development and Evaluation (GRADE) approach * Evolutionary grade, a paraphyletic group of organisms Geology * Graded bedding, a description of the variation in grain size through a bed in a sedimentary rock * Metamorphic grade, an indicatation of the degree of metamorphism of rocks * Ore grade, a measure that describes the concentration of a valuable natural material in the surroundin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gradation (other)
Gradation may refer to: * Gradation (music), gradual change within one parameter, or an overlapping of two blocks of sound * ''Gradation'' (album), 1988 pop album by Shizuka Kudo * Gradation (art), visual technique of gradually transitioning from one colour or texture to another * Consonant gradation, mutation in which consonant sounds alternate between various "grades" * Apophony or vowel gradation, sound change within a word that indicates grammatical information * Calibration, comparison of measurement values of a device with a standard of known accuracy * Production of a graded algebra See also * Color grading, process of altering and enhancing the color of an image * Comparison (grammar), a feature whereby adjectives or adverbs indicate relative degree * Evaluation Evaluation is a systematic determination and assessment of a subject's merit, worth and significance, using criteria governed by a set of standards. It can assist an organization, program, design, project or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fade (other)
Fade or Fading may refer to: Science and technology * Fading, a loss of signal strength at a radio receiver * Color fade, the alteration of color by light * Fade (audio engineering), a gradual change in sound volume * Brake fade, in vehicle braking systems, a reduction in stopping power after repeated use * FADE, a type of anti-piracy software Arts and entertainment Film and television * Fade (filmmaking), a cinematographic technique * ''Fade'', a 2007 film starring Devon Odessa * ''The Fades'' (TV series), a 2011 UK supernatural drama series * "Fade" (''Smallville''), a television episode Literature * ''Fade'' (novel), a 1988 novel by Robert Cormier * ''The Fade'', a 2007 novel by Chris Wooding Music * Dynamics (music), the variation or change in volume in a piece of music Performers * Fade (band), a Japanese alternative rock band * The Fades, a British indie rock band Albums * ''Fade'' (Remove Silence album) or the title song, 2010 * ''Fade'' (Yo La Tengo album), 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Slope
In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is used for slope, but its earliest use in English appears in O'Brien (1844) who wrote the equation of a straight line as and it can also be found in Todhunter (1888) who wrote it as "''y'' = ''mx'' + ''c''". Slope is calculated by finding the ratio of the "vertical change" to the "horizontal change" between (any) two distinct points on a line. Sometimes the ratio is expressed as a quotient ("rise over run"), giving the same number for every two distinct points on the same line. A line that is decreasing has a negative "rise". The line may be practical – as set by a road surveyor, or in a diagram that models a road or a roof either as a description or as a plan. The ''steepness'', incline, or grade of a line is measured by the absolute ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stochastic Gradient Descent
Stochastic gradient descent (often abbreviated SGD) is an iterative method for optimizing an objective function with suitable smoothness properties (e.g. differentiable or subdifferentiable). It can be regarded as a stochastic approximation of gradient descent optimization, since it replaces the actual gradient (calculated from the entire data set) by an estimate thereof (calculated from a randomly selected subset of the data). Especially in high-dimensional optimization problems this reduces the very high computational burden, achieving faster iterations in trade for a lower convergence rate. While the basic idea behind stochastic approximation can be traced back to the Robbins–Monro algorithm of the 1950s, stochastic gradient descent has become an important optimization method in machine learning. Background Both statistical estimation and machine learning consider the problem of minimizing an objective function that has the form of a sum: : Q(w) = \frac\sum_^n Q_i(w), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonlinear Conjugate Gradient Method
