Proximity of Weighted and Layered Least Squares Solutions

TL;DR: Vavasis and Ye proved that the limiting solution exists, and showed that the limit is characterized as the solution of a variant of the least squares problem called the layered least squares (LLS) problem.

Abstract: In this paper, we analyze the limiting behavior of the weighted least squares (WLS) problem $\min_{x\in\Re^n}\sum_{i=1}^p\|D_i(A_ix-b_i)\|^2$, where each $D_i$ is a positive definite diagonal matrix. We consider the situation where the magnitude of the weights differs drastically from one block to the next so that $\max(D_1)\geq\min(D_1)\gg\max(D_2)\geq\min(D_2)\gg\max(D_3)\geq\cdots\gg\max(D_{p-1})\geq\min(D_{p-1})\gg\max(D_p)$. Here $\max(\cdot)$ and $\min(\cdot)$ represent the maximum and minimum entries of diagonal elements, respectively. Specifically, we consider the case when the gap $g\equiv\min_i1/(\|D_i^{-1}\|\|D_{i+1}\|)$ is very large or tends to infinity. Vavasis and Ye proved that the limiting solution exists (when the proportion of diagonal elements within each block $D_i$ is unchanged and only the gap $g$ tends to $\infty$), and showed that the limit is characterized as the solution of a variant of the least squares problem called the layered least squares (LLS) problem. We analyze the difference between the solutions of WLS and LLS quantitatively and show that the norm of the difference of the two solutions is bounded above by $O(\chi_A\bar{\chi}_A^{2(p+1)}g^{-2}\|b\|)$ and $O(\bar{\chi}_A^{2p+3}g^{-2}\|b\|)$ in the variable and the residual spaces, respectively, using the two condition numbers $\chi_A\equiv\max_{B\in{\cal B}}\|B^{-1}\|$ and $\bar{\chi}_A\equiv\max_{B\in{\cal B}}\|AB^{-1}\|$ of $A$, where ${\cal B}$ is the set of all nonsingular $n\times n$ submatrices of $A$, $A=[A_1;\ldots;A_p]$ and $b = [b_1;\ldots;b_p]$. A remarkable feature of this result is that the error bound is represented in terms of $A$, $g$ (and $b$) and independent of the weights $D_i$, $i=1,\ldots,p$. The analysis is carried out by making the change of variables to convert the matrix $A$ into a basis lower-triangular form and then by applying the Sherman-Morrison-Woodbury formula.