Alternating Projection Methods

Escalante, René; Raydan, Marcos

doi:10.1137/9781611971941

Cited by 136 publications

(130 citation statements)

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“…This simple idea is motivated by the fact that it is often easier to compute the individual projections onto the given subspaces than the projection onto their intersection. A detailed exposition of the origins and generalizations of alternating projections methods is given by Escalante and Raydan [15].…”

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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“…It is also known that, under mild regularity conditions, the algorithm converges linearly regardless of the initial point. Interested readers are referred to Escalante and Raydan (2011) for further details.…”

Section: Alternating Projectionmentioning

confidence: 99%

Distance Shrinkage and Euclidean Embedding via Regularized Kernel Estimation

Zhang

Wahba

Yuan

2016

Journal of the Royal Statistical Society Series B: Statistical Methodology

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Although recovering an Euclidean distance matrix from noisy observations is a common problem in practice, how well this could be done remains largely unknown. To fill in this void, we study a simple distance matrix estimate based upon the so-called regularized kernel estimate. We show that such an estimate can be characterized as simply applying a constant amount of shrinkage to all observed pairwise distances. This fact allows us to establish risk bounds for the estimate implying that the true distances can be estimated consistently in an average sense as the number of objects increases. In addition, such a characterization suggests an efficient algorithm to compute the distance matrix estimator, as an alternative to the usual second order cone programming known not to scale well for large problems. Numerical experiments and an application in visualizing the diversity of Vpu protein sequences from a recent HIV-1 study further demonstrate the practical merits of the proposed method.

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“…To be precise, for all k, the computation of the increments y k i may be omitted from the algorithm. In that case, the algorithm, described in (2), coincides with the classical von Neumann-Halperin alternating projection method (or MAP for short); see [23,31].…”

Section: Dykstra's Algorithmmentioning

confidence: 99%

“…The algorithm has been modified and used for many different applications; see, e. g., [1,11,18,19,27,29,30,35,36,38,39,[46][47][48][49]52]. For a review on Dykstra's method, and many other alternating projection schemes, see [4,5,10,13,20,31,32].…”

Section: Introductionmentioning

confidence: 99%

An acceleration scheme for Dykstra’s algorithm

López

Raydan

2015

Comput Optim Appl

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Dykstra's algorithm is an iterative alternating projection procedure for solving the best approximation problem: find the closest point, to a given one, in the intersection of a finite number of closed and convex sets. The main drawback of Dykstra's algorithm is its frequent slow convergence. In this work we develop an acceleration scheme with a strong geometrical flavor, which guarantees termination at the solution in two cycles of projections in the case of two closed subspaces. The proposed scheme can also be applied to any other alternating projection algorithm that solves the best approximation problem.

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Alternating Projection Methods

Cited by 136 publications

References 0 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

Distance Shrinkage and Euclidean Embedding via Regularized Kernel Estimation

An acceleration scheme for Dykstra’s algorithm

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