Related papers: Sampling the eigenvalues of random orthogonal and …
Matrices with low-rank structure are ubiquitous in scientific computing. Choosing an appropriate rank is a key step in many computational algorithms that exploit low-rank structure. However, estimating the rank has been done largely in an…
We determine the limiting empirical singular value distribution for random unitary matrices with Haar distribution and discrete Fourier transform (DFT) matrices when a random set of columns and rows is removed.
A novel algorithm for the recovery of low-rank matrices acquired via compressive linear measurements is proposed and analyzed. The algorithm, a variation on the iterative hard thresholding algorithm for low-rank recovery, is designed to…
A recursive method is derived to calculate all eigenvalue correlation functions of a random hermitian matrix in the large size limit, and after smoothing of the short scale oscillations. The property that the two-point function is…
It is well known that a family of $n\times n$ commuting matrices can be simultaneously triangularized by a unitary similarity transformation. The diagonal entries of the triangular matrices define the $n$ joint eigenvalues of the family. In…
We review the ideas of how random matrix theory has to be properly applied to quantum physics; particularly we focus on how the spectrum has to be properly prepared and the random matrix correctly identified before the random matrix and the…
We discuss uniform sampling algorithms that are based on stochastic growth methods, using sampling of extreme configurations of polymers in simple lattice models as a motivation. We shall show how a series of clever enhancements to a…
We establish formulae for the moments of the moments of the characteristic polynomials of random orthogonal and symplectic matrices in terms of certain lattice point count problems. This allows us to establish asymptotic formulae when the…
Random sampling has become a critical tool in solving massive matrix problems. For linear regression, a small, manageable set of data rows can be randomly selected to approximate a tall, skinny data matrix, improving processing time…
The eigenvalue distribution of the sum of two large Hermitian matrices, when one of them is conjugated by a Haar distributed unitary matrix, is asymptotically given by the free convolution of their spectral distributions. We prove that this…
In this paper we present an efficient algorithm to compute the eigen decomposition of a matrix that is a weighted sum of the self outer products of vectors such as a covariance matrix of data. A well known algorithm to compute the eigen…
Many latent-variable applications, including community detection, collaborative filtering, genomic analysis, and NLP, model data as generated by low-rank matrices. Yet despite considerable research, except for very special cases, the number…
We present a class of fast subspace tracking algorithms based on orthogonal iterations for structured matrices/pencils that can be represented as small rank perturbations of unitary matrices. The algorithms rely upon an updated data sparse…
This paper is concerned with complex eigenvalues of truncated unitary quaternion matrices equipped with the Haar measure. The joint eigenvalue probability density function is obtained for truncations of any size. We also obtain the spectral…
On unitary compact groups the decomposition of a generic element into product of reflections induces a decomposition of the characteristic polynomial into a product of factors. When the group is equipped with the Haar probability measure,…
We develop a new sampling method to estimate eigenvector centrality on incomplete networks. Our goal is to estimate this global centrality measure having at disposal a limited amount of data. This is the case in many real-world scenarios…
A randomized algorithm for computing a data sparse representation of a given rank structured matrix $A$ (a.k.a. an $H$-matrix) is presented. The algorithm draws on the randomized singular value decomposition (RSVD), and operates under the…
Horn's problem, i.e., the study of the eigenvalues of the sum $C=A+B$ of two matrices, given the spectrum of $A$ and of $B$, is re-examined, comparing the case of real symmetric, complex Hermitian and self-dual quaternionic $3\times 3$…
We analyze properties of non-hermitian matrices of size M constructed as square submatrices of unitary (orthogonal) random matrices of size N>M, distributed according to the Haar measure. In this way we define ensembles of random matrices…
We focus the use of \emph{row sampling} for approximating matrix algorithms. We give applications to matrix multipication; sparse matrix reconstruction; and, \math{\ell_2} regression. For a matrix \math{\matA\in\R^{m\times d}} which…