Related papers: Spectral Density of Sparse Sample Covariance Matri…
We study the dependence of the spectral density of the covariance matrix ensemble on the power spectrum of the underlying multivariate signal. The white noise signal leads to the celebrated Marchenko-Pastur formula. We demonstrate results…
We investigate almost-degenerate perturbation theory of eigenvalue problems, using spectral projectors, also named density matrices. When several eigenvalues are close to each other, the coefficients of the perturbative series become…
We consider complex sample covariance matrices $M_N=\frac{1}{N}YY^*$ where $Y$ is a $N \times p$ random matrix with i.i.d. entries $Y_{ij}, 1\leq i\leq N, 1\leq j \leq p$ with distribution $F$. Under some regularity and decay assumption on…
We study heavy-tailed Hermitian random matrices that are unitarily invariant. The invariance implies that the eigenvalue and eigenvector statistics are decoupled. The motivating question has been whether a freely stable random matrix has…
Gaussian time-series models are often specified through their spectral density. Such models present several computational challenges, in particular because of the non-sparse nature of the covariance matrix. We derive a fast approximation of…
The spectral density of random graphs with topological constraints is analysed using the replica method. We consider graph ensembles featuring generalised degree-degree correlations, as well as those with a community structure. In each case…
Traditional numerical methods for calculating matrix eigenvalues are prohibitively expensive for high-dimensional problems. Iterative random sparsification methods allow for the estimation of a single dominant eigenvalue at reduced cost by…
We consider the problem of approximating the set of eigenvalues of the covariance matrix of a multivariate distribution (equivalently, the problem of approximating the "population spectrum"), given access to samples drawn from the…
Sample covariance matrices from multi-population typically exhibit several large spiked eigenvalues, which stem from differences between population means and are crucial for inference on the underlying data structure. This paper…
The distribution of eigenvalues of N times N random matrices in the limit N to infinity is the solution to a variational principle that determines the ground state energy of a confined fluid of classical unit charges. This fact is a…
We discuss the limiting spectral density of real symmetric random matrices. Other than in standard random matrix theory the upper diagonal entries are not assumed to be independent, but we will fill them with the entries of a stochastic…
This paper investigates a statistical procedure for testing the equality of two independent estimated covariance matrices when the number of potentially dependent data vectors is large and proportional to the size of the vectors, that is,…
The variance--covariance matrix plays a central role in the inferential theories of high-dimensional factor models in finance and economics. Popular regularization methods of directly exploiting sparsity are not directly applicable to many…
We study complex networks under random matrix theory (RMT) framework. Using nearest-neighbor and next-nearest-neighbor spacing distributions we analyze the eigenvalues of adjacency matrix of various model networks, namely, random,…
We evaluate the covariance matrix of the matter power spectrum using perturbation theory up to dominant terms at 1-loop order and compare it to numerical simulations. We decompose the covariance matrix into the disconnected (Gaussian) part,…
We study an unconventional chiral random matrix model with a heavy-tailed probabilistic weight. The model is shown to exhibit chiral symmetry breaking with no bilinear condensate, in analogy to the Stern phase of QCD. We solve the model…
The ensemble covariance matrix of a wide sense stationary signal spatially sampled by a full linear array is positive semi-definite and Toeplitz. However, the direct augmented covariance matrix of an augmentable sparse array is Toeplitz but…
Random matrix theory is used to assess the significance of weak correlations and is well established for Gaussian statistics. However, many complex systems, with stock markets as a prominent example, exhibit statistics with power-law tails,…
We study the eigenvalue spectrum of a large real antisymmetric random matrix $J_{ij}$. Using a fermionic approach and replica trick, we obtain a semicircular spectrum of eigenvalues when the mean value of each matrix element is zero, and in…
We compute analytically the joint probability density of eigenvalues and the level spacing statistics for an ensemble of random matrices with interesting features. It is invariant under the standard symmetry groups (orthogonal and unitary)…