Future talks

Winter 2025

March 26, 2025, 4:00–5:00pm, EH3096

Speaker: Reza Gheissari (Northwestern)

Title: Local geometry and spectral transitions in high-dimensional classification

Abstract: We study the spectral theory of a family of random matrices that are variants on empirical covariance matrices, with entries that can have correlations through projections of the data onto some O(1)-many directions. These matrices arise naturally when looking at the empirical Hessians of many high-dimensional statistical tasks at different points in parameter space, to probe the local geometry of their loss landscapes. We prove limits for the bulk distribution and any outlier eigenvalues, in a way that only depends on the point in parameter space through finitely many “summary statistics” of the parameter. This allows us to probe the evolution of the Hessian as one moves through parameter space via some training dynamics like stochastic gradient descent, establishing interesting phenomena like splitting and emergence of outliers over the course of training in basic classification tasks. Based on joint work with G. Ben Arous, J. Huang, and A. Jagannath.

April 2, 2025, 4:00–5:00pm, EH3096

Speaker: Marcus Michelen (UIC)

Title: New lower bounds for sphere packings and independent sets via randomness

Abstract: We construct new lower bounds for sphere packings in high dimensions and for independent sets in graphs with not-too-large co-degrees. For dimension d, this achieves a sphere packing of density (1 + o(1)) d log d / 2^(d+1). In general dimension this provides the first asymptotically growing improvement for sphere packing lower bounds since Rogers' bound of c*d/2^d in 1947. The proof amounts to a random (very dense) discretization together with a new theorem on constructing independent sets on graphs with not-too-large co-degree. Both steps will be discussed and no knowledge of sphere packings will be assumed or required. Central to the analysis is the study of a random process on a graph.  This is based on joint work with Marcelo Campos, Matthew Jenssen and Julian Sahasrabudhe. 

April 16, 2025, 4:00–5:00pm, EH3096

Speaker: TBA

Title: TBA

April 23, 2025, 4:00–5:00pm, EH3096

Speaker: Félix Parraud (Queen's University) 

Title: TBA