How to solve L1 optimization of Low-Rank matrix?

Question

I am trying to solve the following problem but have some difficulties. Any ideas? Any leads?

$$ \min_{A,C} \|C\|_* +\alpha\|X-A\|_1 \text{ s.t. } A=AC $$ $\|*\|_1$ is the sum of absolute values of the matrix elements

The model is :

• $X$ - is the given data
• $A$ - is clean representative of the data.
• $C$- is the the sparse matrix of the data
• $||C||_*$ is the nuclear norm of matrix C
• $\alpha$ - is some regulator.