Let $A = U\Sigma V^T$ be the singular value decomposition (SVD) of a $n\times m$ matrix $A$. Let $\tilde{A} = A + \epsilon P$ be a perburbation of $A$. It is possible, using tools from Matrix Perturbation Theory, to express the singular vectors and values of $\tilde{A}$ as a Taylor series in $\epsilon$ and the singular vectors and values of $A$. For example, the first-order expansion of the $i$-th perturbed singular value is:

$$ \tilde{\sigma_i} = \sigma_i + \epsilon\cdot \mathbf{u}_i^T P \mathbf{v}_i + O(\epsilon^2). $$

See, for example, *Liu et al., First-Order Perturbation Analysis of Singular Vectors in Singular Value Decomposition*.

Is there a published paper or book detailing the **second-order** expansion for the singular vectors and values?

I am aware of the following papers:

*Vaccaro, A second-order perturbation expansion for the SVD*- This paper analyzes the perturbation in the column space of $A$ and its orthogonal complement, but not of individuals singular vectors or values.*Stewart, Perturbation Theory for the Singular Value Decomposition*- does not include this analysis.