An Algorithm for Restricted Least Squares Regression

Dykstra, Richard L.

doi:10.1080/01621459.1983.10477029

Cited by 368 publications

(213 citation statements)

References 10 publications

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“…Since these are convex sets rather than subspaces the alternating projections method has to be used in a modified form proposed by Dykstra [14], in which each projection incorporates a correction; each correction can be interpreted as a normal vector to the corresponding convex set. This correction is not needed for a translate of a subspace [9], so is only required for the projection onto S n .…”

Section: Accelerating the Alternating Projections Methods For The Nearmentioning

confidence: 99%

Anderson acceleration of the alternating projections method for computing the nearest correlation matrix

Higham

Strabić

2015

Numer Algor

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In a wide range of applications it is required to compute the nearest correlation matrix in the Frobenius norm to a given symmetric but indefinite matrix. Of the available methods with guaranteed convergence to the unique solution of this problem the easiest to implement, and perhaps the most widely used, is the alternating projections method. However, the rate of convergence of this method is at best linear, and it can require a large number of iterations to converge to within a given tolerance. We show that Anderson acceleration, a technique for accelerating the convergence of fixed-point iterations, can be applied to the alternating projections method and that in practice it brings a significant reduction in both the number of iterations and the computation time. We also show that Anderson acceleration remains effective, and indeed can provide even greater improvements, when it is applied to the variants of the nearest correlation matrix problem in which specified elements are fixed or a lower bound is imposed on the smallest eigenvalue. Alternating projections is a general method for finding a point in the intersection of several sets and ours appears to be the first demonstration that this class of methods can benefit from Anderson acceleration.

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Section: Accelerating the Alternating Projections Methods For The Nearmentioning

confidence: 99%

Anderson acceleration of the alternating projections method for computing the nearest correlation matrix

Higham

Strabić

2015

Numer Algor

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“…Moreover, we assume that for all y ∈ IR n , the calculation of P Ω (y) is a difficult task, whereas, for each Ω i , P Ω i (y) is easy to obtain. Dykstra's algorithm [8,13], solves (21) by generating two sequences, {y ℓ i } and {z ℓ i }. These sequences are defined by the following recursive formulae:…”

Section: Algorithm 21: Inexact Variable Metric Methodsmentioning

confidence: 99%

Inexact spectral projected gradient methods on convex sets

Birgin¹

2003

IMA Journal of Numerical Analysis

166

178

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A new method is introduced for large scale convex constrained optimization. The general model algorithm involves, at each iteration, the approximate minimization of a convex quadratic on the feasible set of the original problem and global convergence is obtained by means of nonmonotone line searches. A specific algorithm, the Inexact Spectral Projected Gradient method (ISPG), is implemented using inexact projections computed by Dykstra's alternating projection method and generates interior iterates. The ISPG method is a generalization of the Spectral Projected Gradient method (SPG), but can be used when projections are difficult to compute. Numerical results for constrained least-squares rectangular matrix problems are presented.

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“…Clearly, the solution is a matrix which is symmetric not only with respect to the main diagonal but also to the SW-NE diagonal. Since the feasibility region is the intersection of two subspace and because we know how to solve the persymmetric Procrustes problem [5] and the general Toeplitz problem, see section 2, we develop an algorithm based on the alternating projection method [3,1,6,7]. Therefore the solution of (16) is obtained projecting alternating on the subspaces T y S. Given the problem…”

Section: The Symmetric Toeplitz Problemmentioning

confidence: 99%

Finding the closest Toeplitz matrix

Eberle¹,

Maciel²

2003

Mat. apl. comput.

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Abstract. The constrained least-squares n × n-matrix problem where the feasibility set is the subspace of the Toeplitz matrices is analyzed. The general, the upper and lower triangular cases are solved by making use of the singular value decomposition. For the symmetric case, an algorithm based on the alternate projection method is proposed. The implementation does not require the calculation of the eigenvalue of a matrix and still guarantees convergence. Encouraging preliminary results are discussed.Mathematical subject classification: 65F20, 65F30, 65H10.

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An Algorithm for Restricted Least Squares Regression

Cited by 368 publications

References 10 publications

Anderson acceleration of the alternating projections method for computing the nearest correlation matrix

Anderson acceleration of the alternating projections method for computing the nearest correlation matrix

Inexact spectral projected gradient methods on convex sets

Finding the closest Toeplitz matrix

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