A positive map between Euclidean Jordan algebras is a (symmetric cone) order preserving linear map. We show that the norm of such a map is attained at the unit element, thus obtaining an analog of the operator/matrix theoretic Russo-Dye theorem. A doubly stochastic map between Euclidean Jordan algebras is a positive, unital, and trace preserving map. We relate such maps to Jordan algebra automo...

Given an approximate invariant subspace we discuss the effectiveness of majorization bounds for assessing the accuracy of the resulting Rayleigh-Ritz approximations to eigenvalues of Hermitian matrices. We derive a slightly stronger result than previously for the approximation of k extreme eigenvalues, and examine some advantages of these majorization bounds compared with classical bounds. From...

We study the Majorization arrow in a big class of quantum adiabatic algorithms. In a quantum adiabatic algorithm, the ground state of the Hamiltonian is a guide state around which the actual state evolves. We prove that for any algorithm of this class, step-by-step majorization of the guide state holds perfectly. We also show that step-by-step majorization of the actual state appears if the run...

Most of the statistical estimation procedures are based on a quite simple principle: find the distribution that, within a certain class, is as similar as possible to the empirical distribution, obtained from the sample observations. This leads to the minimization of some statistical functionals, usually interpreted ad measures of distance or divergence between distributions. In this paper we st...

Majorization is a powerful, easy-to-use and exible tool which arises frequently in quantum mechanics as a consequence of fundamental connections between unitarity and the majorization relation. Entanglement theory does not escape from its in uence. Thus the interconversion of bipartite pure states by means of local manipulations turns out to be ruled to a great extend by majorization relations....

DIAGONAL ELEMENTS, EIGENVALUES, AND SINGULAR VALUES DRAFT AS OF April 30, 2013 SHENG-JHIH WU AND MOODY T. CHU Abstract. Diagonal entries and eigenvalues of a Hermitian matrix, diagonal entries and singular values of a general matrix, and eigenvalues and singular values of a general matrix satisfy necessarily some majorization relationships which turn out also to be sufficient conditions. The in...

We describe an algorithm for radial layout of undirected graphs, in which nodes are constrained to the circumferences of a set of concentric circles around the origin. Such constraints frequently occur in the layout of social or policy networks, when structural centrality is mapped to geometric centrality, or when the primary intention of the layout is the display of the vicinity of a distingui...

This paper addresses the estimation of the latent dimensionality in nonnegative matrix factorization (NMF) with the -divergence. The -divergence is a family of cost functions that includes the squared euclidean distance, Kullback-Leibler (KL) and Itakura-Saito (IS) divergences as special cases. Learning the model order is important as it is necessary to strike the right balance between data fid...

This paper addresses the estimation of the latent dimensionality in nonnegative matrix factorization (NMF) with the β-divergence. The β-divergence is a family of cost functions that includes the squared euclidean distance, Kullback-Leibler (KL) and Itakura-Saito (IS) divergences as special cases. Learning the model order is important as it is necessary to strike the right balance between data f...