A Sequential Semismooth Newton Method for the Nearest Low-rank Correlation Matrix Problem

Li, Qingna; Qi, Houduo

doi:10.1137/090771181

Cited by 40 publications

(38 citation statements)

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“…We next establish a global convergence result regarding the outer iterations of the above method for solving problem (17). After that, we will study the convergence of its inner iterations.…”

Section: Endmentioning

confidence: 99%

Penalty decomposition methods for rank minimization

Zhang

2014

Optimization Methods and Software

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In this paper we consider general rank minimization problems with rank appearing in either objective function or constraint. We first show that a class of matrix optimization problems can be solved as lower dimensional vector optimization problems. As a consequence, we establish that a class of rank minimization problems have closed form solutions. Using this result, we then propose penalty decomposition methods for general rank minimization problems in which each subproblem is solved by a block coordinate descend method. Under some suitable assumptions, we show that any accumulation point of the sequence generated by our method when applied to the rank constrained minimization problem is a stationary point of a nonlinear reformulation of the problem. Finally, we test the performance of our methods by applying them to matrix completion and nearest low-rank correlation matrix problems. The computational results demonstrate that our methods generally outperform the existing methods in terms of solution quality and/or speed.

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Section: Endmentioning

confidence: 99%

Penalty decomposition methods for rank minimization

Zhang

2014

Optimization Methods and Software

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“…This has led us to use ∂F (y) instead (to be developed in part (c) in this subsection and also see (41)). The other issue is how to solve the linear equation in (29). Direct evaluation of V would need O(n 4 ) flops and hence direct methods are very expensive.…”

Section: ])mentioning

confidence: 99%

“…Optimization with spherical constraints has recently attracted much attention of researchers, see, e.g., [32,31,16,17,29,55] and the references therein. Such a problem can be cast as a more general optimization problem over the Stiefel manifold [52,25].…”

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confidence: 99%

“…Such a problem can be cast as a more general optimization problem over the Stiefel manifold [52,25]. One important example is the nearest low-rank correlation matrix problem, where the unit diagonals of the correlation matrix yields the spherical constraints [17,29,52,25]. It is noted that the sequential second-order methods in [17,29] as well as the feasibility-preserving methods in [52,25] all rely on the fact that the radius is known (e.g., R = 1).…”

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confidence: 99%

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Constrained Best Euclidean Distance Embedding on a Sphere: A Matrix Optimization Approach

Bai

Xiu³

2015

SIAM J. Optim.

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Abstract. The problem of data representation on a sphere of unknown radius arises from various disciplines such as Statistics (spatial data representation), Psychology (constrained multidimensional scaling), and Computer Science (machine learning and pattern recognition). The best representation often needs to minimize a distance function of the data on a sphere as well as to satisfy some Euclidean distance constraints. It is those spherical and Euclidean distance constraints that present an enormous challenge to the existing algorithms. In this paper, we reformulate the problem as an Euclidean distance matrix optimization problem with a low rank constraint. We then propose an iterative algorithm that uses a quadratically convergent Newton-CG method at its each step. We study fundamental issues including constraint nondegeneracy and the nonsingularity of generalized Jacobian that ensure the quadratic convergence of the Newton method. We use some classic examples from the spherical multidimensional scaling to demonstrate the flexibility of the algorithm in incorporating various constraints. We also present an interesting application to the circle fitting problem.Key words. Euclidean distance matrix, Matrix optimization, Lagrangian duality, Spherical multidimensional scaling, Semismooth Newton-CG method.AMS subject classifications. 49M45, 90C25, 90C331. Introduction. In this paper, we are mainly concerned with placing n points {x 1 , . . . , x n } in a best way on a sphere in IR r . The primary information that we use is an incomplete/complete set of pairwise Euclidean distances (often with noises) among the n points. In such a setting, IR r is often a low-dimensional space (e.g., r takes 2 or 3 for data visualization) and is known as the embedding space. The center of the sphere is unknown. For some applications, the center can be put at origin in IR r . Furthermore, the radius of the sphere is also unknown. In our matrix optimization formulation of the problem, we treat both the center and the radius as unknown variables. We develop a fast numerical method for this problem and present a few of interesting applications taken from existing literature.The problem described above has long appeared in the constrained Multi-Dimensional Scaling (MDS) when r ≤ 3, which is mainly for the purpose of data visualization, see [9, Sect. 4.6] and [4, Sect. 10.3] for more details. In particular, it is known as the spherical MDS when r = 3 and the circular MDS when r = 2. Most numerical methods in this part took advantages of r being 2 or 3. For example, two of the earliest circular MDS were by Borg and Lingoes [5] and Lee and Bentler [28], where they introduced a new point x 0 ∈ IR r as the center of the sphere (i.e., circles in their case) and further forced the following constraints to hold:

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“…Here and in the sequel, S n denotes the space of all n × n real symmetric matrices, S n + is the semidefinite matrix cone consisting of all positive semidefinite matrices in S n and rank(X) is the rank of X which is the number of all nonzero eigenvalues of X. Problem (P ) has gained plenty of recent attention in both mathematical ELA 684 Z. Luo and N. Xiu and engineering fields, owing to its wide applications in system control [3,15,16,18], statistics [4,13,20], network localization [10], econometrics, signal processing, quantum information, and many others [2].…”

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confidence: 99%

Exact relaxation for the semidefinite matrix rank minimization problem with extended Lyapunov equation constraint

Luo

Xiu

2014

ELA

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Abstract. The semidefinite matrix rank minimization, which has a broad range of applications in system control, statistics, network localization, econometrics and so on, is computationally NPhard in general due to the noncontinuous and non-convex rank function. A natural way to handle this type of problems is to substitute the rank function into some tractable surrogates, most popular ones of which include the convex trace norm and the non-convex Schatten p-norm relaxations with p ∈ (0, 1). The corresponding exactness of these relaxations have absorbed great attention and interest from researchers both in mathematics and engineering fields. In this paper, a special semidefinite matrix rank minimization problem with the extended Lyapunov equation constraint arising from low-order optimal control is considered and shown to possess the desired exact relaxation properties by exploiting the special structures of the involved linear transformation and by developing some essential properties and features on rank function and the semidefinite matrix cone.

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A Sequential Semismooth Newton Method for the Nearest Low-rank Correlation Matrix Problem

Cited by 40 publications

References 35 publications

Penalty decomposition methods for rank minimization

Penalty decomposition methods for rank minimization

Constrained Best Euclidean Distance Embedding on a Sphere: A Matrix Optimization Approach

Exact relaxation for the semidefinite matrix rank minimization problem with extended Lyapunov equation constraint

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