Kronecker-product random matrices and a matrix least squares problem

TL;DR

The paper analyzes a Kronecker-product random matrix $Q=A\otimes I+I\otimes B+\Theta\otimes\Xi$ with $A,B$ Wigner and $\Theta,\Xi$ diagonal, establishing quantitative deterministic equivalents for the resolvent $G(z)=(Q-zI)^{-1}$ and its Stieltjes transform $m(z)$ via operator-valued fixed-point equations.A two-stage Schur-complement approach, augmented by fluctuation averaging and an operator-algebra framework, yields sharp global bounds on resolvent blocks $G_{ii},G_{ij}$ and precise three-scale entry behavior, namely $O(1)$, $O(n^{-1/2})$, and $O(n^{-1}) depending on block location.These resolvent results underpin an asymptotic characterization of a related matrix-valued least-squares problem, providing deterministic equivalents for the minimum value and linear projections of the minimizer, expressed through traces of the operator-algebra fixed points.The work advances the understanding of Kronecker-structured random models, connecting mean-field-type operator fixed points to traditional resolvent techniques, and offers practical tools for high-dimensional matrix optimization under random data.

Abstract

We study the eigenvalue distribution and resolvent of a Kronecker-product random matrix model $A \otimes I_{n \times n}+I_{n \times n} \otimes B+Θ\otimes Ξ\in \mathbb{C}^{n^2 \times n^2}$, 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 $n \times n$ 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 $\min_{X \in \mathbb{R}^{n \times n}} \frac{1}{2}\|XA+BX\|_F^2+\frac{1}{2}\sum_{ij} ξ_iθ_j x_{ij}^2$ 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 $n \to \infty$.

Figures (2)

• Figure 1: Entrywise modulus of the resolvent $G(z)=(A \otimes I+I \otimes B+\Theta \otimes \Xi-z)^{-1}$ at $z=i$, where $n=20$, $A,B$ are independent GOE matrices of size $n$, and $\Theta,\Xi$ have independent $\text{Uniform}(-1,1)$ diagonal entries.
• Figure 2: Values of $1/(2f(\widehat{X}))$ obtained from solving (\ref{['eq:opt']}) across 10 independent realizations (solid dots, with vertical lines indicating 1 standard deviation) versus the theoretical prediction $T(\mathcal{P},\mathcal{Q})$ computed from Proposition \ref{['prop:computation']} with $M=12$ (dashed lines). Here $A,B$ are GOE matrices of size $n=1000$, we take $\theta_i^{-1},\xi_\alpha^{-1},u_i^2,v_\alpha^2 \sim \text{Uniform}(0.05,0.5)/k$ with $k \in \{1,1.3,3\}$, and the horizontal axis indicates the correlation between coordinate pairs $(\theta_i^{-1},u_i^2)$ and between $(\xi_\alpha^{-1},v_\alpha^2)$.

Theorems & Definitions (63)

• Proposition 2.3
• Theorem 2.4
• Theorem 2.5
• Corollary 2.6
• Proposition 2.7
• Definition 3.1: Stochastic domination
• Lemma 3.2
• proof
• Remark 3.3
• Lemma 3.4