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Symmetric Rank-one
The Symmetric Rank 1 (SR1) method is a quasi-Newton method to update the second derivative (Hessian) based on the derivatives (gradients) calculated at two points. It is a generalization to the secant method for a multidimensional problem. This update maintains the ''symmetry'' of the matrix but does ''not'' guarantee that the update be ''positive definite''. The sequence of Hessian approximations generated by the SR1 method converges to the true Hessian under mild conditions, in theory; in practice, the approximate Hessians generated by the SR1 method show faster progress towards the true Hessian than do popular alternatives ( BFGS or DFP), in preliminary numerical experiments. The SR1 method has computational advantages for sparse or partially separable problems. A twice continuously differentiable function x \mapsto f(x) has a gradient (\nabla f) and Hessian matrix B: The function f has an expansion as a Taylor series at x_0, which can be truncated ::f(x_0+\Delta x) \approx ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-Newton Method
In numerical analysis, a quasi-Newton method is an iterative numerical method used either to find zeroes or to find local maxima and minima of functions via an iterative recurrence formula much like the one for Newton's method, except using approximations of the derivatives of the functions in place of exact derivatives. Newton's method requires the Jacobian matrix of all partial derivatives of a multivariate function when used to search for zeros or the Hessian matrix when used for finding extrema. Quasi-Newton methods, on the other hand, can be used when the Jacobian matrices or Hessian matrices are unavailable or are impractical to compute at every iteration. Some iterative methods that reduce to Newton's method, such as sequential quadratic programming, may also be considered quasi-Newton methods. Search for zeros: root finding Newton's method to find zeroes of a function g of multiple variables is given by x_ = x_n - _g(x_n) g(x_n), where _g(x_n) is the left inverse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rank (linear Algebra)
In linear algebra, the rank of a matrix is the dimension of the vector space generated (or spanned) by its columns. p. 48, § 1.16 This corresponds to the maximal number of linearly independent columns of . This, in turn, is identical to the dimension of the vector space spanned by its rows. Rank is thus a measure of the " nondegenerateness" of the system of linear equations and linear transformation encoded by . There are multiple equivalent definitions of rank. A matrix's rank is one of its most fundamental characteristics. The rank is commonly denoted by or ; sometimes the parentheses are not written, as in .Alternative notation includes \rho (\Phi) from and . Main definitions In this section, we give some definitions of the rank of a matrix. Many definitions are possible; see Alternative definitions for several of these. The column rank of is the dimension of the column space of , while the row rank of is the dimension of the row space of . A fundamental resul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newton's Method In Optimization
In calculus, Newton's method (also called Newton–Raphson) is an iterative method for finding the roots of a differentiable function f, which are solutions to the equation f(x)=0. However, to optimize a twice-differentiable f, our goal is to find the roots of f'. We can therefore use Newton's method on its derivative f' to find solutions to f'(x)=0, also known as the critical points of f. These solutions may be minima, maxima, or saddle points; see section "Several variables" in Critical point (mathematics) and also section "Geometric interpretation" in this article. This is relevant in optimization, which aims to find (global) minima of the function f. Newton's method The central problem of optimization is minimization of functions. Let us first consider the case of univariate functions, i.e., functions of a single real variable. We will later consider the more general and more practically useful multivariate case. Given a twice differentiable function f:\mathbb\to \math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Broyden's Method
In numerical analysis, Broyden's method is a quasi-Newton method for finding roots in variables. It was originally described by C. G. Broyden in 1965. Newton's method for solving uses the Jacobian matrix, , at every iteration. However, computing this Jacobian can be a difficult and expensive operation; for large problems such as those involving solving the Kohn–Sham equations in quantum mechanics the number of variables can be in the hundreds of thousands. The idea behind Broyden's method is to compute the whole Jacobian at most only at the first iteration, and to do rank-one updates at other iterations. In 1979 Gay proved that when Broyden's method is applied to a linear system of size , it terminates in steps, although like all quasi-Newton methods, it may not converge for nonlinear systems. Description of the method Solving single-variable nonlinear equation In the secant method, we replace the first derivative at with the finite-difference approximation: : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasi-Newton Method
