Kronecker-product random matrices and a matrix least squares problem
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| author | Fan, Zhou Ma, Renyuan |
| author_facet | Fan, Zhou Ma, Renyuan |
| contents | 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 $n \to \infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_00961 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Kronecker-product random matrices and a matrix least squares problem Fan, Zhou Ma, Renyuan Probability Statistics Theory 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 $n \to \infty$. |
| title | Kronecker-product random matrices and a matrix least squares problem |
| topic | Probability Statistics Theory |
| url | https://arxiv.org/abs/2406.00961 |