Generalized Ghabries-Abbas-Mourad log-majorization

Generalized Majorization-Minimization

Non-convex optimization is ubiquitous in machine learning. The MajorizationMinimization (MM) procedure systematically optimizes non-convex functions through an iterative construction and optimization of upper bounds on the objective function. The bound at each iteration is required to touch the objective function at the optimizer of the previous bound. We show that this touching constraint is u...

On properties of the generalized majorization

Ela on Properties of the Generalized Majorization

Abstract. In this paper, a complete solution of a problem involving generalized majorization of partitions is given: for two pairs of partitions (d, a) and (c,b) necessary and sufficient conditions for the existence of a partition g that is majorized by both pairs is determined. The obtained conditions are explicit, the solution is constructive and it uses novel techniques and indices. Although...

Majorization properties of generalized thermal distributions

We examine the majorization properties of general thermal-like mixed states depending on a set of parameters. Sufficient conditions which ensure the increase in mixedness, and hence of any associated entropic form, when these parameters are varied, are identified. We then discuss those exhibiting a power law distribution, showing that they can be characterized by two distinct mixing parameters,...

Weak log-majorization inequalities of singular values between normal matrices and their absolute values

‎This paper presents two main results that the singular values of the Hadamard product of normal matrices $A_i$ are weakly log-majorized by the singular values of the Hadamard product of $|A_{i}|$ and the singular values of the sum of normal matrices $A_i$ are weakly log-majorized by the singular values of the sum of $|A_{i}|$‎. ‎Some applications to these inequalities are also given‎. ‎In addi...