We prove a moment majorization principle for matrix-valued functions with domain {−1, 1}m, m ∈ N. The principle is an inequality between higher-order moments of a non-commutative multilinear polynomial with different random matrix ensemble inputs, where each variable has small influence and the variables are instantiated independently. This technical result can be interpreted as a noncommutativ...

In mathematical chemistry, the median eigenvalues of the adjacency matrix of a molecular graph are strictly related to orbital energies and molecular orbitals. In this regard, the difference between the occupied orbital of highest energy (HOMO) and the unoccupied orbital of lowest energy (LUMO) has been investigated (see Fowler and Pisansky in Acta Chim. Slov. 57:513-517, 2010). Motivated by th...

Miranda-Thompson majorization is a group-induced cone ordering on $mathbb{R}^{n}$ induced by the group of generalized permutation with determinants equal to 1. In this paper, we generalize Miranda-Thompson majorization on the matrices. For $X$, $Yin M_{m,n}$, $X$ is said to be  Miranda-Thompson majorized by $Y$ (denoted by $Xprec_{mt}Y$) if there exists some $Din rm{Conv(G)}$ s...

This paper presents applications of a remarkable majorization inequality due to Bapat and Sunder in three different areas. The first application is a proof of Hiroshima’s 2003 result which arises in quantum information theory. The second one is an extension of some eigenvalue inequalities that have been used to bound chromatic number of graphs. The third application is a simplified proof of a m...

Two new elementary proofs are given of Landau's Theorem on necessary and sufficient conditions for a sequence of integers to be the score sequence for some tournament. The first is related to existing proofs by majorization, but it avoids depending on any facts about majorization. The second is natural and direct. Both proofs are constructive, so they each provide an algorithm for obtaining a t...

We prove that majorization relations hold step by step in the Quantum Fourier Transformation (QFT) for phase-estimation algorithms considered in the canonical decomposition. Our result relies on the fact that states which are mixed by Hadamard operators at any stage of the computation only differ by a phase. This property is a consequence of the structure of the initial state and of the QFT, ba...

Majorization methods solve minimization problems by replacing a complicated problem by a sequence of simpler problems. Solving the sequence of simple optimization problems guarantees convergence to a solution of the complicated original problem. Convergence is guaranteed by requiring that the approximating functions majorize the original function at the current solution. The leading examples of...

The Rayleigh-Ritz (RR) method finds the stationary values, called Ritz values, of the Rayleigh quotient on a given trial subspace as approximations to eigenvalues of a Hermitian operator A. If the trial subspace is A-invariant, the Ritz values are exactly some of the eigenvalues of A. Given two subspaces X and Y of the same finite dimension, such that X is A-invariant, the absolute changes in t...

This paper describes algorithms for nonnegative matrix factorization (NMF) with the β-divergence (β-NMF). The β-divergence is a family of cost functions parametrized by a single shape parameter β that takes the Euclidean distance, the Kullback-Leibler divergence and the Itakura-Saito divergence as special cases (β = 2, 1, 0 respectively). The proposed algorithms are based on a surrogate auxilia...