Kronecker-product random matrices and a matrix least squares problem

Abstract

We study the eigenvalue distribution and resolvent of a Kronecker-product random matrix model A⊗In×n+In×n⊗B+Θ⊗Ξ∈Cn2×n2, where A,B are independent Wigner matrices and Θ,Ξ 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×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 minX∈Rn×n1 2‖XA+BX‖F2+1 2∑ ijξiθjxij2 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→∞.