« search calendars« DIMACS Workshop on Modeling Randomness in Neural Network Training: Mathematical, Statistical, and Numerical Guarantees

« Kronecker-product Random Matrices and a Matrix Least-squares Problem

June 05, 2024, 11:05 AM - 11:50 AM

Location:

DIMACS Center

Rutgers University

CoRE Building

96 Frelinghuysen Road

Piscataway, NJ 08854

Click here for map.

Zhou Fan, Yale University

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 $min_{X in mathbb{R}^{n times n}} frac{1}{2}|XA+BX|_F^2+frac{1}{2}sum_{ij} xi_itheta_j x_{ij}^2$ 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 $n to infty$.

This is joint work with Jack (Renyuan) Ma.