In numerical analysis, a quasi-Newton method is an iterative numerical method used either to find zeroes or to find local maxima and minima of functions via an iterative recurrence formula much like the one for Newton's method, except using approximations of the derivatives of the functions in place of exact derivatives. Newton's method requires the Jacobian matrix of all partial derivatives of a multivariate function when used to search for zeros or the Hessian matrix when used for finding extrema. Quasi-Newton methods, on the other hand, can be used when the Jacobian matrices or Hessian matrices are unavailable or are impractical to compute at every iteration. Some iterative methods that reduce to Newton's method, such as sequential quadratic programming, may also be considered quasi-Newton methods. Search for zeros: root finding Newton's method to find zeroes of a function g of multiple variables is given by x_ = x_n - _g(x_n) g(x_n), where _g(x_n) is the left inverse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trust Region
In mathematical optimization, a trust region is the subset of the region of the objective function that is approximated using a model function (often a quadratic). If an adequate model of the objective function is found within the trust region, then the region is expanded; conversely, if the approximation is poor, then the region is contracted. The fit is evaluated by comparing the ratio of expected improvement from the model approximation with the actual improvement observed in the objective function. Simple thresholding of the ratio is used as the criterion for expansion and contraction—a model function is "trusted" only in the region where it provides a reasonable approximation. Trust-region methods are in some sense dual to line-search methods: trust-region methods first choose a step size (the size of the trust region) and then a step direction, while line-search methods first choose a step direction and then a step size. The general idea behind trust region methods is kn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Quasi-Newton Representation
The compact representation for quasi-Newton methods is a matrix decomposition, which is typically used in gradient based optimization (mathematics), optimization algorithms or for solving nonlinear systems. The decomposition uses a low-rank representation for the direct and/or inverse Hessian matrix, Hessian or the Jacobian matrix and determinant, Jacobian of a nonlinear system. Because of this, the compact representation is often used for large problems and constrained optimization. Definition The compact representation of a quasi-Newton matrix for the inverse Hessian H_k or direct Hessian B_k of a nonlinear loss function, objective function f(x):\mathbb^n \to \mathbb expresses a sequence of recursive rank-1 or rank-2 matrix updates as one rank-k or rank-2k update of an initial matrix. Because it is derived from quasi-Newton updates, it uses differences of iterates and gradients \nabla f(x_k) = g_k in its definition \_^k . In particular, for r=k or r=2k the rectangular ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limited-memory BFGS
Limited-memory BFGS (L-BFGS or LM-BFGS) is an optimization algorithm in the family of quasi-Newton methods that approximates the Broyden–Fletcher–Goldfarb–Shanno algorithm (BFGS) using a limited amount of computer memory. It is a popular algorithm for parameter estimation in machine learning. The algorithm's target problem is to minimize f(\mathbf) over unconstrained values of the real-vector \mathbf where f is a differentiable scalar function. Like the original BFGS, L-BFGS uses an estimate of the inverse Hessian matrix to steer its search through variable space, but where BFGS stores a dense n\times n approximation to the inverse Hessian (''n'' being the number of variables in the problem), L-BFGS stores only a few vectors that represent the approximation implicitly. Due to its resulting linear memory requirement, the L-BFGS method is particularly well suited for optimization problems with many variables. Instead of the inverse Hessian H''k'', L-BFGS maintains a history of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Taylor Series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the 18th century. The partial sum formed by the first terms of a Taylor series is a polynomial of degree that is called the th Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally more accurate as increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hessian Matrix
In mathematics, the Hessian matrix, Hessian or (less commonly) Hesse matrix is a square matrix of second-order partial derivatives of a scalar-valued Function (mathematics), function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Otto Hesse, Ludwig Otto Hesse and later named after him. Hesse originally used the term "functional determinants". The Hessian is sometimes denoted by H or \nabla\nabla or \nabla^2 or \nabla\otimes\nabla or D^2. Definitions and properties Suppose f : \R^n \to \R is a function taking as input a vector \mathbf \in \R^n and outputting a scalar f(\mathbf) \in \R. If all second-order partial derivatives of f exist, then the Hessian matrix \mathbf of f is a square n \times n matrix, usually defined and arranged as \mathbf H_f= \begin \dfrac & \dfrac & \cdots & \dfrac \\[2.2ex] \dfrac & \dfrac & \cdots & \dfrac \\[2.2ex] \vdots & \vdot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
