Related papers: Matrix Concentration Inequalities and Free Probabi…
We present a new method for obtaining norm bounds for random matrices, where each entry is a low-degree polynomial in an underlying set of independent real-valued random variables. Such matrices arise in a variety of settings in the…
Covariance matrix estimation concerns the problem of estimating the covariance matrix from a collection of samples, which is of extreme importance in many applications. Classical results have shown that $O(n)$ samples are sufficient to…
We address the problem of learning an unknown smooth function and its derivatives from noisy pointwise evaluations under the supremum norm. While classical nonparametric regression provides a strong theoretical foundation, traditional…
Motivated by the recent work on asymptotic independence relations for random matrices with non-commutative entries, we investigate the limit distribution and independence relations for large matrices with identically distributed and Boolean…
Matrix concentration inequalities give bounds for the spectral-norm deviation of a random matrix from its expected value. These results have a weak dimensional dependence that is sometimes, but not always, necessary. This paper identifies…
Gaussian process (GP) regression is a non-parametric, Bayesian framework to approximate complex models. Standard GP regression can lead to an unbounded model in which some points can take infeasible values. We introduce a new GP method that…
The asymptotic analysis of covariance parameter estimation of Gaussian processes has been subject to intensive investigation. However, this asymptotic analysis is very scarce for non-Gaussian processes. In this paper, we study a class of…
We prove that independent families of permutation invariant random matrices are asymptotically free over the diagonal, both in probability and in expectation, under a uniform boundedness assumption on the operator norm. We can relax the…
We establish a lower bound on the entropy of weighted sums of (possibly dependent) random variables $(X_1, X_2, \dots, X_n)$ possessing a symmetric joint distribution. Our lower bound is in terms of the joint entropy of $(X_1, X_2, \dots,…
We study the Gaussian hermitian random matrix ensemble with an external matrix which has an arbitrary number of eigenvalues with arbitrary multiplicity. We compute the limiting eigenvalues correlations when the size of the matrix goes to…
Asymptotic freeness of independent Haar distributed unitary matrices was discovered by Voiculescu. Many refinements have been obtained, including strong asymptotic freeness of random unitaries and strong asymptotic freeness of random…
We show that the eigenvalue density of a product X=X_1 X_2 ... X_M of M independent NxN Gaussian random matrices in the large-N limit is rotationally symmetric in the complex plane and is given by a simple expression rho(z,\bar{z}) =…
In this article the statistical properties of symmetrical random matrices whose elements are drawn from a q-parameterized non-extensive statistics power-law distribution are investigated. In the limit as q->1 the well known Gaussian…
Random matrix theory has become a cornerstone in modern statistics and data science, providing fundamental tools for understanding high-dimensional covariance structures. Within this framework, the Wishart matrix plays a central role in…
Regression trees and random forests are popular and effective non-parametric estimators in practical applications. A recent paper by Athey and Wager shows that the random forest estimate at any point is asymptotically Gaussian; in this…
We prove concentration inequalities for several models of non-linear random matrices. As corollaries we obtain estimates for linear spectral statistics of the conjugate kernel of neural networks and non-commutative polynomials in (possibly…
Several proofs of the monotonicity of the non-Gaussianness (divergence with respect to a Gaussian random variable with identical second order statistics) of the sum of n independent and identically distributed (i.i.d.) random variables were…
This note attempts to revisit the classical results on Laplace approximation in a modern non-asymptotic and dimension free form. Such an extension is motivated by applications to high dimensional statistical and optimization problems. The…
We prove the universal asymptotically almost sure non-singularity of general Ginibre and Wigner ensembles of random matrices when the distribution of the entries are independent but not necessarily identically distributed and may depend on…
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…