Pablo Tarazaga

The cone of distance matrices

Abstract The geometry of the cone of Euclidean distance matrices (EDMs) is analyzed using a new characterization of an EDM. The facial structure and the angle that EDMs of embedding dimension one make with the center ray are found. This result follows from a complete analysis of the critical points of the distance function in Frobenius norm from the matrix E consisting of zero diagonal and ones elsewhere to the EDMs of embedding dimension one.

Circum-Euclidean distance matrices and faces

Abstract We study the structure of circum-Euclidean distance matrices, those Euclidean distance matrices generated by points lying on a hypersphere. We show, for example, that such Euclidean distance matrices are characterized as having constant row sums and they constitute the interior of the cone of all Euclidean distance matrices. Also, we provide a formula for computing the radius of a representing configuration in the smallest embedding dimension r and show that rk D = r + 1. Finally we obtain a geometric characterization of the faces of this cone. Given a configuration of points and its Euclidean distance matrix D , any matrix in the minimal face containing D comes from a configuration that is a linear perturbation of the points that generate D .

Connections between the real positive semidefinite and distance matrix completion problems

Abstract Though the real symmetric positive semidefinite (PSD) matrices and the Euclidean distance matrices are closely related, exact relationships between the corresponding completion problems are not apparent. We establish strong partial relationships of two types. This permits the transfer of some insights from one problem to the other and allows computational tools for the PSD problem to be used for the distance problem.

Perron–Frobenius theorem for matrices with some negative entries

Abstract We extend the classical Perron–Frobenius theorem to matrices with some negative entries. We study the cone of matrices that has the matrix of 1s ( ee t ) as the central ray, and such that any matrix inside the cone possesses a Perron–Frobenius eigenpair. We also find the maximum angle of any matrix with ee t that guarantees this property.

Distance matrices and regular figures

Abstract A regular figure (which includes all regular polygons) is a set of points on a hypersphere whose center coincides with their centroid. We characterize all regular figures as those whose points generate a Euclidean distance matrix (EDM) with eigenvector e , the vector of all ones. Restricting the classical maps of Schoenberg, Gower, and Critchley for all EDMs to the subcone of EDMs with eigenvector e yields new geometrical information about the generating points and a simple formula for the radius of the hypersphere.

POSITIVE DEFINITE COMPLETIONS AND DETERMINANT MAXIMIZATION

Abstract A method is described for determining whether a positive definite completion of a given partial Hermitian matrix exists and, if so, for finding the determinant maximizing positive definite completion. No assumption is made about the arrangement of the specified entries. The method employs iterative application to a modified problem of an explicit formula for the maximum determinant in case there is only one symmetrically placed pair of unspecified entries. A robust and fast algorithm based upon this approach is shown to have global convergence to the necessarily unique solution.

Computing the Nearest Diagonally Dominant Matrix

We solve the problem of minimizing the distance from a given matrix to the set of symmetric and diagonally dominant matrices. First, we characterize the projection onto the cone of diagonally dominant matrices with positive diagonal, and then we apply Dykstra’s alternating projection algorithm on this cone and on the subspace of symmetric matrices to obtain the solution. We discuss implementation details and present encouraging preliminary numerical results.

Primal and polar approach for computing the symmetric diagonally dominant projection

We solve the problem of minimizing the distance from a given matrix to the cone of symmetric and diagonally dominant matrices with positive diagonal (SDD+). Using the extreme rays of the polar cone we project onto the supporting hyperplanes of the faces of SDD+ and then, applying the cyclic Dykstras algorithm, we solve the problem. Similarly, using the extreme rays of SDD+ we characterize the projection onto the polar cone, which also allows us to obtain the projection onto SDD+ by means of the orthogonal decomposition theorem for convex cones. In both cases the symmetry and the sparsity pattern of the given matrix are preserved. Preliminary numerical experiments indicate that the polar approach is a promising one. Copyright

Block matrices and multispherical structure of distance matrices

Abstract The block structure of a matrix and its relation to the block structure of the corresponding eigenvectors is investigated. A set of points is said to have multispherical structure if they lie on a collection of concentric spheres. When the centroid of each of the clusters lies at the common center, the associated distance matrix has a block structure with simple relations between the blocks. Further, such block structure may be recognized from the structure of the eigenvectors of the distance matrix. A computational procedure is proposed to find the least number of concentric spheres containing the points represented by a distance matrix.

Euclidean distance matrices: new characterization and boundary properties

In this article we give a new characterization of Euclidean distance matrices using known necessary conditions. We also relate this characterization with the faces of the cone and give new properties for the boundary. Finally, a new characterization of spherical/non-spherical matrices is proposed.