Derivative of the nuclear norm The nuclear norm is defined in the following way
$$\|X\|_*=\mathrm{tr} \left(\sqrt{X^T X} \right)$$
I'm trying to take the derivative of the nuclear norm with respect to its argument
$$\frac{\partial \|X\|_*}{\partial X}$$
Note that $\|X\|_*$ is a norm and is convex. I'm using this for some coordinate descent optimization algorithm. Thank you for your help.

The challenges of calculating the gradients of $||X||_{*}$ comes from the non-smooth processing of calculating the singular values of $X$. 
Thus, I usually first transform $X$ into a symmetric positive semi-definite matrix $\hat{X}$, and then we have $||\hat{X}||_{*}=tr(\hat{X})$. The intuitive is that the eignvalues are equals to singular values, when $\hat{X}$ is a symmetric  positive semi-definite matrix, and $tr(\hat{X})=\sum eignvalues$.
Finally, we have $\frac{\partial ||\hat{X}||_{*}}{\partial \hat{X}}=\frac{\partial tr(\hat{X})}{\partial \hat{X}}=I$.
I am not sure whether it is helpful to you.
