F. Critchley. On certain linear mappings between inner-product and squared distance matrices. Linear Algebra Appl., 105:91-107, 1988. Home/Search Document Not in Database Summary Related Articles Check |
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Better Initial Configurations for Metric Multidimensional.. - Malone, Tarazaga, Trosset (2001) (Correct)
....transformation s de ned by s (D) 1 I es D I es is an inverse of . Johnson and Tarazaga [11] demonstrated that the image of s is a face of the polyhedral cone n (n) The inverse transformation 1 obtained by setting s = e=n was introduced by Torgerson [21] See Critchley [1] for a detailed study of the properties of and 1 . Implicit in Torgerson s [21] and Gower s [5] pioneering studies of metric MDS was an optimization problem. Given a dissimilarity matrix = ij ] let r denote the matrix with entries ( ij ) Let k k F denote the Frobenius norm, ....
F. Critchley. On certain linear mappings between inner-product and squared-distance matrices. Linear Algebra and Its Applications, 105:91-107, 1988.
Computational Surface Flattening: A Voxel-based Approach - Grossmann, Kiryati, Kimmel (2002) (Correct)
.... kD(X) D(X) Gamma Delta Deltak 2 F where X is the n Theta 2 matrix of point coordinates in 2 , D(X) is a n Theta n matrix of Euclidean distances between the points in 2 and k Delta k F denotes the Frobenius norm, i.e. kAk 2 F = trace(A 0 A) The double centering operator [4] is defined as: T(A) Gamma 1 2 JAJ : Here A is a square n Theta n matrix and J = I Gamma 1 n ee 0 , where e = 1; 1) 0 is an n vector of ones and I is the n Theta n identity matrix. The double centering operator is a linear operator on square matrices. Note that T(A) is ....
F. Critchley, "On Certain Linear Mappings Between Inner-Product and Squared-Distance Matrices", Linear Algebra and its Applications, Vol. 105, pp. 91-107, 1988.
Optimal Dilations for Metric Multidimensional Scaling - Malone, Trosset (2000) (Correct)
....In fact, for any s 2 p such that s 0 e = 1, the linear transformation s defined by s (D) Gamma 1 2 (I Gamma es 0 ) D (I Gamma es 0 ) is an inverse of , where I denotes the n Theta n identity matrix. We are interested in the inverse 1 obtained by setting s = e=n. See Critchley (1988) for a detailed study of the properties of and 1 . Let k Delta k denote the Frobenius norm. Classical MDS can be defined by the optimization problem minimize kB Gamma 1 ( Delta 2 )k 2 subject to B 2 Omega n (p) 1) which is implicit in Torgerson (1952) The objective function was ....
Critchley, F. (1988). On certain linear mappings between inner-product and squared-distance matrices. Linear Algebra and Its Applications, 105:91-- 107.
A Study of the Stationary Configurations of the SStress.. - Samuel Malone Michael (Correct)
....s 2 p such that s 0 e = 1, the linear transformation s defined by s (D) Gamma 1 2 Gamma I Gamma es 0 Delta D Gamma I Gamma es 0 Delta is an inverse of , where I denotes the n Theta n identity matrix. We are interested in the inverse 1 obtained by setting s = e=n. See Critchley (1988) for a detailed study of the properties of and 1 . Classical MDS can be defined by the optimization problem minimize kB Gamma 1 ( Delta 2 )k 2 subject to B 2 Omega n (p) 11) which is implicit in Torgerson (1952) The objective function was subsequently dubbed the strain criterion. The ....
Critchley, F. (1988). On certain linear mappings between inner-product and squared-distance matrices. Linear Algebra and Its Applications, 105:91--107.
On the Embeddability of Weighted Graphs in Euclidean Spaces - Alfakih, Wolkowicz (1998) (3 citations) (Correct)
....complement of e, the vector of all ones. Let S n denote the space of symmetric matrices of order n. Define the centered and hollow subspaces of S n as SC : fB 2 S n : Be = 0g; SH : fD 2 S n : diag(D) 0g; 2) where diag(D) denotes the column vector formed from the diagonal of D. Following [5], define the two linear operators acting on S n K(B) diag(B) e T e diag(B) T Gamma 2B; 3) and T (D) Gamma 1 2 JDJ; 4) where J = I Gamma ee T n is the orthogonal projection matrix onto subspace M . Theorem 2.1 The linear operators satisfy K(SC ) SH ; T (S H ) SC ; and ....
