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.
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. :(
