Zhou Fan, Yale University

Abstract

We study the eigenvalue distribution and resolvent of a Kronecker-product random matrix model $A otimes I_{n times n}+I_{n times n} otimes B+Theta otimes Xi in mathbb{C}^{n^2 times n^2}$, where $A,B$ are independent Wigner matrices and $Theta,Xi$ are deterministic and diagonal. For fixed spectral arguments, we establish a quantitative approximation for the Stieltjes transform by that of an approximating free operator, and a diagonal deterministic equivalent approximation for the resolvent. We further obtain sharp estimates in operator norm for the $n times n$ resolvent blocks, and show that off-diagonal resolvent entries fall on two differing scales of $n^{-1/2}$ and $n^{-1}$ depending on their locations in the Kronecker structure.

Our study is motivated by consideration of a matrix-valued least-squares optimization problem $min_{X in mathbb{R}^{n times n}} frac{1}{2}|XA+BX|_F^2+frac{1}{2}sum_{ij} xi_itheta_j x_{ij}^2$ subject to a linear constraint. For random instances of this problem defined by Wigner inputs $A,B$, our analyses imply an asymptotic characterization of the minimizer $X$ and its associated minimum objective value as $n to infty$.

This is joint work with Jack (Renyuan) Ma.

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