Both "eigenvalue perturbation" and "matrix perturbation" seems to study this scenario that given a matrix $A$, if we add something to it like $\tilde{A} = A+E$, how will the eigenvalues of $\tilde{A}$ distribute differently from the eigenvalues of $A$.

I want to ask a reverse question here: given $A=USV^T$, and we change $\tilde{S} = SD$, then how will $\tilde{A}=U\tilde{S}V^T$ distribute differently from $A$?. **What are some relevant studies that I can look into?**

I guess this is very likely to be a hard problem in general, but I'm currently only hoping to find some solution to a more practical problem.

To reiterate the problem here briefly: truncated SVD (TSVD) is well known as an effective method for denoising data, which basically means that: for $\tilde{A} = A + E$, we take the SVD of $ \tilde{A} = U\tilde{S}V^T$, and then reconstruct with first $k$ dominant singular values $\hat{A} = \sum_i^kU_i\tilde{S}_iV_i^T$, then empirically $\hat{A}$ is a better approximation of $A$ than $\tilde{A}$. **I wonder how to show this conclusion officially, given whatever properties we need to assume about $A$, $E$, and $k$.**

I tried to make some progress in a more specific setting, say $A$ is symmetric, p.s.d, block diagonal structure, with all elements to be positive, and the expectation of these values (denoted as $a$) is greater than the expectation of values in $E$ (with a gap denoted as $c$). $E$ is a dense matrix with all elements to be positive. Now I can work with $\hat{A}_{ij} = \tilde{A}_{ij} - \delta_{ij}$, then I got stuck and need some help:

**With large enough $k$, can I just assume $\delta$ are uniformly distributed across the entire matrix?****If not, is there any relation between the $\delta$ within block-diagonal and the $\delta$ off-diagonal that can be represented by $A, E, a,c,k$?**

Since this is a relatively long post and I asked a hierarchy of questions, I put my questions in bold, any insights of any of these questions will be helpful!