presentations

2026

1. Generalization, memorization, and overfitting for diffusion models trained in the lazy high-dimensional regime

   Seminar talk

   Sep 2026

   Yale Theory Student Seminar, Yale University, New Haven, CT

   Abstract Website

   Diffusion models, which are widely used for tasks such as image generation, reduce distribution learning to a sequence of denoising regression problems. In theory, solving these regression problems perfectly on a finite dataset leads to memorization of the training samples, while in practice diffusion models can generate genuinely new samples. This raises the question: what in the training and sampling pipeline prevents memorization, and what do diffusion models actually learn?
   
   To partially address this question, we consider a class of score estimators obtained by running gradient flow in a reproducing kernel Hilbert space in the high-dimensional regime. I will present results describing the evolution of the learned score during training, from generalization to overfitting and eventually memorization, and then study what happens when these learned scores are composed through the reverse diffusion process.
   
   This is joint work with Sinho Chewi, Aram-Alexandre Pooladian, John Sous, and Theodor Misiakiewicz.

2. Generalization, Memorization, and Overfitting in Kernel Denoising Score Matching in High Dimensions

   Poster presentation

   Aug 2026

   Princeton Machine Learning Theory Summer School, Princeton University, Princeton, NJ

2024

1. Introduction to the Matrix Dyson Equation

   Talk

   Mar 2024

   ETH Zurich

   Notes

2023

1. Matrix Dyson Equation for Correlated Linearizations

   Workshop talk

   Dec 2023

   The Many Facets of Random Matrix Theory workshop, 2023 Canadian Mathematical Society Winter Meeting, Montreal, QC

   Abstract Slides Website

   The exploration of large random matrices through asymptotic deterministic equivalents has been approached by a multitude of techniques. One approach employs the matrix Dyson equation to establish an asymptotic equivalence between a random resolvent and the solution of a matrix fixed point equation. Another, the linearization trick, has proven effective in studying rational functions of random matrices. This trick involves embedding a matrix expression into a larger random matrix, known as a linear matrix pencil, with a simplified correlation structure. In this presentation, we introduce an extension of the matrix Dyson equation framework tailored specifically for linearizations. This extends previous work which has focused primarily on the case of pencils with blocks of canonical Wigner or Circular type. Within this framework, we derive an anisotropic global law for a broad class of pseudo-resolvents with general correlation structures. To highlight the practical implications of our framework, we apply it to a problem coming from machine learning. Specifically, we apply it to derive an exact asymptotic expression for the validation error of random features ridge regression and establish a general Gaussian equivalence result.

2. Matrix Dyson Equation for Correlated Linearizations

   Seminar talk

   Sep 2023

   Seminar in Machine Learning, Optimization and Random Matrix Theory, McGill University, Montreal, QC

   Abstract Website

   Random matrix theory has proved to be a valuable tool for understanding puzzling aspects of machine learning models. There are various methods in random matrix theory, each depending on different structural assumptions for the given problem. One of these methods essentially suggests that we can analyze the spectrum of a large random matrix by substituting the associated random resolvent with the solution of a deterministic fixed-point equation known as the matrix Dyson equation. In this talk, based on collaborative work with KC Tsiolis and Elliot Paquette, I will discuss the extension of the matrix Dyson equation framework to analyze rational expressions in random matrices using a linearization trick. I will also provide an illustrative example by applying our framework to study the test error of a random feature model.

2021

1. GD and Large Linear Regression: Concentration and Asymptotics for a Spiked Model

   Conference talk

   Aug 2021

   4th Undergraduate Student Research Conference, McGill University, Montreal, QC

   Abstract Slides Website

   Machine learning algorithms are often trained using a large set of samples in a high-dimensional feature space. The traditional methods that performed well for low-dimensional problems can run into several problems in analyzing such high-dimensional models. One difficulty is that the analysis can depend on the probability distribution of the inputs. However, we show that this is not the case for a model of large-scale spiked random least squares problem trained with gradient descent. In fact, we show and verify experimentally that the halting time exhibits a universality property: it is independent of the probability distribution of the inputs. Furthermore, we provide explicit asymptotic results and discuss the effect of various parameters on those asymptotics.