Perturbation analysis for the matrix least squares problem $AXB=C$

Abstract

Let $S$ and $\widehat{S}$ be two sets of solutions to matrix least squares problem (LSP) $AXB=C$ and the perturbed matrix LSP $\widehat{A}\widehat{X}\widehat{B}=\widehat{C}$, respectively, where $\widehat{A}=A+\Delta A$, $\widehat{B}=B+\Delta B$, $\widehat{C}=C+\Delta C$, and $\Delta A$, $\Delta B$, $\Delta C$ are all small perturbation matrices. For any given $X\in S$, we deduce general formulas of the least squares solutions $\widehat{X}\in \widehat{S}$ that are closest to $X$ under appropriated norms, meanwhile, we present the corresponding distances between them. With the obtained results, we derive perturbation bounds for the nearest least squares solutions. At last, a numerical example is provided to verify our analysis.