I am interested in characterizing the subdifferential of $f=|X^TX|_*$, i.e., the nuclear norm of $X^T X$.
Two ways I am looking at it right now. Certainly, the singular values of $X^T X$ are the square of singular values of $X$. So the subdifferential of $f$ has to be somewhere along the line.
I suspect there gotta be something like a chain rule since $f$ is just a composition of the nuclear norm function and the function $X^T X$, which is a non-linear operator. There is such a formula but then it gets involved with differentiability of non-linear operator which I am not an expert. Any help? Thanks a lot.