First-Order Perturbation Analysis of Singular Vectors in Singular Value Decomposition

The perturbation analysis of singular value decomposition (SVD) has been well documented in the literature within the context of subspace decomposition. The contribution of the signal subspace to the perturbation of the singular vectors that span the signal subspace is often ignored as it is treated as second and higher order terms, and thus the first-order perturbation is typically given as the column span of the noise subspace. In this paper, we show that not only the noise subspace, but also the signal subspace, contribute to the first-order perturbation of the singular vectors. We further show that the contribution of the signal subspace does not impact on the performance analysis of algorithms that rely on the signal subspace for parameter estimation, but it affects the analysis of algorithms that depends on the individual basis vectors. For the latter, we also give a condition under which the contribution of the signal subspace to the perturbation of singular vectors may be ignored in the statistical sense.