A workshop on random matrix theory, algorithms and applications that focuses on the theory and applications of random matrices. It is unique in the sense the “users” and “producers” of random matrix theory from the entire breadth of science and engineering are being brought together.
Our speakers will speak about the role of random matrices in a variety of areas that include statistical signal processing, machine learning, numerical algorithms, analysis, wireless communications, array processing, computational electromagnetics, uncertainty quantification, optics, and multivariate statistics.
Participants will interact with the workshops on Stochastic Computation, Special Functions and Orthogonal Polynomials, Numerical Linear Algebra, Foundations of Data Science and Machine Learning, inter alia.
Organizers
TU Munich
University of California, San Diego
Caltech
Speakers
Semi-plenary speakers
New York University
IST Austria
Yale University
Invited speakers
NYU Abu Dhabi
The University of Hong Kong
Duke University
University of Michigan
MIT
Johns Hopkins University
Korea Advanced Institute of Science and Technology
University of Michigan
Imperial College
Princeton University
NYU Abu Dhabi
University of Washington
Hong Kong University of Science and Technology
Monday, 13 July
![[HS 11 – 2nd Floor]](https://focm2026.univie.ac.at/wp-content/uploads/2026/06/HS11.png){kind=link}
14:00-14:30
Tom Trogdon (University of Washington)
We discuss an efficient computational approach to estimating limiting spectral distributions for fairly wide class of random matrices. The approach probes the distributions using only matrix-vector products. The method is provably accurate and can be used for statistical tasks such as spike detection. The estimator can also be used to accelerate existing algorithms for shrinkage and population covariance estimation. The error analysis relies on the understanding of the asymptotics of orthogonal polynomials, a relatively-new Riemann–Hilbert-based perturbation theory and local laws from random matrix theory.
14:30-15:00
Ke Wang (Hong Kong University of Science and Technology)
A central challenge in high-dimensional statistics is tracking the evolution of a deterministic signal when it is corrupted by general, non-uniform noise. While classical perturbation theory provides powerful tools for homogeneous noise, these bounds can become loose or inadequate in modern, heterogeneous environments. In this talk, I will present a unified framework for analyzing additive random perturbations for generalized heterogeneous noise. By relaxing standard structural conditions such as eigenvector incoherence, we establish precise, non-asymptotic bounds for both eigenvalues and eigenspaces under minimal assumptions. This approach naturally accommodates general signal matrices beyond standard finite-rank spiked models. Our work extends classical matrix perturbation theory to the heterogeneous regime, offering a flexible and rigorous toolset for modern spectral analysis.
15:00-15:30
Liza Rebrova (Princeton University)
Randomized tensor algorithms often compress data mode by mode, producing a smaller tensor rather than unstructured measurements. In this talk, I will discuss concentration for Kronecker-structured random sketches. These sketches are compatible with multilinear computations, but their norm-preservation behavior can depend strongly on the geometry of the input tensor. I will first explain their role in low-memory algorithms for Tucker approximation and related tensor-reduction methods, and then turn to the underlying problem of understanding the norm of a fixed tensor after independent sub-Gaussian matrices act along its modes. I will discuss recent bounds that recover standard worst-case guarantees while allowing sharper concentration when the tensor unfoldings have sufficiently diffuse spectra.
15:30-16:00
Michal Derezinski (University of Michigan)
Perturbing a deterministic n-dimensional matrix with small Gaussian noise is a cornerstone of smoothed analysis of algorithms [Spielman and Teng, JACM 2004], as it reduces the condition number of the input to O(n), and with it the complexity of many matrix algorithms. However, when deployed algorithmically, these perturbations are expensive due to the cost of generating and storing n^2 Gaussian random variables. We propose a perturbation that requires generating and storing O(n) random numbers in O(log n) bits of precision, and reduces the condition number of any deterministic matrix to O(n), matching Gaussian perturbations. Our result in particular implies a better complexity for the perturbed conjugate gradient algorithm, showing that we can solve an n x n linear system in linear space to within an arbitrarily small constant backward error using O(n) matrix-vector products. In our construction, we introduce the concept of a pattern matrix, which is a dense deterministic matrix that maps all sparse vectors into dense vectors, and we combine it with a sparse perturbation whose entries are dependent and located in a non-uniform fashion. In order to analyze this construction, we develop new techniques for lower bounding the smallest singular value of a random matrix with dependent entries. Joint work with Shabarish Chenakkod, Xiaoyu Dong, and Mark Rudelson (preprint at arxiv.org/abs/2604.23193).
