Perturbation analysis for subspace decomposition with applications in subspace-based algorithms

Subspace decomposition has been exploited in different applications. Due to perturbations from various sources such as finite data samples and measurement noise, perturbations arise in subspaces. Therefore, some loss is introduced to performance of subspace-based algorithms. Although first-order perturbation results have been proposed in the literature and applied to various problems, up to second-order perturbation analysis can provide more accurate analytical results and is studied in this paper. Based on the orthogonality principle, perturbations of subspaces and singular values (or eigenvalues) are derived explicitly as functions of a perturbation in the objective matrix up to the second order, respectively, all in closed forms. It is shown that by keeping only the first-order terms, the derived results reduce to those from existing approaches. Examples to apply the proposed results to both matrix computation and subspace-based channel estimation are provided to verify our analysis.