In numerical optimization, the nonlinear conjugate gradient method generalizes the conjugate gradient method to nonlinear optimization. For a quadratic function \displaystyle f(x) :: \displaystyle f(x)=\, Ax-b\, ^2, the minimum of f is obtained when the gradient is 0: :: \nabla_x f=2 A^T(Ax-b)=0. Whereas linear conjugate gradient seeks a solution to the linear equation \displaystyle A^T Ax=A^T b, the nonlinear conjugate gradient method is generally used to find the local minimum of a nonlinear function using its gradient \nabla_x f alone. It works when the function is approximately quadratic near the minimum, which is the case when the function is twice differentiable at the minimum and the second derivative is non-singular there. Given a function \displaystyle f(x) of N variables to minimize, its gradient \nabla_x f indicates the direction of maximum increase. One simply starts in the opposite (steepest descent) direction: :: \Delta x_0=-\nabla_x f (x_0) with an adjustable s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gradient Method
In optimization (mathematics), optimization, a gradient method is an algorithm to solve problems of the form :\min_\; f(x) with the search directions defined by the gradient of the function at the current point. Examples of gradient methods are the gradient descent and the conjugate gradient. See also * Gradient descent * Stochastic gradient descent * Coordinate descent * Frank–Wolfe algorithm * Landweber iteration * Random coordinate descent * Conjugate gradient method * Derivation of the conjugate gradient method * Nonlinear conjugate gradient method * Biconjugate gradient method * Biconjugate gradient stabilized method References * First order methods Optimization algorithms and methods Numerical linear algebra Gradient methods, {{linear-algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gradient Sro
Gradient sro is a Czech aircraft manufacturer based in Prague and founded in 1997. The company specializes in the design and manufacture of paragliders in the form of ready-to-fly aircraft.Bertrand, Noel; Rene Coulon; et al: ''World Directory of Leisure Aviation 2003-04'', page 19. Pagefast Ltd, Lancaster UK, 2003. The company is organized as a společnost s ručením omezeným (sro), a Czech private limited company. The company has produced a wide range of paragliders, including the intermediate sport Aspen, the Avax competition wing, the two-place tandem BiOnyx, the intermediate performance Bliss, the beginner and flight training Bright and the intermediate Golden The company has ceased publishing performance specifications for its gliders, stating: Aircraft Summary of aircraft built by Gradient: * Gradient Agility * Gradient Aspen * Gradient Avax * Gradient BiGolden * Gradient BiOnyx * Gradient Bliss *Gradient Bright The Gradient Bright is a Czech single-place, p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gradient Theorem
The gradient theorem, also known as the fundamental theorem of calculus for line integrals, says that a line integral through a gradient field can be evaluated by evaluating the original scalar field at the endpoints of the curve. The theorem is a generalization of the second fundamental theorem of calculus to any curve in a plane or space (generally ''n''-dimensional) rather than just the real line. For as a differentiable function and as any continuous curve in which starts at a point and ends at a point , then \int_ \nabla\varphi(\mathbf)\cdot \mathrm\mathbf = \varphi\left(\mathbf\right) - \varphi\left(\mathbf\right) where denotes the gradient vector field of . The gradient theorem implies that line integrals through gradient fields are path-independent. In physics this theorem is one of the ways of defining a ''conservative'' force. By placing as potential, is a conservative field. Work done by conservative forces does not depend on the path followed by the obje ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gradient Descent
In mathematics, gradient descent (also often called steepest descent) is a first-order iterative optimization algorithm for finding a local minimum of a differentiable function. The idea is to take repeated steps in the opposite direction of the gradient (or approximate gradient) of the function at the current point, because this is the direction of steepest descent. Conversely, stepping in the direction of the gradient will lead to a local maximum of that function; the procedure is then known as gradient ascent. Gradient descent is generally attributed to Augustin-Louis Cauchy, who first suggested it in 1847. Jacques Hadamard independently proposed a similar method in 1907. Its convergence properties for non-linear optimization problems were first studied by Haskell Curry in 1944, with the method becoming increasingly well-studied and used in the following decades. Description Gradient descent is based on the observation that if the multi-variable function F(\mathbf) is def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
