can someone point me to the direction how to calculate the derivatives of a sum of singular values of a matrix?
I am trying to minimize
$$\min\_A \parallel A \parallel\_\*+ \cdots $$ where $\parallel A \parallel\_\*=\sum\_i \sigma\_i$ is the sum of the singular values of $A$.


This is motivated by this [paper](https://papers.nips.cc/paper/3704-robust-principal-component-analysis-exact-recovery-of-corrupted-low-rank-matrices-via-convex-optimization.pdf).


Thank you very much!


**Update**:
Here is the full problem:
$$\min\_A \parallel A \parallel\_\* + \lambda \parallel \nabla\_A \parallel D-A-E\parallel\_F^2 \parallel\_F^2$$
with targeted decomposition $D=A+E$


The authors of the mentioned paper use the so called "singular value thresholding" to solve it:


$$u,s,v=SVD(A-\frac{1}{2}(A+E-D))$$
after setting $S=$diag($(s\_i-\tau)\_+$)
$$A\_{k+1}=u\cdot S \cdot v^T$$


However, I dont see how to get there. :(



derivative of sum of singular values
This function is not differentiable (consider $A=0$). If you are interested in learning about its subdifferential (and more on subdifferential of spectral functions), please refer to the excellent papers:


1. [A. Lewis. *Nonsmooth analysis of singular values: Part I*](http://link.springer.com/article/10.1007/s11228-004-7197-7)
2. [A. Lewis. *Nonsmooth analysis of singular values: Part II*](http://link.springer.com/article/10.1007/s11228-004-7198-6)


