Georgia Institute of Technology College of Sciences

In this talk, I will present the results of a collaboration with Benjamin McKenna on the injective norm of large random Gaussian tensors and uniform random quantum states, and describe some of the context underlying this work. The injective norm is a natural generalization to tensors of the operator norm of a matrix and appears in multiple fields. In quantum information, the injective norm is one important measure of genuine multipartite entanglement of quantum states, known as geometric entanglement. In our preprint, we provide a high-probability upper bound on the injective norm of real and complex Gaussian random tensors, which corresponds to a lower bound on the geometric entanglement of random quantum states, and to a bound on the ground-state energy of a particular multispecies spherical spin glass model.

Suppose that particles are randomly distributed in $R^d$, and they are subject to identical stochastic motion independently of each other. The Smoluchowski process describes fluctuations of the number of particles in an observation region over time. The goal is to infer on particle displacement process from such count data. We discuss probabilistic properties of the Smoluchowski processes and consider related statistical problems for two different models of the particle displacement process: the undeviated uniform motion (when a particle moves with random constant velocity along a straight line) and the Brownian motion displacement. In these settings we develop estimators with provable accuracy guarantees.

Data imbalance is a fundamental challenge in data analysis, where minority classes account for a small fraction of the training data compared to majority classes. Many existing techniques attempt to compensate for the underrepresentation of minority classes, which are often critical in applications such as rare disease detection and anomaly detection. Notably, in empirical deep learning, the large model size exacerbates the issue. However, despite extensive empirical heuristics, the statistical foundations of these methods remain underdeveloped, which poses an issue to the reliability of these machine learning models.

In this talk, I will examine imbalanced classification problems in high dimensions, focusing on support vector machine (SVMs) and logistic regression. I will introduce a "truncation" phenomenon---which we verifed across single-cell tabular data, image data, and text data---where overfitting in high dimensions distorts the distribution of logits on training data. I will provide a theoretical foundation by characterizing the asymptotic distribution via a variational formulation. This analysis formalizes the intuition that overfitting disproportionately harms minority classes and reveals how margin rebalancing---a widely used deep learning heuristic---mitigates data imbalance. As a consequence, the theory offers both qualitative and quantitative insights into generalization errors and uncertainty measures such as calibration.

This talk is based on a joint work with Jingyang Lyu (3rd-year Stats PhD student) and Kangjie Zhou (Columbia Statistics): arXiv:2502.11323.

N. Zabusky coined the word "soliton" in 1965 to describe a curious feature he and M. Kruskal observed in their numerical simulations of the initial-value problem for a simple nonlinear PDE. The first part of the talk will be a broad introduction to the theory of solitons/solitary waves and integrable PDEs (the Korteweg-de Vries equation in particular), describing classical results in the field. The second (and main) part of the talk will focus on some new developments and growing interest into a special case of solutions defined as "soliton gas".

We study random configurations of N soliton solutions q_N(x,t) of the KdV equation. The randomness appears in the scattering (linear) problem, which is used to solve the PDE: the complex eigenvalues are chosen to be (1) i.i.d. random variables sampled from a probability distribution with compact support on the complex plane, or (2) sampled from a random matrix law. 

Next, we consider the scattering problem for the expectation of the random measure associated to the spectral data, in the limit as N -> + infinity. The corresponding solution q(x,t) of the KdV equation is a soliton gas. 

We are then able to prove a Law of Large Numbers and a Central Limit Theorem for the differences q_N(x,t)-q(x,t).

This is a collection of works (and ongoing collaborations) done with K. McLaughlin (Tulane U.), T. Grava (SISSA/Bristol), R. Jenkins (UCF), A. Minakov (U. Karlova), J. Najnudel (Bristol).