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A fundamental result of free probability theory due to Voiculescu and subsequently refined by many authors states that conjugation by independent Haar-distributed random unitary matrices delivers asymptotic freeness. In this paper we…
We study the conjugation action of orthogonal matrices on symmetric random matrices. Given a fixed orthogonal matrix over an algebraic number field and a random matrix with entries sufficiently uniform in the ring of integers, we wonder…
In this article we consider means of positive bounded linear operators on a Hilbert space. We present a complete theory that provides a framework which extends the theory of the Karcher mean, its approximating matrix power means, and a…
We introduce a finite version of free probability and show the link between recent results using polynomial convolutions and the traditional theory of free probability. One tool for accomplishing this is a seemingly new transformation that…
We reconsider randomized algorithms for the low-rank approximation of symmetric positive semi-definite (SPSD) matrices such as Laplacian and kernel matrices that arise in data analysis and machine learning applications. Our main results…
In this paper we review and compare the numerical evaluation of those probability distributions in random matrix theory that are analytically represented in terms of Painlev\'e transcendents or Fredholm determinants. Concrete examples for…
We introduce a Banach space-valued extension of random feature learning, a data-driven supervised machine learning technique for large-scale kernel approximation. By randomly initializing the feature maps, only the linear readout needs to…
Inspired by the possibility that generative models based on quantum circuits can provide a useful inductive bias for sequence modeling tasks, we propose an efficient training algorithm for a subset of classically simulable quantum circuit…
We study the spectra of MANOVA estimators for variance component covariance matrices in multivariate random effects models. When the dimensionality of the observations is large and comparable to the number of realizations of each random…
We study the problem of estimating the diagonal of an implicitly given matrix $A$. For such a matrix we have access to an oracle that allows us to evaluate the matrix vector product $Av$. For random variable $v$ drawn from an appropriate…
Precision matrix estimation is a cornerstone concept in statistics, economics, and finance. Despite advances in recent years, estimation methods that are simultaneously (i) dense, (ii) consistent, and (iii) model-free are lacking. While…
For a Haar random set $\mathcal{S}\subset U(d)$ of quantum gates we consider the uniform measure $\nu_\mathcal{S}$ whose support is given by $\mathcal{S}$. The measure $\nu_\mathcal{S}$ can be regarded as a…
We prove Bernstein-type matrix concentration inequalities for linear combinations with matrix coefficients of binary random variables satisfying certain $\ell_\infty$-independence assumptions, complementing recent results by Kaufman, Kyng…
New family of extended Cauchy type identities is found and related Fermat type matrices are provided ready for applications in extended scope. This is achieved due to the use specifically non-commuting variables of extended finite operator…
In this article we study in detail a family of random matrix ensembles which are obtained from random permutations matrices (chosen at random according to the Ewens measure of parameter $\theta>0$) by replacing the entries equal to one by…
A density matrix describes the statistical state of a quantum system. It is a powerful formalism to represent both the quantum and classical uncertainty of quantum systems and to express different statistical operations such as measurement,…
We give a probabilistic introduction to determinantal and permanental point processes. Determinantal processes arise in physics (fermions, eigenvalues of random matrices) and in combinatorics (nonintersecting paths, random spanning trees).…
For real symmetric matrices that are accessible only through matrix vector products, we present Monte Carlo estimators for computing the diagonal elements. Our probabilistic bounds for normwise absolute and relative errors apply to Monte…
Derivative-free algorithms seek the minimum of a given function based only on function values queried at appropriate points. Although these methods are widely used in practice, their performance is known to worsen as the problem dimension…
Many random combinatorial objects have a component structure whose joint distribution is equal to that of a process of mutually independent random variables, conditioned on the value of a weighted sum of the variables. It is interesting to…