Function minimization by conjugate gradients

Fletcher, R.; Reeves, C. M.

doi:10.1093/comjnl/7.2.149

Cited by 4,203 publications

(1,788 citation statements)

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“…Fletcher and Reeves generalized the procedure to nonquadratic functions yielding the non-linear conjugate gradients algorithm [13]. Here, only the function f (x) and its gradient ∇f (x) are available.…”

Section: The Linear and Non-linear Conjugate Gradient Algorithmsmentioning

confidence: 99%

Conjugate Gradient Bundle Adjustment

Byröd

Åström

2010

Computer Vision – ECCV 2010

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Abstract. Bundle adjustment for multi-view reconstruction is traditionally done using the Levenberg-Marquardt algorithm with a direct linear solver, which is computationally very expensive. An alternative to this approach is to apply the conjugate gradients algorithm in the inner loop. This is appealing since the main computational step of the CG algorithm involves only a simple matrix-vector multiplication with the Jacobian. In this work we improve on the latest published approaches to bundle adjustment with conjugate gradients by making full use of the least squares nature of the problem. We employ an easy-to-compute QR factorization based block preconditioner and show how a certain property of the preconditioned system allows us to reduce the work per iteration to roughly half of the standard CG algorithm.

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Section: The Linear and Non-linear Conjugate Gradient Algorithmsmentioning

confidence: 99%

Conjugate Gradient Bundle Adjustment

Byröd

Åström

2010

Computer Vision – ECCV 2010

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“…We propose an algorithm that differs from it in three ways. Firstly, the heuristic updates are replaced by a standard conjugate gradient algorithm [11]. Secondly, the linearisation method from [7] is applied.…”

Section: Variational Bayesian Methodsmentioning

confidence: 99%

State Inference in Variational Bayesian Nonlinear State-Space Models

Raiko

Tornio

Honkela

et al. 2006

Independent Component Analysis and Blind Signal Separation

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Abstract. Nonlinear source separation can be performed by inferring the state of a nonlinear state-space model. We study and improve the inference algorithm in the variational Bayesian blind source separation model introduced by Valpola and Karhunen in 2002. As comparison methods we use extensions of the Kalman filter that are widely used inference methods in tracking and control theory. The results in stability, speed, and accuracy favour our method especially in difficult inference problems.

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“…In Theorem 4.3 of Smith (1993), the following result is stated for the Riemannian Fletcher & Reeves (1964) and Polak & Ribière (1969) conjugate gradient methods. SupposeŶ is a non-degenerate stationary point such that the Hessian atŶ is positive definite.…”

Section: Global Convergencementioning

confidence: 99%

“…Our implementation of geometric optimisation over low-rank correlation matrices 'LRCM min' 5 is an adoption of the 'SG min' template of Edelman & Lippert (2000) (written in MATLAB) for optimisation over the Stiefel and Grassmann manifolds. This template contains four distinct well-known non-linear optimisation algorithms adapted for geometric optimisation over Riemannian manifolds: Newton algorithm; dogleg step or Levenberg (1944) and Marquardt (1963) algorithm;Polak & Ribière (1969) conjugate gradient;and Fletcher & Reeves (1964) conjugate gradient.…”

Section: Where D * Can Be Obtained By Selecting At Most D Nonnegativementioning

confidence: 99%

Efficient Rank Reduction of Correlation Matrices

Grubišić

Pietersz²

2005

SSRN Journal

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Abstract. Geometric optimisation algorithms are developed that efficiently find the nearest low-rank correlation matrix. We show, in numerical tests, that our methods compare favourably to the existing methods in the literature. The connection with the Lagrange multiplier method is established, along with an identification of whether a local minimum is a global minimum. An additional benefit of the geometric approach is that any weighted norm can be applied. The problem of finding the nearest low-rank correlation matrix occurs as part of the calibration of multi-factor interest rate market models to correlation.

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Function minimization by conjugate gradients

Cited by 4,203 publications

References 0 publications

Conjugate Gradient Bundle Adjustment

Conjugate Gradient Bundle Adjustment

State Inference in Variational Bayesian Nonlinear State-Space Models

Efficient Rank Reduction of Correlation Matrices

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