16:00-16:30 Coffee Break
16:30-17:30 semi-plenary talk
Gérard Ben Arous (New York University)
I will survey recent progress in the understanding of the optimization dynamics for important tasks for Machine Learning, or high dimensional statistics. We will see how these very high-dimensional dynamics are in fact ruled by the so-called “effective dynamics” of much lower dimensional systems. This dynamical dimension reduction is related to the BBP spectral transition of Random Matrix Theory, appearing dynamically along the algorithm path. I will illustrate these phenomena in multi-spike Tensor PCA, XOR, and classification of Gaussian mixtures with multi-layer neural nets.
This talk is based on joint works with Reza Gheissari (Northwestern), Jiaoyang Huang (Wharton), Aukosh Jagannath (Waterloo), and on joint works with Cedric Gerbelot (ENS Lyon) and Vanessa Piccolo (EPFL).
17:30-18:00
Sheehan Olver (Imperial College)
Free probability is naturally formulated in terms of the inverse of the Stieltjes transform, which might be multivalued. We explore using this formulation for numerically computing free convolutions including the case where the support is not simply connected. We establish conditions bounding the number of inverses based on properties of the measure which can be combined with contour integral-based root finding algorithms to rigorously compute all inverses, and thereby deduce free convolutions numerically.
18:00-18:30
Ji Oon Lee (Korea Advanced Institute of Science and Technology)
Spiked random matrix models form a central framework for studying “signal-plus-noise” matrices. In these models, when the norm of the low-rank, deterministic “signal, or “spike”, is above a certain threshold, the behavior of the largest eigenvalue changes drastically from the behavior seen in purely random matrices. This phenomenon, known as the BBP transition, has been extended to many other spiked models. In this talk, I will present recent results on the largest eigenvalues of the transformed spiked Wigner matrices. These results show that the law of the fluctuation converges to the Gaussian distribution when the effective signal-to-noise ratio (SNR) is above the critical number, and to the GOE Tracy–Widom distribution when the effective SNR is below it. The talk is based on a joint work with Aro Lee.
Tuesday, 14 July
14:00-14:30
Dmitriy (Tim) Kunisky (Johns Hopkins University)
In a seminal 1999 paper, Lehner derived elegant formulas for the operator norms of series of free semicircular and Rademacher operators with matrix coefficients. These have found recent applications in the study of general correlated Gaussian random matrices and random lifts of graphs, respectively. While Lehner’s formulas are given as somewhat mysterious nonlinear matrix optimization problems, I will show that the semicircular formula can be reformulated as a semidefinite program (SDP) and the Rademacher formula can be relaxed to a bounding SDP. Further, I will show how the dual variables in the semicircular case have a natural sensitivity interpretation, which, in spiked matrix models, describes the fluctuations of their leading eigenvectors around the signal component. Based on this reasoning, I will present a conjecture describing these fluctuations for spiked matrix models with general isotropic but possibly correlated Gaussian noise, predicting that the eigenvector fluctuations are Gaussian but anisotropic and thus non-universal, with a covariance structure determined by the dual optimizers in Lehner’s formula.
14:30-15:00
Tatiana Brailovskaya (Duke University)
The study of diffusions on Lie groups has a rich history dating back to 1950s when they were first introduced by Itô. In this talk, I will focus on a family of multiplicative Brownian motions B^N(t) on the General Linear Group of degree N, i.e. at each time t, B^N(t) is an N by N random matrix. In 1997, Philippe Biane conjectured that as N —> infinity the empirical spectral measure of B^N(t) converges to the Brown measure of a free multiplicative Brownian motion, which is a diffusion on a certain infinite-dimensional space. With Nick Cook, Todd Kemp and Félix Parraud we confirm this conjecture.
