Dykstra's Algorithm for a Constrained Least‐squares Matrix Problem

Escalante, René; Raydan, Marcos

doi:10.1002/(sici)1099-1506(199611/12)3:6<459::aid-nla82>3.3.co;2-j

Cited by 9 publications

(15 citation statements)

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“…Here, we adopt a simple cyclic constraint projection approach known as the Dykstra algorithm [13]. First, we split the constraints into two sets C 1 and C 2 :…”

Section: φ(X) = X 2 /2mentioning

confidence: 99%

Improving clustering by learning a bi-stochastic data similarity matrix

Wang

König

et al. 2011

Knowl Inf Syst

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An idealized clustering algorithm seeks to learn a cluster-adjacency matrix such that, if two data points belong to the same cluster, the corresponding entry would be 1; otherwise, the entry would be 0. This integer (1/0) constraint makes it difficult to find the optimal solution. We propose a relaxation on the cluster-adjacency matrix, by deriving a bi-stochastic matrix from a data similarity (e.g., kernel) matrix according to the Bregman divergence. Our general method is named the Bregmanian Bi-Stochastication (BBS) algorithm. We focus on two popular choices of the Bregman divergence: the Euclidean distance and the Kullback-Leibler (KL) divergence. Interestingly, the BBS algorithm using the KL divergence is equivalent to the Sinkhorn-Knopp (SK) algorithm for deriving bi-stochastic matrices. We show that the BBS algorithm using the Euclidean distance is closely related to the relaxed k-means clustering and can often produce noticeably superior clustering results to the SK algorithm (and other algorithms such as Normalized Cut), through extensive experiments on public data sets.

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“…Here, we adopt a simple cyclic constraint projection approach known as the Dykstra algorithm [13]. First, we split the constraints into two sets C 1 and C 2 :…”

Section: φ(X) = X 2 /2mentioning

confidence: 99%

Improving clustering by learning a bi-stochastic data similarity matrix

Wang

König

et al. 2011

Knowl Inf Syst

View full text Add to dashboard Cite

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“…where I ∈ R n×n is the identity matrix and ε ∈ R + , which is a quadratic problem with linear positive semidefinite constraints (see [5]). The choice of the Frobenius norm is somewhat arbitrary, and a more encompassing approach consists of considering the vector problem…”

Section: Examplesmentioning

confidence: 99%

“…On the other hand, the situation is simpler for weak Pareto minimization, i.e. conditions (2)- (5). (3) and (5) are immediately satisfied because g i (x) = 0 (1 ≤ i ≤ 4).…”

Section: Examplesmentioning

confidence: 99%

On First Order Optimality Conditions for Vector Optimization

Drummond

Iusem

Svaiter

2003

Acta Mathematicae Applicatae Sinica, English Series, English Se

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We develop first order optimality conditions for constrained vector optimization. The partial orders for the objective and the constraints are induced by closed and convex cones with nonempty interior. After presenting some well known existence results for these problems, based on a scalarization approach, we establish necessity of the optimality conditions under a Slater-like constraint qualification, and then sufficiency for the K-convex case. We present two alternative sets of optimality conditions, with the same properties in connection with necessity and sufficiency, but which are different with respect to the dimension of the spaces to which the dual multipliers belong. We introduce a duality scheme, with a point-to-set dual objective, for which strong duality holds. Some examples and open problems for future research are also presented.

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“…In Dykstra's method, as in many other iterative projection schemes, it is assumed that the projections onto each of the individual sets i are relatively simple to compute, e. g., boxes, balls, half-spaces, and subspaces. 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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Dykstra's Algorithm for a Constrained Least‐squares Matrix Problem

Cited by 9 publications

References 0 publications

Improving clustering by learning a bi-stochastic data similarity matrix

Improving clustering by learning a bi-stochastic data similarity matrix

On First Order Optimality Conditions for Vector Optimization

An acceleration scheme for Dykstra’s algorithm

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