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Bregman Method
The Bregman method is an iterative algorithm to solve certain convex optimization problems involving regularization. The original version is due to Lev M. Bregman, who published it in 1967. The algorithm is a row-action method accessing constraint functions one by one and the method is particularly suited for large optimization problems where constraints can be efficiently enumerated. The algorithm works particularly well for regularizers such as the \ell_1 norm, where it converges very quickly because of an error-cancellation effect. Algorithm In order to be able to use the Bregman method, one must frame the problem of interest as finding \min_u J(u) + f(u), where J is a regularizing function such as \ell_1. The ''Bregman distance'' is defined as D^p(u, v) := J(u) - (J(v) + \langle p, u - v\rangle) where p belongs to the subdifferential of J at u (which we denoted \partial J(u)). One performs the iteration u_:= \min_u(\alpha D(u, u_k) + f(u)), with \alpha a constant to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iterative Algorithm
In computational mathematics, an iterative method is a mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''i''-th approximation (called an "iterate") is derived from the previous ones. A specific implementation with termination criteria for a given iterative method like gradient descent, hill climbing, Newton's method, or quasi-Newton methods like BFGS, is an algorithm of an iterative method or a method of successive approximation. An iterative method is called '' convergent'' if the corresponding sequence converges for given initial approximations. A mathematically rigorous convergence analysis of an iterative method is usually performed; however, heuristic-based iterative methods are also common. In contrast, direct methods attempt to solve the problem by a finite sequence of operations. In the absence of rounding errors, direct methods would deliver an exact solution (for exampl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Least Absolute Deviations
Least absolute deviations (LAD), also known as least absolute errors (LAE), least absolute residuals (LAR), or least absolute values (LAV), is a statistical optimality criterion and a statistical optimization technique based on minimizing the sum of absolute deviations (also ''sum of absolute residuals'' or ''sum of absolute errors'') or the ''L''1 norm of such values. It is analogous to the least squares technique, except that it is based on ''absolute values'' instead of squared values. It attempts to find a function which closely approximates a set of data by minimizing residuals between points generated by the function and corresponding data points. The LAD estimate also arises as the maximum likelihood estimate if the errors have a Laplace distribution. It was introduced in 1757 by Roger Joseph Boscovich. Formulation Suppose that the data set consists of the points (''x''''i'', ''y''''i'') with ''i'' = 1, 2, ..., ''n''. We want to find a function ''f'' such that f(x_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wirtinger Derivatives
In complex analysis of one and several complex variables, Wirtinger derivatives (sometimes also called Wirtinger operators), named after Wilhelm Wirtinger who introduced them in 1927 in the course of his studies on the theory of functions of several complex variables, are partial differential operators of the first order which behave in a very similar manner to the ordinary derivatives with respect to one real variable, when applied to holomorphic functions, antiholomorphic functions or simply differentiable functions on complex domains. These operators permit the construction of a differential calculus for such functions that is entirely analogous to the ordinary differential calculus for functions of real variables. Historical notes Early days (1899–1911): the work of Henri Poincaré Wirtinger derivatives were used in complex analysis at least as early as in the paper , as briefly noted by and by . In the third paragraph of his 1899 paper, Henri Poincaré first define ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature, "imaginary" complex numbers have a mathematical existence as firm as that of the real numbers, and they are fundamental tools in the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nesterov Method
Nesterov (), until 1938 known by its German name (; ) and in 1938-1946 as Ebenrode, is a town and the administrative center of Nesterovsky District in Kaliningrad Oblast, Russia, located east of Kaliningrad, near the Russian-Lithuanian border on the railway connecting Kaliningrad Oblast with Moscow. Population figures: History In the Middle Ages, the area in Old Prussia had been settled by the Nadruvian tribe of the Baltic Prussians. It was conquered by the Teutonic Knights in about 1276 and incorporated into the State of the Teutonic Order. From the 15th century onwards, the Knights largely resettled the lands with Samogitian and Lithuanian colonists. Since 1466, it was part of the Kingdom of Poland as a fief held by the Teutonic Order. The settlement itself was first mentioned as ''Stallupoenen'', or ''Stallupönen'', in 1539, named after a nearby river called ''Stalupė'' in Lithuanian. At that time, with the secularization of the Order's Prussian lands in 1525, Stal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