F. CRITCHLEY. On certain linear mappings between inner-product and squared distance matrices. Linear Algebra Appl., 105:91--107, 1988.
Computing Distances Between Convex Sets and Subsets of the.. - Trosset (1997) (5 citations) (Correct)
....are 1. Then double centering can also be represented by the matrix equation (A) Gamma 1 2 I n Gamma 1 n 1 n 1 0 n A I n Gamma 1 n 1 n 1 0 n : 1) Notice that (A) is symmetric if A is symmetric. A detailed study of and related mappings was made by Critchley [3]. Let Omega n denote the set of symmetric positive semidefinite n Theta n matrices. Let Omega n (p) denote the matrices in Omega n whose rank is no greater than p. We now state the embedding theorem on which our work is based: Theorem 1 Let Delta be an n Theta n dissimilarity matrix. Then ....
F. Critchley. On certain linear mappings between inner-product and squared-distance matrices. Linear Algebra and Its Applications, 105:91--107, 1988.
Measure aspects of cut polyhedra: l_1-embeddability and.. - Deza, Laurent (1993) (Correct)
....if one uses relation (3.7) for computing p(x; x 0 ) then one obtains that p(x; x 0 ) 0 for all x 2 X . This explains why we consider p as being defined only on the pairs of elements from X n fx 0 g. The covariance mapping appeared in many different areas of mathematics. See, for instance, [Cri88], CP93] where, for a metric space (X; d) and its image p = d) the quantity p(x; y) is known as the Gromov product of x; y 2 X n fx 0 g) Fic87] where it is called a linear generalized similarity function) The connection between cut and correlation polyhedra, which is formulated in (3.6) ....
F. Critchley. On certain linear mappings between inner-product and squareddistance matrices. Linear Algebra and its Applications, 105:91--107, 1988.
Solving Euclidean Distance Matrix Completion Problems Via.. - Alfakih, al. (1997) (11 citations) (Correct)
....ee t n (3) EUCLIDEAN DISTANCE MATRIX COMPLETION PROBLEMS 5 is the orthogonal projection onto M . Now define the centered and hollow subspaces of Sn SC : fB 2 Sn : Be = 0g; SH : fD 2 Sn : diag(D) 0g; 4) where diag(D) denotes the column vector formed from the diagonal of D. Following [10], we define the two linear operators acting on Sn K(B) diag(B) e t e diag(B) t Gamma 2B; 5) and T (D) Gamma 1 2 JDJ: 6) The operator Gamma2T is an orthogonal projection onto SC ; thus it is a self adjoint idempotent. Theorem 1 The linear operators satisfy K(SC ) SH ; T (SH ....
F. CRITCHLEY. On certain linear mappings between inner-product and squared distance matrices. Linear Algebra Appl., 105:91--107, 1988.
An Approximate Solution to the Metric SSTRESS Problem in.. - Tarazaga, Trosset (1998) (Correct)
....0 . Given an n Theta n matrix B, let b denote the n vector diag(B) Define a linear function : Omega n (n) n (n) by (B) be 0 eb 0 Gamma 2B: Then D 2 n (p) if and only if there exists B 2 Omega n (p) for which (B) D. The linear function was extensively analyzed by Critchley [1]. The linear function does not have a unique inverse. The null space of is the set of all matrices of the form e 0 x xe 0 for some x 2 n . Let I denote the n Theta n identity matrix. Given any s 2 p such that s 0 e = 1, the linear function s defined by s (D) Gamma 1 2 ....
F. Critchley. On certain linear mappings between inner-product and squared-distance matrices. Linear Algebra and Its Applications, 105:91--107, 1988.
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F. Critchley. On certain linear mappings between inner-product and squared distance matrices. Linear Algebra Appl., 105:91-107, 1988.
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F. CRITCHLEY. On certain linear mappings between inner-product and squared distance matrices. Linear Algebra Appl., 105:91--107, 1988.
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F. Critchley, On certain linear mappings between innerproduct and squared distance matrices, Linear Algebra Appl., 105 (1998), pp. 91--107.
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Critchley, F. (1988). On certain linear mappings between inner-product and squared-distance matrices. Linear Algebra and Its Applications, 105:91-107.
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F. CRITCHLEY. On certain linear mappings between inner-product and squared distance matrices. Linear Algebra Appl., 105:91--107, 1988.
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