Related papers: Random Laplacian matrices and convex relaxations

In this work we present a framework for studying the eigenvalues of a family of matrices with a particular displacement structure. The family admits a specific decomposition as the product of an upper and a lower triangular matrices having…

We consider a product of an arbitrary number of independent rectangular Gaussian random matrices. We derive the mean densities of its eigenvalues and singular values in the thermodynamic limit, eventually verified numerically. These…

Preconditioned iterative methods for numerical solution of large matrix eigenvalue problems are increasingly gaining importance in various application areas, ranging from material sciences to data mining. Some of them, e.g., those using…

The goal of this paper is to survey the properties of the eigenvalue relaxation for least squares binary problems. This relaxation is a convex program which is obtained as the Lagrangian dual of the original problem with an implicit compact…

We consider the eigenvalues and eigenvectors of finite, low rank perturbations of random matrices. Specifically, we prove almost sure convergence of the extreme eigenvalues and appropriate projections of the corresponding eigenvectors of…

We compute analytically the probability density function (pdf) of the largest eigenvalue $\lambda_{\max}$ in rotationally invariant Cauchy ensembles of $N\times N$ matrices. We consider unitary ($\beta = 2$), orthogonal ($\beta =1$) and…

The largest eigenvalue of random tensors is an important feature of systems involving disorder, equivalent to the ground state energy of glassy systems or to the injective norm of quantum states. For symmetric Gaussian random tensors of…

In this paper, we consider the log-concave ensemble of random matrices, a class of covariance-type matrices $XX^*$ with isotropic log-concave $X$-columns. A main example is the covariance estimator of the uniform measure on isotropic convex…

We calculate analytically the probability of large deviations from its mean of the largest (smallest) eigenvalue of random matrices belonging to the Gaussian orthogonal, unitary and symplectic ensembles. In particular, we show that the…

We present an analytical technique to compute the probability of rare events in which the largest eigenvalue of a random matrix is atypically large (i.e.\ the right tail of its large deviations). The results also transfer to the left tail…

This work concerns the minimization of the pseudospectral abscissa of a matrix-valued function dependent on parameters analytically. The problem is motivated by robust stability and transient behavior considerations for a linear control…

Given a square complex matrix $A$, we tackle the problem of finding the nearest matrix with multiple eigenvalues or, equivalently when $A$ had distinct eigenvalues, the nearest defective matrix. To this goal, we extend the general framework…

In this paper, we show that the largest and smallest eigenvalues of a sample correlation matrix stemming from $n$ independent observations of a $p$-dimensional time series with iid components converge almost surely to $(1+\sqrt{\gamma})^2$…

The graph Laplacian, a typical representation of a network, is an important matrix that can tell us much about the network structure. In particular its eigenpairs (eigenvalues and eigenvectors) incubate precious topological information…

We prove a general result demonstrating the power of Lagrangian relaxation in solving constrained maximization problems with arbitrary objective functions. This yields a unified approach for solving a wide class of {\em subset selection}…

We observe that the Laplacian of a random graph G on N vertices represents and explicitly solvable model in the limit of infinitely increasing N. Namely, we derive recurrent relations for the limiting averaged moments of the adjacency…

Typically, graph structures are represented by one of three different matrices: the adjacency matrix, the unnormalised and the normalised graph Laplacian matrices. The spectral (eigenvalue) properties of these different matrices are…

We give an upper bound on the total variation distance between the linear eigenvalue statistic, properly scaled and centred, of a random matrix with a variance profile and the standard Gaussian random variable. The second order Poincar\'e…

Determining the effect of structural perturbations on the eigenvalue spectra of networks is an important problem because the spectra characterize not only their topological structures, but also their dynamical behavior, such as…

We show several sharp upper and lower bounds for the sum of the largest eigenvalues of the signless Laplacian matrix. These bounds improve and extend previously known bounds.