Lectures & long talks

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Ioana Dumitriu

Talk 1 (Wednesday)

Title: Random Matrices, the Stochastic Block Model, and Community Detection in Networks

Abstract: We will explain the basics of community detection, the connection to graphs and the (random) Stochastic Block Model (SBM), and survey the different regimes of community recovery and the different methodologies used. We note that often, recovery algorithms are based on deep random matrix theory, whether in the form of concentration results or through special operator (i.e., nonbacktracking).

Talk 2 (Thursday)

Title: Exact Recovery in the non-uniform Hypergraph Stochastic Block Model

Abstract: Graphs are a useful model in community detection, but they are sometimes insufficient; enter the more general hypergraphs. We will define and then explain how the theory of community detection extends to the (random) Hypergraph Stochastic Block Model (HSBM), mention some of the new regimes, and show how one can obtain exact recovery thresholds for the non-uniform HSBM (joint work with Haixiao Wang).

Ramon van Handel

Title: A new approach to nonasymptotic random matrix theory

Abstract: Classical random matrix theory is largely concerned with the asymptotic
properties very special random matrix models, such as matrices with i.i.d.
entries, invariant ensembles, and the like. On the other hand, matrix
concentration inequalities, which are widely used in applied mathematics
to obtain nonasymptotic bounds on very general random matrices, can only
provide crude and often suboptimal information on the spectrum. Very
recently, however, a new approach to nonasymptotic random matrix theory
has resulted in a drastically improved understanding of arbitrarily
structured random matrices. This theory opens the door to applications
that were beyond the reach of previous methods. My aim is to introduce
some of the basic ingredients of this theory, and to illustrate by means
of concrete examples some ways in which it can be used.

Kyle Luh

Title: Random Laplacian Matrices

Abstract: Classical random matrix theory has been primarily concerned with matrices with independent entries (up to symmetry). In these lectures, we survey the understudied model of random Laplacian matrices, which are motivated by graph theory and computer science. These matrices have significant algorithmic applications and offer novel mathematical challenges since their entries are correlated.

Elizaveta Rebrova

Title: On randomized iterative sketching

Abstract: Generally speaking, sketching is multiplying a given matrix by a generic rectangular matrix for the dimension reduction purposes. Random matrices are known to successfully preserve the structure of the data and thus widely used for dimension reduction and in downstream numerical linear algebra tasks, such that, to perform fast randomized SVD decomposition. A special interesting case of sketching is when we are willing to use a collection, or iterative sequence of small sketches that collectively preserve enough information about the data to perform some task, e.g., to solve a linear system of equations. I will talk about such results and respective proof techniques, emphasizing high-dimensional probabilistic properties of the sketching matrices, or of the data models, that are responsible for the convergence properties of the numerical algorithms.

Nikhil Srivastava

Title: Random Matrix Theory for the Eigenvalue Problem (Part II)

Abstract: Random matrix theory has been a key ingredient in numerical linear algebra over the past two decades, mainly in the context of algorithms for low rank approximation, regression, and the solution of linear equations. In this sequence of lectures we will discuss non-asymptotic random matrix phenomena relevant in the design of numerical algorithms for solving the eigenvalue problem Ax=\lambda x. In the setting where A is an arbitrary (possibly non-Hermitian) deterministic square matrix and M is a random matrix with independent (tiny) entries, we will show that with high probability, the perturbed matrix A+M enjoys certain spectral stability properties enabling the design and analysis of such algorithms.

This second part of the series will focus on showing that the eigenvectors of A+M are well-conditioned with high probability for various models M. A key role is played by the area of the pseudospectra of A+M, which we will control using old and new least least singular value estimates for noncentered random matrices.

Konstantin Tikhomirov

Title: Gaussian Elimination with Partial Pivoting in randomized setting

Abstract: Gaussian Elimination with Partial Pivoting is a classical method of solving systems of linear equations. It is known that in floating point arithmetic, the worst-case relative error of the solution obtained by Gaussian Elimination depends exponentially on the dimension of the problem, and hence it can be very significant even for relatively small systems. At the same time, the method usually works very well in practice. In this talk, I will discuss a joint work with Han Huang in which we modeled the coefficients by independent standard Gaussians and showed that in the setting the relative error of the solution depends on the problem size polynomially rather than exponentially. The proof illustrates some of the standard techniques of non-asymptotic random matrix theory: estimating distances between random vectors and subspaces, anti-concentration, decouplings and union bound estimates.

Jorge Garza Vargas

Title: Random Matrix Theory for the Eigenvalue Problem (Part I)

Abstract: Random matrix theory has been a key ingredient in numerical linear algebra over the past two decades, mainly in the context of algorithms for low rank approximation, regression, and the solution of linear equations. In this sequence of lectures we will discuss non-asymptotic random matrix phenomena relevant in the design of numerical algorithms for solving the eigenvalue problem Ax=\lambda x. In the setting where A is an arbitrary (possibly non-Hermitian) deterministic square matrix and M is a random matrix with independent (tiny) entries, we will show that with high probability, the perturbed matrix A+M enjoys certain spectral stability properties enabling the design and analysis of such algorithms.

The first part of this series will focus on understanding the singular values and eigenvalue gaps of the random matrix A+M, which are crucial when analyzing eigenvector stability.

(to be updated)

Short talks

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Tatiana Brailovskaya

Title: Optimal bounds on the largest eigenvalue of inhomogeneous random matrices

(Joint work with Ramon van Handel)

Abstract: In this talk, I will discuss sharp non-asymptotic bounds on the expectation of the largest eigenvalue of a Gaussian random matrix with independent entries and arbitrary variance structure. Bounds of this kind are of significant interest to applied mathematicians and computer scientists, however classical random matrix theory fails to accurately capture the behavior of such objects. Our methods build upon prior work on this subject by Bandeira & Van Handel (2016) and allow us to improve their results by replacing universal constants with optimal numerical values.

