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 $A\otimes I_{n\times n}+I_{n\times n}\otimes B+\Theta \otimes \Xi \in {\mathbb{C}}^{n^2\times n^2}$, 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 ${\min }_{X\in {\mathbb{R}}^{n\times n}}\frac{1}{2}\Vert XA+BX{\Vert }_F^2+\frac{1}{2}{\sum }_{ij}{\xi }_i{\theta }_j{x}_{ij}^2$ 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 $n\to \infty$.

References in corpus (10)

• Spectral statistics of Erdős-Rényi graphs I: Local semicircle law
• Averaging Fluctuations in Resolvents of Random Band Matrices
• Edge Universality for Deformed Wigner Matrices
• Bulk universality for deformed Wigner matrices
• Stability of the Matrix Dyson Equation and Random Matrices with Correlations
• Ridge regression and asymptotic minimax estimation over spheres of growing dimension
• Matrix Concentration Inequalities and Free Probability
• Local law for random Gram matrices
• A Precise High-Dimensional Asymptotic Theory for Boosting and Minimum-${\ell }_1$-Norm Interpolated Classifiers
• On the operator norm of non-commutative polynomials in deterministic matrices and iid GUE matrices