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Hessian matrix

About: Hessian matrix is a(n) research topic. Over the lifetime, 7209 publication(s) have been published within this topic receiving 223267 citation(s).

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Journal ArticleDOI
John A. Nelder, R. Mead1Institutions (1)
TL;DR: A method is described for the minimization of a function of n variables, which depends on the comparison of function values at the (n 41) vertices of a general simplex, followed by the replacement of the vertex with the highest value by another point.

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Abstract: A method is described for the minimization of a function of n variables, which depends on the comparison of function values at the (n 41) vertices of a general simplex, followed by the replacement of the vertex with the highest value by another point. The simplex adapts itself to the local landscape, and contracts on to the final minimum. The method is shown to be effective and computationally compact. A procedure is given for the estimation of the Hessian matrix in the neighbourhood of the minimum, needed in statistical estimation problems.

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25,414 citations


Journal ArticleDOI
TL;DR: An algorithm for solving large nonlinear optimization problems with simple bounds is described, based on the gradient projection method and uses a limited memory BFGS matrix to approximate the Hessian of the objective function.

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Abstract: An algorithm for solving large nonlinear optimization problems with simple bounds is described. It is based on the gradient projection method and uses a limited memory BFGS matrix to approximate the Hessian of the objective function. It is shown how to take advantage of the form of the limited memory approximation to implement the algorithm efficiently. The results of numerical tests on a set of large problems are reported.

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4,483 citations


Book ChapterDOI
11 Oct 1998-
TL;DR: The multiscale second order local structure of an image (Hessian) is examined with the purpose of developing a vessel enhancement filter and a vesselness measure is obtained.

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Abstract: The multiscale second order local structure of an image (Hessian) is examined with the purpose of developing a vessel enhancement filter. A vesselness measure is obtained on the basis of all eigenvalues of the Hessian. This measure is tested on two dimensional DSA and three dimensional aortoiliac and cerebral MRA data. Its clinical utility is shown by the simultaneous noise and background suppression and vessel enhancement in maximum intensity projections and volumetric displays.

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3,622 citations


Journal ArticleDOI
Abstract: Quasi-Newton methods accelerate the steepest-descent technique for function minimization by using computational history to generate a sequence of approximations to the inverse of the Hessian matrix. This paper presents a class of approximating matrices as a function of a scalar parameter. The problem of optimal conditioning of these matrices under an appropriate norm as a function of the scalar parameter is investigated. A set of computational results verifies the superiority of the new methods arising from conditioning considerations to known methods.

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2,988 citations


Book
01 Apr 1988-
Abstract: Preface MATRICES: Basic Properties of Vectors and Matrices Kronecker Products, the Vec-Operator and the Moore- Penrose Inverse Miscellaneous Matrix Results DIFFERENTIALS: THE THEORY: Mathematical Preliminaries Differentials and Differentiability The Second Differential Static Optimization DIFFERENTIALS: THE PRACTICE: Some Important Differentials First- Order Differentials and Jacobian Matrices Second-Order Differentials and Hessian Matrices INEQUALITIES: Inequalities THE LINEAR MODEL: Statistical Preliminaries The Linear Regression Model Further Topics in the Linear Model APPLICATIONS TO MAXIMUM LIKELIHOOD ESTIMATION: Maximum Likelihood Estimation Simultaneous Equations Topics in Psychometrics Subject Index Bibliography.

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2,774 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202213
2021460
2020518
2019471
2018388
2017366

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Topic's top 5 most impactful authors

Omar Ghattas

30 papers, 749 citations

Michael W. Mahoney

23 papers, 776 citations

Alejandro Ribeiro

20 papers, 444 citations

Miroslav Krstic

18 papers, 457 citations

Walter Murray

17 papers, 3.2K citations