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Kronecker-product random matrices and a matrix least squares problem

3 June 2024
Zhou Fan
Renyuan Ma
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Abstract

We study the eigenvalue distribution and resolvent of a Kronecker-product random matrix model A⊗In×n+In×n⊗B+Θ⊗Ξ∈Cn2×n2A \otimes I_{n \times n}+I_{n \times n} \otimes B+\Theta \otimes \Xi \in \mathbb{C}^{n^2 \times n^2}A⊗In×n​+In×n​⊗B+Θ⊗Ξ∈Cn2×n2, where A,BA,BA,B are independent Wigner matrices and Θ,Ξ\Theta,\XiΘ,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×nn \times nn×n resolvent blocks, and show that off-diagonal resolvent entries fall on two differing scales of n−1/2n^{-1/2}n−1/2 and n−1n^{-1}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∈Rn×n12∥XA+BX∥F2+12∑ijξiθjxij2\min_{X \in \mathbb{R}^{n \times n}} \frac{1}{2}\|XA+BX\|_F^2+\frac{1}{2}\sum_{ij} \xi_i\theta_j x_{ij}^2minX∈Rn×n​21​∥XA+BX∥F2​+21​∑ij​ξi​θj​xij2​ subject to a linear constraint. For random instances of this problem defined by Wigner inputs A,BA,BA,B, our analyses imply an asymptotic characterization of the minimizer XXX and its associated minimum objective value as n→∞n \to \inftyn→∞.

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