I.J. SCHOENBERG. Remarks to Maurice Frechet's article: Sur la definition axiomatique d'une classe d'espaces vectoriels distancies applicables vectoriellement sur l'espace de Hilbert. Ann. Math., 36:724--732, 1935.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....[9] 31] multidimensional scaling and multivariate analysis problems in statistics [25] 26] genetics, geography, and others, 1] Many of these applications require a low embedding dimension, e.g. r = 3. Theoretical properties of EDMs can be found in e.g. 7] 11] 15] 16] 21] 24] [34]. This includes characterizations as well as graph theoretic conditions for existence of completions. More information can be found in the recent survey article by Laurent [24] Generalizations of EDM arise in [36] An interesting discussion on algorithms for EDMCP appears in [41] The point is ....

....the problem. In Section 4 we derive the algorithm and the Slater constraint qualification result. We conclude with several remarks and numerical tests in Section 5. In addition, we include Section 5.1 with some technical details on the SDP algorithm. 2. DISTANCE GEOMETRY It is well known, e.g. [34], 15] 16] 38] that a pre distance matrix D is a EDM if and only if D is negative semidefinite on the orthogonal complement of e, where e is the vector of all ones. Thus the set of all EDMs is a convex cone, which we denote by . We exploit this result to translate the cone to the cone of ....

[Article contains additional citation context not shown here]

I.J. SCHOENBERG. Remarks to Maurice Frechet's article: Sur la definition axiomatique d'une classe d'espaces vectoriels distancies applicables vectoriellement sur l'espace de Hilbert. Ann. Math., 36:724-732, 1935.

....only if f(D ) is zero. Furthermore, 1) can be posed as a semidefinite programming problem be exploiting the relation between the cone E and P , the cone of positive semidefinite matrices. This relation is established in the following subsection. 2. 1 Distance Geometry It is well known, e.g. [21, 7, 9, 22], that a symmetric matrix D with nonnegative elements and with zero diagonal is a EDM if and only if D is negative semidefinite on M : n x 2 n : x T e = 0 o ; the orthogonal complement of e, the vector of all ones. Let S n denote the space of symmetric matrices of order n. Define the ....

....B. If B = XX T for some n Theta r matrix X , then the rows of X are the coordinates of the points x 1 ; x 2 ; x n that generate D. Furthermore, since Be = 0; it follows that the origin coincides with the centroid of these points. For these and other basic results on EDM see e.g. [8, 9, 10, 11, 21]. Let V be an n Theta (n Gamma 1) matrix such that V T e = 0 ; V T V = I ; 6) where I 2 S n Gamma1 is the identity matrix. Hence, the subspace M can be represented as the range of V . J , the orthogonal projection matrix onto M , is then given by J : V V T = I Gamma ee T n . Now ....

I.J. SCHOENBERG. Remarks to Maurice Frechet's article: Sur la definition axiomatique d'une classe d'espaces vectoriels distancies applicables vectoriellement sur l'espace de Hilbert. Ann. Math., 36:724--732, 1935.

.... [9] 31] multidimensional scaling and multivariate analysis problems in statistics [25] 26] genetics, geography, and others, 1] Many of these applications require a low embedding dimension, e.g. r = 3: Theoretical properties of EDMs can be found in e.g. 7] 11] 15] 16] 21] 24] [34]. This includes characterizations as well as graph theoretic conditions for existence of completions. More information can be found in the recent survey article by Laurent [24] Generalizations of EDM arise in [36] An interesting discussion on algorithms for EDMCP appears in [41] The point is ....

....the problem. In Section 4 we derive the algorithm and the Slater constraint qualification result. We conclude with several remarks and numerical tests in Section 5. In addition, we include Section 5.1 with some technical details on the SDP algorithm. 2. DISTANCE GEOMETRY It is well known, e.g. [34], 15] 16] 38] that a pre distance matrix D is a EDM if and only if D is negative semidefinite on M : Phi x 2 n : x t e = 0 Psi ; the orthogonal complement of e, where e is the vector of all ones. Thus the set of all EDMs is a convex cone, which we denote by E : We exploit this ....

[Article contains additional citation context not shown here]

I.J. SCHOENBERG. Remarks to Maurice Frechet's article: Sur la definition axiomatique d'une classe d'espaces vectoriels distancies applicables vectoriellement sur l'espace de Hilbert. Ann. Math., 36:724--732, 1935.

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I.J. SCHOENBERG. Remarks to Maurice Frechet's article: Sur la definition axiomatique d'une classe d'espaces vectoriels distancies applicables vectoriellement sur l'espace de Hilbert. Ann. Math., 36:724--732, 1935.

No context found.

I.J. SCHOENBERG. Remarks to Maurice Frechet's article: Sur la definition axiomatique d'une classe d'espaces vectoriels distancies applicables vectoriellement sur l'espace de Hilbert. Ann. Math., 36:724--732, 1935.

No context found.

I.J. SCHOENBERG. Remarks to Maurice Frechet's article: Sur la definition axiomatique d'une classe d'espaces vectoriels distancies applicables vectoriellement sur l'espace de Hilbert. Ann. Math., 36:724--732, 1935.

No context found.

I. J. Schoenberg, Remarks to Maurice Frechet's article: Sur la definition axiomatique d'une classe d'espaces vectoriels distancies applicables vectoriellement sur l'espace de Hilbert, Ann. Math., 36 (1935), pp. 724-- 732.

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I.J. SCHOENBERG. Remarks to Maurice Frechet's article: Sur la definition axiomatique d'une classe d'espaces vectoriels distancies applicables vectoriellement sur l'espace de Hilbert. Ann. Math., 36:724--732, 1935.

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