15:00-15:30
Marwa Banna (NYU Abu Dhabi)
Random matrices provide a natural and fundamental bridge to free probability: in the large-N limit, many independent matrix ensembles converge to free noncommutative random variables. Beyond limiting spectral distributions, recent advances in strong convergence methods have made it possible to control operator norms and spectral behavior in increasingly general settings.
In this talk, I will discuss a strong convergence result for a family of multiplicative Brownian motions on the general linear group. We prove that, as N→∞, these processes converge to free multiplicative Brownian motion, in the strong sense of almost sure convergence of ∗-moments and operator norms. The talk will outline the motivation from random matrices and free probability, the role of multiplicative Brownian motions, and the main ideas behind the proof.
15:30-16:00
Patrick Santos (NYU Abu Dhabi)
In this talk, I will discuss the spectral distribution of tensor products of random matrices. While random matrices are naturally viewed as noncommutative operators, tensor products introduce additional commutation relations that significantly alter the resulting spectral behavior.
We encode these commutation relations using graphs. In the high-order regime, the limiting spectral distribution can be characterized in terms of the graphon limits of the associated graphs. These graphon limiting laws are universal and encompass a wide variety of distributions, including the Gaussian, semicircular, and $q$-Gaussian laws.
From a different perspective, we use this graph-based framework to derive upper bounds on their norms. Somewhat surprisingly, this leads to connections between norm estimation problems and classical questions in extremal graph theory, including the Turán problem.
If time permits, I will also mention several open problems and ongoing projects related to this framework. This talk is based on joint work with Guillaume Cébron, Cécilia Lancien, Raghavendra Tripathi, and Pierre Youssef.
16:00-16:30 Coffee Break
16:30-17:30 semi-plenary talk
Theo McKenzie (Yale University)
Universality is one of the central principles of random matrix theory: many spectral statistics are insensitive to the fine details of the model and instead depend only on a small number of structural parameters. A striking recent development is that this phenomenon extends beyond classical matrix ensembles to highly constrained sparse models. In particular, although the adjacency matrices of fixed-degree sparse random graphs have dependent entries and are generated from far fewer independent random variables than Wigner matrices, their spectral statistics can nevertheless agree with those of the Gaussian Orthogonal Ensemble, the canonical model of chaotic spectral behavior. In this talk, we will discuss some of these recent results and explain the main ideas behind this progress. Central to the analysis are combinatorial analogues of self-consistent equations, edge-switching arguments that recover approximate independence, and new approaches that move beyond local laws to establish different forms of universality through more directly combinatorial methods.
17:30-18:00
Alan Edelman (MIT)
ALAN EDELMAN, SUNGWOO JEONG, AND SIMEON SCHAUB
We present an exact sampling algorithm for Pfaffian point processes based on a skew- symmetric analogue of the Cholesky factorization. This algorithm enables efficient sampling of a wide range of statistics arising in random matrix theory and combinatorics. For instance, we can sample eigenvalues of the orthogonal and symplectic ensembles (β = 1, 4).
In addition, we introduce a symplectic Arnoldi method for computing skew-orthogonal polynomials associated with a general weight function. This method can be used to efficiently construct the 2 × 2 matrix valued skew-symmetric kernels that arise in β = 1, 4 polynomial ensembles. We illustrate our approach with several numerical examples and experiments, including the symmetric corner growth model, the finite-N Gaussian (Hermite) orthogonal and symplectic ensembles, and the β = 1, 4 Airy point processes and Tracy–Widom distributions.
Wednesday, 15 July
14:00-15:00 semi-plenary talk
László Erdős (IST Austria)
The resolvents of large random matrices, even very close to the imaginary axis, concentrate with a small fluctuation as the dimension increases. We extend this phenomenon to alternating products of resolvents and deterministic matrices using a new strategy consisting of an alternating tandem of two matrix flows. We focus on the applications, including (i) the precise identification of the blow up rate of the solution to a large system of ODE, (ii) eigenstate thermalisation hypothesis in physics and statistics and (iii) decorrelation phenomena of eigenvectors.
