I am trying to solve the matrix completion problem with proximal gradient method: $$\min_{||X||_* \leq \theta} \frac{1}{2}\sum_{(i, j) \in \Omega} (X_{ij} - M_{ij})^2$$ or in terms of the projection operator: $$\min_{||X||_* \leq \theta} \frac{1}{2} ||P_\Omega(X) - P_\Omega(M)||_F^2$$
Now, I know how to solve $$\min_{||X||_* \leq \theta} \frac{1}{2} ||X-Y||_F^2$$ since that is just $X = U D^* V^T$ where $Y = UDV^T$ and $D^* = diag(d^*)$ and $D = diag(d)$ where $d^*$ is the solution to $$\min_{||x||_1 \leq \theta} ||x-d||^2$$. Can someone point in the right direction on how to apply this to solve the matrix completion problem using proximal gradient method?