Related papers: No-gaps delocalization for general random matrices
We propose a second-order accurate method to estimate the eigenvectors of extremely large matrices thereby addressing a problem of relevance to statisticians working in the analysis of very large datasets. More specifically, we show that…
Let M be an arbitrary Hermitian matrix of order n, and k be a positive integer less than or equal to n. We show that if k is large, the distribution of eigenvalues on the real line is almost the same for almost all principal submatrices of…
We consider the statistics of the extreme eigenvalues of sparse random matrices, a class of random matrices that includes the normalized adjacency matrices of the Erd{\H o}s-R{\'e}nyi graph $G(N,p)$. Recently, it was shown by Lee, up to an…
In this paper we consider $N \times N $ matrices $D_{N}$ with i.i.d. entries all following an $a-$stable law divided by $N^{1/a}$. We prove that the least singular value of $D_{N}$, multiplied by $N$, tends to the same law as in the…
We investigate the distribution of eigenvalues of weighted adjacency matrices from a specific ensemble of random graphs. We distribute $N$ vertices across a fixed number $\kappa$ of components, with asymptotically $\alpha_j \dot N$ vertices…
In this paper, we derive entrywise error bounds for low-rank approximations of kernel matrices obtained using the truncated eigen-decomposition (or singular value decomposition). While this approximation is well-known to be optimal with…
The largest eigenvalue of a matrix is always larger or equal than its largest diagonal entry. We show that for a large class of random Laplacian matrices, this bound is essentially tight: the largest eigenvalue is, up to lower order terms,…
Let $G$ be a random $d$-regular graph. We prove that for every constant $\alpha > 0$, with high probability every eigenvector of the adjacency matrix of $G$ with eigenvalue less than $-2\sqrt{d-2}-\alpha$ has $\Omega(n/$polylog$(n))$ nodal…
Let $\Gamma$ be an $N\times n$ random matrix with independent entries and such that in each row entries are i.i.d. Assume also that the entries are symmetric, have unit variances, and satisfy a small ball probabilistic estimate uniformly.…
We prove improved bounds on how localized an eigenvector of a high girth regular graph can be, and present examples showing that these bounds are close to sharp. This study was initiated by Brooks and Lindenstrauss (2009) who relied on the…
This paper provides a construction of an uncountable family of i.i.d. random vectors, indexed by the points of a nonatomic measure space, such that (a) a sample is a measurable function from the index space, and (b) an idealization of the…
We develop a new method for deriving local laws for a large class of random matrices. It is applicable to many matrix models built from sums and products of deterministic or independent random matrices. In particular, it may be used to…
We completely describe all integer symmetric matrices that have all their eigenvalues in the interval [-2,2]. Along the way we classify all signed graphs, and then all charged signed graphs, having all their eigenvalues in this same…
In this work we show that, using the eigen-decomposition of the adjacency matrix, we can consistently estimate latent positions for random dot product graphs provided the latent positions are i.i.d. from some distribution. If class labels…
A Random SubMatrix method (RSM) is proposed to calculate the low-rank decomposition of large-scale matrices with known entry percentage \rho. RSM is very fast as the floating-point operations (flops) required are compared favorably with the…
We study the universality of spectral statistics of large random matrices. We consider $N\times N$ symmetric, hermitian or quaternion self-dual random matrices with independent, identically distributed entries (Wigner matrices) where the…
Eigenvector localization refers to the situation when most of the components of an eigenvector are zero or near-zero. This phenomenon has been observed on eigenvectors associated with extremal eigenvalues, and in many of those cases it can…
The class of norm-dependent Random Matrix Ensembles is studied in the presence of an external field. The probability density in those ensembles depends on the trace of the squared random matrices, but is otherwise arbitrary. An exact…
An elliptic random matrix $X$ is a square matrix whose $(i,j)$-entry $X_{ij}$ is independent of the rest of the entries except possibly $X_{ji}$. Elliptic random matrices generalize Wigner matrices and non-Hermitian random matrices with…
We consider the universality of the nearest neighbour eigenvalue spacing distribution in invariant random matrix ensembles. Focussing on orthogonal and symplectic invariant ensembles, we show that the empirical spacing distribution…