Related papers: Right-left asymmetry of the eigenvector method: A …
The difference between left- and right-handed particles is perhaps one of the most puzzling aspects of the Standard Model (SM). In left-right models (LRMs) the symmetry between left- and right-handed particles can be restored at high…
Subspace methods are commonly used for finding approximate eigenvalues and singular values of large-scale matrices. Once a subspace is found, the Rayleigh-Ritz method (for symmetric eigenvalue problems) and Petrov-Galerkin projection (for…
We present two algorithms for constructing orthonormal bases of rational function vectors with respect to a discrete inner product, and discuss how to use them for a rational approximation problem. Building on the pencil-based formulation…
In the paper we show that there is an analogy between the operations occurring in a layer of a transformer (projections and layer normalizations, disregarding the feedforward neural network) and a step in the power method. Coherently with…
In prioritization schemes, based on pairwise comparisons, such as the Analytical Hierarchy Process, it is necessary to extract a cardinal ranking vector from a reciprocal matrix that is unlikely to be consistent. It is natural to choose…
Spectral methods include a family of algorithms related to the eigenvectors of certain data-generated matrices. In this work, we are interested in studying the geometric landscape of the eigendecomposition problem in various spectral…
We develop an iterative refinement method that improves the accuracy of a user-chosen subset of $k$ eigenvectors ($k\ll n$) of an $n\times n$ real symmetric matrix. Using an orthogonal matrix represented in compact WY form, the method…
This paper is to give a new understanding and applications of the subspace projection method for selfadjoint eigenvalue problems. A new error estimate in the energy norm, which is induced by the stiff matrix, of the subspace projection…
We endeavour to estimate numerous multi-dimensional means of various probability distributions on a common space based on independent samples. Our approach involves forming estimators through convex combinations of empirical means derived…
Pairwise comparisons between alternatives are a well-known method for measuring preferences of a decision-maker. Since these often do not exhibit consistency, a number of inconsistency indices has been introduced in order to measure the…
We examine the possibility of using a reinforcement learning (RL) algorithm to solve large-scale eigenvalue problems in which the desired the eigenvector can be approximated by a sparse vector with at most $k$ nonzero elements, where $k$ is…
This paper combines two ingredients in order to get a rather surprising result on one of the most studied, elegant and powerful tools for solving convex feasibility problems, the method of alternating projections (MAP). Going back to names…
The Method of Alternating Projections (MAP), a classical algorithm for solving feasibility prob- lems, has recently been intensely studied for nonconvex sets. However, intrinsically available are only local convergence results: convergence…
We analyze gene co-expression network under the random matrix theory framework. The nearest neighbor spacing distribution of the adjacency matrix of this network follows Gaussian orthogonal statistics of random matrix theory (RMT). Spectral…
We present a five-step method for the calculation of eigenvalue correlation functions for various ensembles of real random matrices, based upon the method of (skew-) orthogonal polynomials. This scheme systematises existing methods and also…
The history of research on eigenvalue problems is rich with many outstanding contributions. Nonetheless, the rapidly increasing size of data sets requires new algorithms for old problems in the context of extremely large matrix dimensions.…
The comparison of alternative rankings of a set of items is a general and prominent task in applied statistics. Predictor variables are ranked according to magnitude of association with an outcome, prediction models rank subjects according…
The Rayleigh-Ritz method is widely used for eigenvalue approximation. Given a matrix $X$ with columns that form an orthonormal basis for a subspace $\X$, and a Hermitian matrix $A$, the eigenvalues of $X^HAX$ are called Ritz values of $A$…
We prove the universality of the joint distribution of an eigenvalue and the corresponding diagonal eigenvector overlap, in the bulk and at the edge, for eigenvalues of complex matrices and real eigenvalues of real matrices. As part of the…
The classical AM-GM inequality has been generalized in a number of ways. Generalizations which incorporate variance appear to be the most useful in economics and finance, as well as mathematically natural. Previous work leaves unanswered…