In this paper, we use known bounds on the smallest eigenvalue of a symmetric matrix and Schoenberg’s Theorem to provide both necessary as well as sufficient trace inequalities that guarantee a matrix D is a Euclidean distance matrix, EDM. We also provide necessary and sufficient trace inequalities that guarantee a matrix D is an EDM generated by a regular figure.

Consider the problem of determining whether or not a partial dissimilarity matrix can be completed to a Euclidean distance matrix. The dimension of the distance matrix may be restricted and the known dissimilarities may be permitted to vary subject to bound constraints. This problem can be formulated as an optimization problem for which the global minimum is zero if and only if completion is po...

Euclidean Distance Matrix Analysis (EDMA) of form is a coordinate free approach to the analysis o f form using landmark data. In this paper, the problem of estimation of mean form, variancecovariance matrix, and mean form difference under the Gaussian perturbation model is considered using EDMA. The suggested estimators are based on the method of moments. They are shown to be consistent, that i...

Non-negative Matrix Factorization (NMF) has received considerable attentions in various areas for its psychological and physiological interpretation of naturally occurring data whose representation may be parts-based in the human brain. Despite its good practical performance, one shortcoming of original NMF is that it ignores intrinsic structure of data set. On one hand, samples might be on a m...

In this paper we consider some geometric questions relating to a higher dimensional analogon of a triangle, called simplex. In particular, we shall be concerned with the distance matrix of its vertices. We shall also majorize the volume of a simplex in terms of the distances between vertices. As consequences, we shall derive some inequalities for determinants and, in particular, an improvement ...

Nonnegative Matrix Factorization (NMF) has been a popular representation method for pattern classification problems. It tries to decompose a nonnegative matrix of data samples as the product of a nonnegative basis matrix and a nonnegative coefficient matrix. The columns of the coefficient matrix can be used as new representations of these data samples. However, traditional NMF methods ignore cl...

We refer to [4], [8] for basic definitions and terminology in graph theory. A tree is a simple connected graph without any circuit. We consider trees in which each edge is replaced by two arcs in either direction. In this paper, such trees are called bidirected trees. We now introduce some notation. Let e,0 be the column vectors consisting of all ones and all zeros, respectively, of the appropr...

We review the results on the characteristic polynomial, spectrum, spectral radius and energy of the distance matrix of graphs.

Graham, Hoffman and Hosoya gave a very nice formula about the determinant of the distance matrix DG of a graph G in terms of the distance matrix of its blocks. We generalize this result to a q-analogue of DG. Our generalization yields results about the equality of the determinant of the mod-2 (and in general mod-k) distance matrix (i.e. each entry of the distance matrix is taken modulo 2 or k) ...