Based upon joint works with Zhigang Bao, Giorgio Cipolloni, Joscha Henheik, Oleksii Kolupaiev, Yuanyuan Xu.
15:00-15:30
Raj Rao (University of Michigan)
We propose a new regularization scheme for deep autoencoders based on matricial free energy, which is closely related to the negative of Voiculescu’s free entropy. This functional yields a differentiable loss expressed directly in terms of the singular values of the code matrix (code dimension × batch size).
From the perspective of free probability and random matrix theory, the loss is minimized when the singular value distribution of the code matches that of a suitably scaled random matrix with i.i.d. Gaussian entries.
Empirically, this regularizer reliably drives autoencoders to produce Gaussian-like codes, enabling applications such as solving underdetermined inverse problems and detecting out-of-distribution inputs.
This is joint work with Rishi Sonthalia (Boston College).
15:30-16:00
Zhigang Bao (The University of Hong Kong)
In this talk, we introduce two representations of the eigenvector distribution for the spiked Gaussian beta ensemble in the critical regime of the BBP transition. The first representation is expressed in terms of an extended Airy process, based on the eigenvector–eigenvalue identity, while the second is given in terms of the Airy–Green function. This talk is based on joint work with Dong Wang and Yue Zhu.
16:00-16:30 Coffee Break
Posters
![[PZ11 – 2nd Floor]](https://focm2026.univie.ac.at/wp-content/uploads/2026/06/poster_zones_2-stock.png){kind=link}
Shuyang Ling
(NYU Shanghai)
The spiked matrix model plays a central role in high-dimensional probability and statistics and has long been a fundamental object of study in random matrix theory. A key question concerns the distribution of leading eigenvectors. While these questions have been extensively analyzed using probabilistic techniques, we propose a complementary, computationally motivated framework based on Riemannian optimization.
Exploiting the variational characterization of leading eigenvectors, we analyze a one-step Riemannian Newton iteration on the Stiefel (or Grassmannian) manifold associated with the spectral optimization problem. This approach yields an explicit approximation of the leading eigenspace together with a sharp second-order expansion that quantifies its deviation from the global optimizer. The analysis is purely algebraic and does not rely on specific distributional assumptions, making the framework applicable to a broad class of noise models.
Beyond the eigenvalue problem, the framework also naturally extends to the characterization of singular values. Our results provide an algebraic and computational perspective on eigenvector fluctuations and offer an alternative route toward distributional characterization under the spiked matrix model.
John Lopez Santander
(University of Manitoba)
The Jacobi ensemble arises in statistical mechanics as a model for so-called \textit{log (Coulomb) gas} of charged particles confined to the interval $[0,1]$
and interacting via logarithmic repulsion. The \textit{$\beta$-Jacobi ensemble} is the family of probability density functions:
\begin{equation}\label{e:joint_eigenvalue_pdf_beta}
f_{p}(\boldsymbol{\theta})= {\mathcal Z}{p,\beta}\cdot \prod{i=1}^p (1-\theta_i)^{c_1\beta/2}\theta_i^{c_2 \beta/2}\prod_{i 0$, $c_1,c_2 > -2/\beta$, and ${\mathcal Z}_{p,\beta}$ is a normalization constant.
For a particular choice of the parameters $c_1$ and $c_2$, it also appears in connection with multivariate statistics as the joint eigenvalue distribution of the matrix $J=(A+B)^{-1}B$, where $A=X^X$ and $B=Y^Y$, with $X$ and $Y$ being $m\times p$ and $n\times p$ ($m,n \ge p$) random matrices, respectively, having i.i.d.\ standard Gaussian entries. The distribution of the largest eigenvalue has been characterized in terms of the Tracy–Widom distribution in the case that $m$ and $n$ grow proportionally with $p$, for multiple values of $\beta > 0$.
