I have an overdetermined system $A\vec{x}=\vec{b}$. I solve it using SVD pseudoinverse trick: $\vec{x}=VS^{-1}U^{T}\vec{b}$. Also, I truncate some small singular values. Now let's say I want to understand what is happening with my solution under random perturbations of matrix $A$ (matrix $A$ is some measurements and then we add an experimental noise matrix $E$, getting a new matrix $A+E$). There are two theorems about perturbed SVD, namely Mirsky and Wedin. The former claims an upper bound for a singular value change: $\sum_{i}(\sigma_{i}^{perturbed}-\sigma_{i})^2<||E||_{f}$. The latter claims an upper bound for a singular vector change: $\sin(v_{perturbed},v)<\frac{C||E||_{f}}{\delta}$, where $\delta=\sigma_{i}-\sigma_{i-1}$ is the difference between the singular values for $\vec{v}$ and its closest neighbor. Knowing this, I have questions:
- How can I estimate the change in my solution $\vec{x}_{perturbed}=\vec{x}+\delta{\vec{x}}$, based on these theorems or maybe by some other tricks?
- Let's say I know the probability distribution of the noise $E$ (e.g. gaussian). Is it possible to directly calculate $\vec{x}_{perturbed}$ based only on $E$ and $\vec{x}$ without actually solving the system?