Conference Agenda

In this paper, we develop nonlinear higher-order shrinkage estimators for both covariance and precision matrices. Our framework applies to settings in which the sample size n is either larger or smaller than p, the dimensionality of the data-generating process. The proposed estimators incorporate higher-order moments up to an arbitrary order and therefore encompass linear shrinkage estimators as special cases. We derive recursive representations of these higher-order nonlinear shrinkage estimators using partial exponential Bell polynomials.

Through simulation studies, the proposed methods are compared with the oracle nonlinear shrinkage estimator and are shown to be particularly effective in settings where no closed-form expressions for nonlinear shrinkage estimators are available. The theoretical derivations rely on mild assumptions on the underlying model, including the existence of fourth moments and a bounded spectrum of the true population covariance matrix. The finite-sample performance of the proposed estimators is evaluated in an extensive simulation study and benchmarked against existing approaches. Our main finding is that the higher-order shrinkage estimators can outperform well-established nonlinear shrinkage methods, particularly when the concentration ratio p/n is large.

We develop methodology and theory for the detection of a phase transition in a time-series of high-dimensional random matrices. In the model we study, at each time point $ t = 1,2,ldots $, we observe a deformed Wigner matrix $ mathbf{M}_t $, where the unobservable deformation represents a latent signal. This signal is detectable only in the supercritical regime, and our objective is to detect the transition to this regime in real time, as new matrix--valued observations arrive.
Our approach is based on a partial sum process of extremal eigenvalues of $mathbf{M}_t$, and its theoretical analysis combines state-of-the-art tools from random-matrix-theory and Gaussian approximations. The resulting detector is self-normalized, which ensures appropriate scaling for convergence and a pivotal limit,,without any additional parameter estimation. Simulations show excellent performance for varying dimensions. Applications to pollution monitoring and social interactions in primates illustrate the usefulness of our approach.

Random geometric graphs provide a fundamental model for spatially embedded networks, yet their spectral fluctuations remain poorly understood. In this talk, I will present the first rigorous results on Gaussian fluctuations of linear eigenvalue statistics for such graphs. Specifically, we establish central limit theorems for quantities of the form $mathrm{Tr}[phi(A)]$, where $A$ denotes the adjacency matrix and $phi$ belongs to a broad class of test functions, including non-polynomial functions. In the polynomial setting, we go further and prove a quantitative central limit theorem with an explicit rate of convergence to the limiting Gaussian distribution. I will also discuss extensions of these results to other canonical spatial networks, such as $k$-nearest neighbor graphs and relative neighborhood graphs. Together, these results highlight new mechanisms governing spectral fluctuations in random spatial structures and reveal a delicate interplay between geometry, local dependence, and spectral behavior.

The talk is based on joint work with Christian Hirsch (Aarhus) and Kyeongsik Nam (Seoul).