相关论文: On near and the nearest correlation matrix
We obtain general, exact formulas for the overlaps between the eigenvectors of large correlated random matrices, with additive or multiplicative noise. These results have potential applications in many different contexts, from quantum…
Using Morse theory and a new relative homological linking of pairs, we prove a ``homological linking principle'', thereby generalizing many well known results in critical point theory.
This work deals with the generation of theoretical correlation matrices with specific sparsity patterns, associated to graph structures. We present a novel approach based on convex optimization, offering greater flexibility compared to…
We describe a randomized algorithm for producing a near-optimal hierarchical off-diagonal low-rank (HODLR) approximation to an $n\times n$ matrix $\mathbf{A}$, accessible only though matrix-vector products with $\mathbf{A}$ and…
The paper develops Newton's method of finding multiple eigenvalues with one Jordan block and corresponding generalized eigenvectors for matrices dependent on parameters. It computes the nearest value of a parameter vector with a matrix…
Matrices are typically considered over fields or rings. Motivated by applications in parametric differential equations and data-driven modeling, we suggest to study matrices with entries from a Hilbert space and present an elementary theory…
V. I. Arnold [Russian Math. Surveys 26 (2) (1971) 29-43] constructed a simple normal form for a family of complex n-by-n matrices that smoothly depend on parameters with respect to similarity transformations that smoothly depend on the same…
We approximate an elliptic problem with oscillatory coefficients using a problem of the same type, but with constant coefficients. We deliberately take an engineering perspective, where the information on the oscillatory coefficients in the…
In this paper we explore a family of congruences over $\N^\ast$ from which one builds a sequence of symmetric matrices related to the Mertens function. From the results of numerical experiments, we formulate a conjecture about the growth of…
In this paper a new fast algorithm for the computation of the distance of a matrix to a nearby defective matrix is presented. The problem is formulated following Alam & Bora (Linear Algebra Appl., 396 (2005), pp.~273--301) and reduces to…
Correlation clustering is a widely studied framework for clustering based on pairwise similarity and dissimilarity scores, but its best approximation algorithms rely on impractical linear programming relaxations. We present faster…
We provide an algorithm to check whether two rational space curves are related by a similarity. The algorithm exploits the relationship between the curvatures and torsions of two similar curves, which is formulated in a computer algebra…
Low rank approximation of matrices has been well studied in literature. Singular value decomposition, QR decomposition with column pivoting, rank revealing QR factorization (RRQR), Interpolative decomposition etc are classical deterministic…
We study the problem of approximating a matrix $\mathbf{A}$ with a matrix that has a fixed sparsity pattern (e.g., diagonal, banded, etc.), when $\mathbf{A}$ is accessed only by matrix-vector products. We describe a simple randomized…
We present novel, deterministic, efficient algorithms to compute the symmetries of a planar algebraic curve, implicitly defined, and to check whether or not two given implicit planar algebraic curves are similar, i.e. equal up to a…
We consider the problem of matching two shapes assuming these shapes are related by an elastic deformation. Using linearized elasticity theory and the finite element method we seek an elastic deformation that is caused by simple external…
This work deals with the generation of theoretical correlation matrices with specific sparsity patterns, associated to graph structures. We present a novel approach based on convex optimization, offering greater flexibility compared to…
Many idealized problems in signal processing, machine learning and statistics can be reduced to the problem of finding the symmetric canonical decomposition of an underlying symmetric and orthogonally decomposable (SOD) tensor. Drawing…
We consider real symmetric or complex hermitian random matrices with correlated entries. We prove local laws for the resolvent and universality of the local eigenvalue statistics in the bulk of the spectrum. The correlations have fast decay…
We calculate within a semiclassical approximation the autocorrelation function of cross sections. The starting point is the semiclassical expression for the diagonal matrix elements of an operator. For general operators with a smooth…