On Stewart's Perturbation Theorem for SVD
Ren-Cang Li, Ninoslav Truhar, Lei-Hong Zhang
TL;DR
This work develops a spectral-norm–centered variant of Stewart's perturbation theorem for SVD, enabling the use of any unitarily invariant norm to obtain sharper bounds on perturbed singular subspaces. By fixing a decomposition $G = U\begin{pmatrix} G_1 & 0 \\ 0 & G_2 \end{pmatrix}V^{\mathrm{H}}$ and analyzing $\widetilde{G}=G+E$, the authors formulate a pair of coupled Sylvester equations whose solution $(\Gamma,\Omega)$ yields a near-block-diagonal representation of $\widetilde{G}$ into $\check G_1$ and $\check G_2$, with the singular values of $\widetilde{G}$ equal to the union of those of $\check G_1$ and $\check G_2$. Under a spectral-gap condition $\bar{\delta}>0$ and a contraction bound on the perturbation, explicit bounds on $\| (\Gamma,\Omega) \|_{\mathrm{ui}}$ and on $\sigma_{\min}(\check G_1)$ and $\sigma_{\max}(\check G_2)$ are derived, along with a bound on the subspace perturbation measured by unitarily invariant norms. The spectral-norm specialization yields $m$- and $n$-independent results and sharper sin$\Theta$-type theorems for SVD, with Frobenius-norm variants and applications to reduced-order modeling and balanced truncation highlighted. Overall, the paper provides a versatile framework for perturbation analysis of SVD that improves on Stewart's original bounds by exploiting the spectral norm and UI norms.
Abstract
This paper establishes a variant of Stewart's theorem (Theorem~6.4 of Stewart, {\em SIAM Rev.}, 15:727--764, 1973) for the singular subspaces associated with the SVD of a matrix subject to perturbations. Stewart's original version uses both the Frobenius and spectral norms, whereas the new variant uses the spectral norm and any unitarily invariant norm that offer choices per convenience of particular applications and lead to sharper bounds than that straightforwardly derived from Stewart's original theorem with the help of the well-known equivalence inequalities between matrix norms. Of interest in their own right, bounds on the solution to two couple Sylvester equations are established for a few different circumstances.