Related papers: The $\beta$ Maps: Strong Clustering and Distributi…
This paper presents a diffusion based probabilistic interpretation of spectral clustering and dimensionality reduction algorithms that use the eigenvectors of the normalized graph Laplacian. Given the pairwise adjacency matrix of all…
In a recent work Killip and Nenciu gave random recurrences for the characteristic polynomials of certain unitary and real orthogonal upper Hessenberg matrices. The corresponding eigenvalue p.d.f.'s are beta-generalizations of the classical…
In \cite{Diaz} beta type I and II doubly singular distributions were introduced and their densities and the joint densities of nonzero eigenvalues were derived. In such matrix variate distributions $p$, the dimension of two singular Wishart…
Quantifying the eigenvalue spectra of large random matrices allows one to understand the factors that contribute to the stability of dynamical systems with many interacting components. This work explores the effect that the interaction…
We develop a theory which describes the behaviour of eigenvalues of a class of one-dimensional random non-Hermitian operators introduced recently by Hatano and Nelson. Under general assumptions on random parameters we prove that the…
Using numerical exact diagonalization, we study matrix elements of a local spin operator in the eigenbasis of two different nonintegrable quantum spin chains. Our emphasis is on the question to what extent local operators can be represented…
We present efficient numerical techniques for calculation of eigenvalue distributions of random matrices in the beta-ensembles. We compute histograms using direct simulations on very large matrices, by using tridiagonal matrices with…
In this paper, the study of bivariate generalised beta type I and II distributions is extended to the complex matrix variate case, for which the corresponding density functions are found. In addition, for complex bimatrix variate beta type…
In data science, individual observations are often assumed to come independently from an underlying probability space. Kernel matrices formed from large sets of such observations arise frequently, for example during classification tasks. It…
We explore clustering the softmax predictions of deep neural networks and introduce a novel probabilistic clustering method, referred to as k-sBetas. In the general context of clustering discrete distributions, the existing methods focused…
We extend classical time-frequency limiting analysis, historically applied to one-dimensional finite signals, to the multidimensional discrete setting. This extension is relevant for images, videos, and other multidimensional signals, as it…
In this paper, the densities of the doubly singular beta type I and II distributions are found, and the joint densities of their corresponding nonzero eigenvalues are provided. As a consequence, the density function of a singular inverted…
We compute the joint eigenvalue distribution for the rank one Hermitian and non-Hermitian perturbations of chiral Gaussian $\beta$-ensembles ($\beta>0$) of random matrices.
Random Matrix Theory (RMT) has successfully modeled diverse systems, from energy levels of heavy nuclei to zeros of $L$-functions; this correspondence has allowed RMT to successfully predict many number theoretic behaviors. However there…
Spectral clustering is one of the most popular, yet still incompletely understood, methods for community detection on graphs. This article studies spectral clustering based on the Bethe-Hessian matrix $H_r = (r^2-1)I_n + D-rA$ for sparse…
In this paper, we consider the singular values and singular vectors of finite, low rank perturbations of large rectangular random matrices. Specifically, we prove almost sure convergence of the extreme singular values and appropriate…
We compute the limiting statistical distribution of the eigenvalues of sequences of matrices whose entries satisfy what we call a vanishing mean variation condition and are $\mu$-distributed for some probability measure. As an application…
Spectral properties of random matrices play an important role in statistics, machine learning, communications, and many other areas. Engaging results regarding the convergence of the empirical spectral distribution (ESD) and the…
In the present work, eigenvalue distributions defined by a random rectangular matrix whose components are neither independently nor identically distributed are analyzed using replica analysis and belief propagation. In particular, we…
The eigenvalue distribution of Hoppe's two matrix model is investigated in detail as a function of the model's coupling. For small couplings it is a perturbed Wigner semicircle, while for large couplings it is a parabolic distribution which…