Related papers: A Note on Equivalent Conditions for Majorization
In this thesis, I studied a mathematical development to define and quantify the uncertainty inherent in classical channels. This thesis starts with the introduction and background on how to formally think about uncertainty in the domain of…
In a previous paper, we have given an algebraic model to the set of intervals. Here, we apply this model in a linear frame. We define a notion of diagonalization of square matrices whose coefficients are intervals. But in this case, with…
We introduce the triangulant of two matrices, and relate it to the existence of orthogonal eigenvectors. We also use it for a new characterization of mutually unbiased bases. Generalizing the notion, we introduce higher order triangulants…
We discuss the notion of linearization through examples, which include the Price map, PageRank, representation theory, the Euler characteristic and quantum invariants. We also review categorification, which adds an additional layer of…
The purpose of this paper is two-fold: we present some matrix inequalities of log-majorization type for eigenvalues indexed by a sequence; we then apply our main theorem to generalize and improve the Hua-Marcus' inequalities. Our results…
This paper presents a unified approach for localizing some relevant graph topological indices via majorization techniques. Through this method, old and new bounds are derived and numerical examples are provided, showing how former results…
A natural link between the notions of majorization and strongly Sperner posets is elucidated. It is then used to obtain a variety of consequences, including new R\'enyi entropy inequalities for sums of independent, integer-valued random…
The purpose of this article is to investigate triangularization and simultaneous triangularization of matrices over max algebras using graph theoretic methods. We establish a connection between commutators and commutants with simultaneous…
As an alternative to the well-known methods of "chaining" and "bracketing" that have been developed in the study of random fields, a new method, which is based on a stochastic maximal inequality derived by using It\^o's formula and on a new…
The notion of matrix entropy was introduced by Tropp and Chen with the aim of measuring the fluctuations of random matrices. It is a certain entropy functional constructed from a representing function with prescribed properties, and Tropp…
There has been much discussion in the literature about rival measures of classical polarization in three dimensions. We gather and compare the various proposed measures of polarization, creating a geometric representation of the…
In this short expository note, we describe a unified algorithmic perspective on several classical problems which have traditionally been studied in different communities. This perspective views the main characters -- the problems of Optimal…
The aim of this paper is twofold. First, we introduce a new class of linearizations, based on the generalization of a construction used in polynomial algebra to find the zeros of a system of (scalar) polynomial equations. We show that one…
In recent years, many recommender systems have utilized textual data for topic extraction to enhance interpretability. However, our findings reveal a noticeable deficiency in the coherence of keywords within topics, resulting in low…
We revisit the classical problem of inverting dimension-reducing linear mappings using the maximum entropy (MaxEnt) criterion. In the literature, solutions are problem-dependent, inconsistent, and use different entropy measures. We propose…
In these notes we will give an overview and road map for a definition and characterization of (relative) entropy for both classical and quantum systems. In other words, we will provide a consistent treatment of entropy which can be applied…
This paper is concentrated on the classification of permutation matrix with the permutation similarity relation, mainly about the canonical form of a permutational similar equivalence class, the cycle matrix decomposition of a permutation…
We study regularization in the context of small sample-size learning with over-parameterized neural networks. Specifically, we shift focus from architectural properties, such as norms on the network weights, to properties of the internal…
Majorisation, also called rearrangement inequalities, yields a type of stochastic ordering in which two or more distributions can be compared. In this paper we argue that majorisation is a good candidate as a theory for uncertainty. We…
This paper aims to generalize and unify classical criteria for comparisons of balanced lattice designs, including fractional factorial designs, supersaturated designs and uniform designs. We present a general majorization framework for…