Consider functions as $f:\mathbb{R}^{m\times n}\rightarrow\mathbb{R}^{m\times n}$. I am trying to find examples of these functions that are cheaper to evaluate (numerically) on low-rank matrices (for which the SVD is known) than on full-rank ones.
Can you suggest some examples?
More precisely, suppose that to compute $f(A)$ we need time $t$. Let $\Pi_r(A) = USV^T$ be the rank $r$ truncated SVD of $A$. What has to satisfy $f$ in order to have that $f(USV^T)$ can be evaluated in a time $t'<t$ for every $A$?