This poster presents recent work on the unbalanced regime where $m$ grows faster than linearly with $p$, while $n$ grows still proportionally to $p$, for the case $\beta=2$. In contrast to the case where parameters grow proportionally, where the support of the limiting spectral measure remains fixed, in our case the spectral measure exhibits the distinctive feature that its support shrinks, with both endpoints approaching zero. We prove that the distribution of the largest eigenvalue is still governed by the Tracy–Widom law and establish a lower-order correction term.
This poster is based on joint work with Katerina Gkogkou, Gustavo Didier and Kenneth McLaughlin.
Petar Nizic-Nikolac
(ETH Zurich)
Matrix concentration inequalities and their recently discovered sharp counterparts provide powerful tools to bound the spectrum of random matrices whose entries are linear functions of independent random variables. However, in many applications in theoretical computer science and in other areas one encounters more general random matrix models, called matrix chaoses, whose entries are polynomials of independent random variables. This poster outlines how to develop general matrix concentration inequalities for matrix chaoses, allowing these models to be treated in a systematic and unified way.
Joint work with Afonso S. Bandeira, Kevin Lucca, and Ramon van Handel.
Haixiao Wang
(University of Wisconsin-Madison)
In modern machine learning applications, data matrices are always assumed to admit the signal-plus-noise structure. Typically, we assume that the spectra of signal and noise matrices are well-separated and that noise subspaces only produce a marginal influence. While these assumptions are readily verified for dense matrices via classical random matrix theory, real-world data is often sparse, posing significant theoretical challenges. We investigate the spectral properties of X with i.i.d. Bernoulli(p) entries. Previous literature proved that when, the singular values of X remain within the compact support of the Marčenko-Pastur (MP) distribution. However, we identify a critical sparsity regime p = b\log(n)/\sqrt{mn} where this classical result fails. We identify a quantitative characterization of the emergence of outlier singular values. For explicit thresholds $b_{}$ and $b^{}$ as functions of the aspect ratio $\gamma = n/m \geq 1$, we prove a three-stage phase transition: (1) for $b > b_{}$, no outliers exist; (2) for $b^{} < b < b_{}$, outliers emerge only beyond the right edge of the MP law; and (3) for $b < b^{}$, outliers appear on both sides of the bulk, all with high probability. The locations of those outliers are precisely determined by the largest and smallest degree vertices of the underlying random graph. Besides, behavior of singular vectors corresponding to bulk and edge singular values can be characterized precisely. Our approach follows the framework established by Alt, Ducatez, and Knowles (2021), which can be extended to sparse matrices with general bounded entries.
Michael Wibmer
(University of Applied Sciences Munich)
Authors: Georg Schlüchtermann (1,2), Michael Wibmer (1)
Affiliation: (1) University of Applied Sciences Munich, (2) LMU Munich
Free stochastic differential equations are stochastic differential equations in the context of non-commutative random variables (e.g. large random matrices). In the paper [1] the authors developed a free analogue of the Euler-Maruyama method and proved strong convergence with order 1/2 and weak convergence with order 1. An important tool to achieve the results is the application of Multiple Operator Integrals, allowing to differentiate functions of noncommutative random variables.
In this poster we show how to extend the free Euler Maruyama method to strong convergence with order 1. We present an iterated Stochastic It{o}-Taylor Expansion, which enables to define those terms, which are needed to be discretized in order to achieve order 1. We will show, that these terms consists of free iterated stochastic integrals. One can follow, that to resolve these integrals into a numerical algorithm is more difficult compared to the commutative case. A free analog of the classical Milstein Method is only possible for linear, single diffusion. If one restricts the method to the von Neumann algebra of NxN random matrices, we develop a numerical algorithm allowing for nonlinearity in the diffusion term. The situation changes with multiple diffusion terms appear, for which it is required to discretize the iterated integrals directly. The poster finally shows examples, demonstrating the theoretical results.
[1] G. Schlüchtermann, M. Wibmer. Numerical Solution of Free Stochastic Differential equations. SIAM Journal on Numerical Analysis, Vol 61, 2023.
[2] G. Schlüchtermann, M. Wibmer. On Milstein Type Methods for Free Stochastic Differential Equations. In preparation.