Say we've got $A\in\mathbb{R}^{N\times N}$, and $A=U\textrm{diag}(\sigma_1,\sigma_2,\ldots,\sigma_N) V^\top$ is its singular value decomposition. For a function $f\colon\mathbb{R}_{+}\to\mathbb{R}_{+}$, I am curious if there is a method for efficiently evaluating
$$\hat{A}=U\textrm{diag}(f(\sigma_1),\sigma_2,\ldots,\sigma_n)V^\top,$$ without having to compute the full SVD of $A$ (which can be quite expensive).
My motivation in thinking this might be possible is because for eigenvalues, one can compute the largest eigenvalue-eigenvector pair much faster (numerically speaking) than performing a full eigendecomposition. However, I feel like despite this, the answer is probably still no?