The nearest ‘doubly stochastic’ matrix to a real matrix with the same first moment

Glunt, W.; Hayden, T. L.; Reams, Robert

doi:10.1002/(sici)1099-1506(199811/12)5:6<475::aid-nla155>3.0.co;2-5

Cited by 18 publications

(17 citation statements)

References 7 publications

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“…Xu and Zikatonov discuss how alternating projection can be used to solve the linear systems that arise in the discretization of partial differential equations [80]. In the matrix analysis community, alternating projection has been used as a computational method for solving IEPs [35], [37] and for solving matrix nearness problems [36], [103]. In statistics, one may view the expectation maximization (EM) algorithm as an alternating projection [104].…”

Section: G Literature On Alternating Projectionmentioning

confidence: 99%

Designing structured tight frames via an alternating projection method

Tropp

Dhillon

Heath

et al. 2005

IEEE Trans. Inform. Theory

426

315

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Abstract-Tight frames, also known as general Welch-boundequality sequences, generalize orthonormal systems. Numerous applications-including communications, coding, and sparse approximation-require finite-dimensional tight frames that possess additional structural properties. This paper proposes an alternating projection method that is versatile enough to solve a huge class of inverse eigenvalue problems (IEPs), which includes the frame design problem. To apply this method, one needs only to solve a matrix nearness problem that arises naturally from the design specifications. Therefore, it is the fast and easy to develop versions of the algorithm that target new design problems. Alternating projection will often succeed even if algebraic constructions are unavailable.To demonstrate that alternating projection is an effective tool for frame design, the paper studies some important structural properties in detail. First, it addresses the most basic design problem: constructing tight frames with prescribed vector norms. Then, it discusses equiangular tight frames, which are natural dictionaries for sparse approximation. Finally, it examines tight frames whose individual vectors have low peak-to-average-power ratio (PAR), which is a valuable property for code-division multiple-access (CDMA) applications. Numerical experiments show that the proposed algorithm succeeds in each of these three cases. The appendices investigate the convergence properties of the algorithm.

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Section: G Literature On Alternating Projectionmentioning

confidence: 99%

Designing structured tight frames via an alternating projection method

Tropp

Dhillon

Heath

et al. 2005

IEEE Trans. Inform. Theory

426

315

View full text Add to dashboard Cite

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“…Although Theorem 1 and Corollary 2 resemble results proved in [3] and [6], the results herein present a different formulation. We include them for clarity and because we will use them later.…”

Section: The Closest Generalized Doubly Stochastic Matrixmentioning

confidence: 62%

“…The motivation for this problem comes from an application in [2] where it is desired to approximate a certain matrix T , where T comes from a linear system corresponding to a large linear network, subject to the approximating matrix satisfying certain conditions. We outlined these applications in [3] and (among other things there) used a computational algorithm to find the closest matrix A to T , subject to A being generalized doubly stochastic and M 1 , or subject to A being doubly stochastic and M 1 . See the references [2] and [3] for more details about the applications.…”

Section: Introductionmentioning

confidence: 99%

“…We outlined these applications in [3] and (among other things there) used a computational algorithm to find the closest matrix A to T , subject to A being generalized doubly stochastic and M 1 , or subject to A being doubly stochastic and M 1 . See the references [2] and [3] for more details about the applications. Our extended work here takes only an analytic approach to the problem, includes the second moment condition, and explicitly finds the closest matrix which is generalized doubly stochastic, M 1 and M 2 , having dropped the requirement that the closest matrix be nonnegative.…”

Section: Introductionmentioning

confidence: 99%

“…It is worth emphasizing that despite dropping the nonnegativity requirement our solution is still relevant to the original problem. Previous approaches to this problem, in both [3] and [8], did not include the second moment, although they did include the nonnegative condition. A survey of matrix nearness problems and their applications, which include areas of control theory, numerical analysis and statistics, was given by Higham [4], see also [7].…”

Section: Introductionmentioning

confidence: 99%

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The nearest generalized doubly stochastic matrix to a real matrix with the same firstand second moments

2008

Self Cite

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Abstract. Let T be an arbitrary n × n matrix with real entries. We explicitly find the closest (in Frobenius norm) matrix A to T , where A is n × n with real entries, subject to the condition that A is "generalized doubly stochastic" (i.e. Ae = e and e T A = e T , where e = (1, 1, . . . , 1) T , although A is not necessarily nonnegative) and A has the same first moment as T (i.e. e T 1 Ae 1 = e T 1 T e 1 ). We also explicitly find the closest matrix A to T when A is generalized doubly stochastic has the same first moment as T and the same second moment as T (i.e. e T 1 A 2 e 1 = e T 1 T 2 e 1 ), when such a matrix A exists.Mathematical subject classification: 15A51, 65K05, 90C25.

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Introduction

Cegielski

2012

Lecture Notes in Mathematics

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The nearest ‘doubly stochastic’ matrix to a real matrix with the same first moment

Cited by 18 publications

References 7 publications

Designing structured tight frames via an alternating projection method

Designing structured tight frames via an alternating projection method

The nearest generalized doubly stochastic matrix to a real matrix with the same firstand second moments

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