I'm trying to solve this minimization problem: given a matrix $X$ with at least one singular value greater than 1, find
$ \min_Z\; \langle X-Z,X-Z \rangle$ subject to the constraint $ \|Z\| = \max_i\; \sigma_i(Z) \le 1 $, where $\langle A,B \rangle = \mathrm{tr}(AB^\top)$
I'm tempted to just replace the singular values of $X$ that are greater than $1$ with exactly one, but I can't prove that that's correct. It's certainly right for the equivalent vector case of projection onto the $\ell_\infty$-ball.