Elizabeth Collins-Woodfin

Title: Edge CLT for log determinant of Laguerre beta ensembles

Abstract: Laguerre beta ensembles are a generalization of the Laguerre Orthogonal Ensemble (aka real Wishart) and Laguerre Unitary Ensemble (aka complex Wishart), which correspond to β = 1 and β = 2 respectively. This talk will focus on a central limit theorem for log | det(M_n − s_nI_n)| where M_n is a Laguerre beta ensemble and s_n is a real number near the upper edge of the matrix spectrum (joint work with Han Le). A similar result was proved for Wigner matrices by Johnstone, Klochkov, Onatski, and Pavlyshyn. Obtaining this type of CLT of Laguerre matrices is of interest for statistical testing of critically spiked sample covariance matrices as well as free energy of bipartite spherical spin glasses at critical temperature.

Youyi Huang

Title: Entropy fluctuation formulas of fermionic Gaussian states

Abstract: We study the statistical behavior of quantum entanglement in bipartite systems over fermionic Gaussian states as measured by von Neumann entropy, which is a linear spectrum statistic over the Jacobi unitary ensemble. The formulas of average von Neumann entropy with and without particle number constrains have been recently obtained, whereas the main results of this work are the exact yet explicit formulas of variances for both cases. For the latter case of no particle number constrain, the results resolve a recent conjecture on the corresponding variance. Different than the existing methods in computing variances over other generic state models, proving the results of this work relies on a new simplification framework. The framework consists of a set of new tools in simplifying finite summations of what we refer to as dummy summation and re-summation techniques. As a byproduct, the proposed framework leads to various new transformation formulas of hypergeometric functions. This talk is based on the joint work with Lu Wei available at arXiv:2211.16709.

Hongchang Ji

Title: Condition number and Wegner estimate for eigenvalues of non-centered, non-Hermitian random matrices

Abstract: In this talk, we consider eigenvalues of a non-Hermitian random matrix A+X, where A is a general deterministic matrix and X consists of i.i.d. centered random variables. The spectrum of A can be very unstable in general, in the sense that a small perturbation of A may cause arbitrarily large change in its eigenvalues. In this talk, we show that the noise matrix X stabilizes the spectrum so that (i) eigenvalues of A+X have (almost) bounded density in the complex plane and (ii) their condition number (sensitivity to perturbation) is bounded in expectation by their size. Both results are consequences of (iii) an optimal lower tail estimate for the smallest singular value of A+X for general A, which is of independent interest. This talk is based on a joint work with László Erdös.

Pax Kivimae

Title: Gaussian Multiplicative Chaos for Gaussian Orthogonal and Symplectic Ensembles

Abstract: In recent years, our understanding of the asymptotic behavior of characteristic polynomials of random matrices has seen much progression. A key paradigm in this area is that the asymptotic behavior is often captured by an appropriate family of Gaussian multiplicative chaos (GMC) measures (defined heuristically as the normalized exponential of log-correlated random fields). Indeed, such results have been shown for Harr distributed matrices for U(N), O(N), and Sp(2N), as well as for GUE(N). In this talk we explain an extension of these results to GOE(2N) and GSE(N). The key tool is a new asymptotic relation between the moments of the characteristic polynomials of all three classical ensembles.

Lamia Lamrani

Title: Frobenius error of Validation and Cross-validation Filtering of Large Covariance Matrices

Abstract: The estimation of covariance matrices is of crucial importance in several areas including signal theory, finance, economics, etc. The sample covariance estimator is highly inefficient when the number of features is comparable to the number of data points and thus filtering techniques must be used. One way to approximate the Oracle filtering is by using a cross-validation scheme. An open question remain: how many cross-validation folds one should use? The procedure is then to try computing the expected Frobenius error of the filtered matrices as a function of the data matrix parameters and of the number of cross-validation folds. We report results of the validation error in the case of an Inverse Wishart prior using random matrix theory and show that in this case the optimal split is proportional to the square root of the number of features.

Xiaoyu Xie

Title: Fluctuations in Quantum Unique Ergodicity for Wigner Matrix.

Abstract: I will give a shortintroduction on Quantum Unique Ergodicity (QUE) in the context of quantum chaos and its analogue in random matrix theory. In particular, I will focus on the QUE and the fluctuation of QUE in the context of Wigner matrix. A two moments matching argument will be introduced in some details. This is going to be based on a joint work with Lucas Benigni, Nixia Chen and Patrick Lopatto.

Ping Zhong

Title: Upgrading free convolution to non-normal free random variables

Abstract: Brown measure is a sort of spectral measure for free random variables, not necessarily normal. The Brown measure can predict the limit eigenvalue distribution of suitable random matrix models.I will report some recent progress on the Brown measure of the sum X+Y of two free random variables X and Y, where Y has certain symmetry and somewhat explicit R-transform. Our first result generalizes a model deformation phenomenon obtained by various authors to full generality. It suggests potential application to unify deformed i.i.d. random matrix model and deformed Wigner random matrix model. Our second result concerns the eigenvalue distribution of full rank deformed single ring model.

The procedure relies on Hermitian reduction and subordination functions. The analytic results (due to Voiculescu, Biane, Belinschi and Bercovici) for usual free convolution on the real line are very useful in this approach. The talk is based on my recent work on elliptic operators (arXiv:2108.09844) and joint work with Bercovici (arXiv:2209.12379).