M.W. TROSSET. Computing distances between convex sets and subsets of the positive semidefinite matrices. Technical report, Rice Unversity, Houston, Texas, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....for EDMCP appears in [41] The point is made that there is no definitive general algorithm for EDMCP, i.e. one cannot provide an efficient decision rule for the question of whether a completion exists or not. However, there are many algorithms that find approximate completions. In [41] 39] [40], the author presents results on finding EDM completions based on spectral decompositions. In particular, the computationally hard problem of fixing the rank (the embedding dimension) is discussed. Some work on finding the closest EDM to a given symmetric matrix appears in [12] 47] 2] Another ....

M.W. TROSSET. Computing distances between convex sets and subsets of the positive semidefinite matrices. Technical report Rice Unversity Houston Texas 1997.

.... problems was proposed by Trosset [40] who briefly noted its relevance to molecular conformation problems, on which he subsequently focussed in [38] The same author proposed an analogous extension to order constraints in [41] and unified the mathematical theory common to these extensions in [39]. Question 1 Suppose that we knew all m = n(n Gamma 1) 2 interatomic distances, stored in the n Theta n distance matrix Delta 0 . Can we determine a configuration matrix X such that D(X) Delta 0 A constructive answer to Question 1 is contained in a famous embedding theorem from ....

....as the basic function of the molecular graphics program for building a crystallographic model from an electron density map. 3.2 A Limited Memory Algorithm Problem (7) is a nonlinear optimization problem with simple bound constraints. The objective function F p ffi was studied by Trosset [39], who established the following result: Theorem 3 Let T be a symmetric matrix with eigenvalues 1 (T ) Delta Delta Delta n (T ) The function F p is continuously differentiable at T , unless p (T ) p 1 (T ) 0. To compute the gradient of F p ffi , we write ( Delta) ffi i 1 j 1 ; ....

[Article contains additional citation context not shown here]

M. W. Trosset. Computing Distances Between Convex Sets and Subsets of the Positive Semidefinite Matrices. Technical Report 97-3, Department of Computational & Applied Mathematics---MS 134, Rice University, Houston, TX 77005-1892, 1997. Submitted for publication.

....for EDMCP appears in [41] The point is made that there is no definitive general algorithm for EDMCP, i.e. one cannot provide an efficient decision rule for the question of whether a completion exists or not. However, there are many algorithms that find approximate completions. In [41] 39] [40], the author presents results on finding EDM completions based on spectral decompositions. In particular, the computationally hard problem of fixing the rank (the embedding dimension) is discussed. Some work on finding the closest EDM to a given symmetric matrix appears in [12] 47] 2] Another ....

M.W. TROSSET. Computing distances between convex sets and subsets of the positive semidefinite matrices. Technical report, Rice Unversity, Houston, Texas, 1997.

.... induced by the inner product hA; Bi F = trace(A B) Then classical MDS can be de ned by the optimization problem minimize kB 1 ( 2 )k subject to B n (p) 1) This formulation of classical MDS has been explicitly discussed by Mardia [16] by de Leeuw and Heiser [2] and by Trosset [23, 22]. The objective function is sometimes called the strain criterion. The following explicit solution to Problem (1) is well known: 3 Theorem 2 (Classical MDS) Given , let 1 n denote the eigenvalues of B = 1 ( 2 ) and let v 1 ; v n denote the corresponding eigenvectors. ....

M. W. Trosset. Computing distances between convex sets and subsets of the positive semide - nite matrices. Technical Report 97-3, Department of Computational & Applied Mathematics| MS 134, Rice University, 6100 Main Street, Houston, TX 77005-1892, 1997.

....minimum is zero if and only if completion is possible. The optimization problem is derived in a very natural way from an embedding theorem in classical distance geometry and from the classical approach to multidimensional scaling. It belongs to a general family of problems studied by Trosset [13] and can be formulated as a nonlinear programming problem with simple bound constraints. Thus, this approach provides a constructive technique for obtaining approximate solutions to a general class of distance matrix completion problems. Key words: Euclidean distance matrices, positive ....

....We exploit these results in Section 3 and pose the distance matrix completion problem as an optimization problem. In Section 4 we discuss a reformulation of our optimization problem and some methods available for solving it. The theoretical framework that we describe was established by Trosset [13]; our notation is intended to facilitate comparison with that report. Section 5 presents some examples and Section 6 concludes. 2 Embedding We begin by introducing the embedding problem of classical distance geometry. More detailed discussions of the material in this section, together with ....

[Article contains additional citation context not shown here]

M. W. Trosset. Computing Distances Between Convex Sets and Subsets of the Positive Semidefinite Matrices. Technical Report 97-3, Department of Computational & Applied Mathematics---MS 134, Rice University, Houston, TX 77005-1892, 1997. Submitted for publication.

....purpose of solving distance matrix completion problems was proposed by Trosset [19] who briefly noted its relevance to molecular conformation problems. The same author proposed an analogous extension to order constraints in [20] and unified the mathematical theory common to these extensions in [18]. 3.1 Embedding Question 1 Suppose that we knew all m = n(n Gamma 1) 2 interatomic distances, stored in the n Theta n distance matrix Delta 0 . Can we determine a configuration matrix X such that D(X) Delta 0 A constructive answer to Question 1 is contained in a famous embedding ....

....(one for each atomic coordinate, less six to remove translational and rotational indeterminancy) However, certain advantages appear to accrue from allowing each (squared) interatomic dissimilarity to vary independently of the others. The objective function F p ffi was studied by Trosset [18], who established the following result: Theorem 3 Let T be a symmetric matrix with eigenvalues 1 (T ) Delta Delta Delta n (T ) The function F p is continuously differentiable at T , unless p (T ) p 1 (T ) 0. To compute the gradient of F p ffi , we write ( Delta) ffi i 1 j1 ....

[Article contains additional citation context not shown here]

M. W. Trosset. Computing Distances Between Convex Sets and Subsets of the Positive Semidefinite Matrices. Technical Report 97-3, Department of Computational & Applied Mathematics---MS 134, Rice University, Houston, TX 77005-1892, 1997. Submitted for publication.

.... in his pioneering paper was the formulation of (metric) MDS as the optimization problem minimize kB Gamma 1 ( Delta Delta)k 2 F subject to B 2 Omega n (p) 3) This formulation of classical MDS has been explicitly discussed by Mardia [12] by de Leeuw and Heiser [2] and by Trosset [21, 20]. The following representation of the global solution to Problem (3) is well known: Theorem 1 Let 1 Delta Delta Delta n denote the eigenvalues of B 0 = 1 ( Delta Delta) and let v 1 ; vn denote corresponding eigenvectors. Let i = max( i ; 0) for i = 1; p and i = ....

M. W. Trosset. Computing Distances Between Convex Sets and Subsets of the Positive Semidefinite Matrices. Technical Report 97-3, Department of Computational & Applied Mathematics---MS 134, Rice University, Houston, TX 77005-1892, 1997.

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M.W. TROSSET. Computing distances between convex sets and subsets of the positive semidefinite matrices. Technical report, Rice Unversity, Houston, Texas, 1997.

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Trosset, M.W., Computing distances between convex sets and subsets of the positive semidefinite matrices. Tech. Report 97-3, Dept. Comp. & Appl. Math., Rice Univ., Houston, TX (1997).

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M.W. TROSSET. Computing distances between convex sets and subsets of the positive semidefinite matrices. Technical report, Rice University, Houston, Texas, 1997.

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