Abstract: On the real line, a random walk that can only move in the positive direction is very unlikely to remain close to its starting point. After a fixed number of steps, the left tail has a Gaussian profile, under minimal assumptions. Remarkably, the same phenomenon occurs when we consider a positive random walk on the cone of positive-semidefinite matrices. After a fixed number of steps, the minimum eigenvalue is also described by a Gaussian model.
This talk introduces a new way to make this intuition rigorous, and it presents an application to the sample covariance matrix of a very sparse vector. The key technical ingredient is Stahl's theorem (2012), recently improved by Otte Heinävaara (PhD Princeton, 2024).
Prof. Joel A. Tropp - Bio
Prof. Joel A. Tropp
Joel A. Tropp is Steele Family Professor of Applied & Computational Mathematics at the California Institute of Technology. His research centers on applied mathematics, machine learning, data science, numerical algorithms, and random matrix theory. Some of his best-known contributions include matching pursuit algorithms, randomized SVD algorithms, and matrix concentration inequalities.
Prof. Tropp attained the Ph.D. degree in Computational Applied Mathematics at the University of Texas at Austin in 2004, and he joined Caltech in 2007. He won the PECASE in 2008, and he was recognized as a Highly Cited Researcher in Computer Science each year from 2014–2018. He is co-founder and Section Editor of the SIAM Journal on Mathematics of Data Science (SIMODS), and he was co-chair of the inaugural 2020 SIAM Conference on the Mathematics of Data Science. Prof. Tropp was elected SIAM Fellow in 2019, IEEE Fellow in 2020, and IMS Fellow in 2024.