2024/06/03 by Fan, Zhou, Ma, Renyuan · 1 citation
#FOS: Mathematics #Probability (math.PR) #Statistics Theory (math.ST)
paper · doi:10.48550/arxiv.2406.00961
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 × n (1)/(2)‖XA+BX‖F2+(1)/(2)∑ij ξiθj xij2 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 → ∞.