Probability seminar series
Zhou Fan
Yale University
Room: M3 3127
Kronecker-product random matrices and a matrix least squares problem
In recent years, high-dimensional probabilistic analyses have yielded important insights into the exact asymptotic behavior of many optimization problems arising in statistical learning applications. In most such examples, the optimization variable is a high-dimensional vector whose behavior is characterized by a set of scalar fixed-point equations in the large-system limit, derived via mean-field approximation over an unstructured matrix of random data.
Motivated loosely by optimization problems arising in applications of random graph matching, I will discuss a probabilistic analysis of a different type of least-squares problem, having a matrix-valued optimization variable and a Kronecker-product mean-field structure. The main results are an asymptotic characterization of the optimal solution and its objective value, obtained by a quantitative deterministic-equivalent characterization of the resolvent of a certain Kronecker-structured random matrix model. I will present the main ideas of the analysis of this model, which draw upon tools of free probability and random matrix theory.
Based on https://arxiv.org/pdf/2406.00961, joint work with Renyuan Ma.