Recently a majorization method for optimizing partition functions of log-linear models was proposed alongside a novel quadratic variational upper-bound. In the batch setting, it outperformed state-of-the-art firstand second-order optimization methods on various learning tasks. We propose a stochastic version of this bound majorization method as well as a low-rank modification for highdimensiona...

Many inequality relations between real vector quantities can be succinctly expressed as “weak (sub)majorization” relations using the symbol ≺w. We explain these ideas and apply them in several areas: angles between subspaces, Ritz values, and graph Laplacian spectra, which we show are all surprisingly related. Let Θ(X ,Y) be the vector of principal angles in nondecreasing order between subspace...

We present a straightforward linear algebraic model of greed, based only on extensions of classical majorization and convexity theory. This gives an alternative to other models of greedy-solvable problems such as matroids, greedoids, submodular functions, etc., and it is able to express established examples of greedy-solvable optimization problems that they cannot. The linear algebraic approach...

Let Mn,m be the set of all n × m matrices with entries in F, where F is the field of real or complex numbers. A matrix R ∈ Mn with the property Re=e, is said to be a g-row stochastic (generalized row stochastic) matrix. Let A,B∈ Mn,m, so B is said to be gw-majorized by A if there exists an n×n g-row stochastic matrix R such that B=RA. In this paper we characterize all linear operators that stro...

This paper considers the problem of constructing a normatively significant multidimensional Gini index of relative inequality. The social evaluation relation (SER) from which the index is derived is required to satisfy a weak version of the Pigou-Dalton Bundle Principle (WPDBP) (rather than Uniform Majorization or similar conditions). It is also desired to satisfy a weak form of the condition o...

Horn’s Theorem plays an important role in the theory of matrix majorization and elsewhere. We give a simple proof of it, as well as a related theorem of Mirsky.

In this paper we extend the majorization theorem from convex to covexifiable functions, in particular to smooth functions with Lipschitz derivative, twice continuously differentiable functions and analytic functions. AMS subject classifications: 26B25, 52A40, 26D15