Iterative Thresholding - The Low-rank component derivation

Question

I am trying to understand the derivation of iterative thresholding from this paper. A convex problem defined as, $$\min_{A,E}||A||_*+\lambda||E||_1+\frac{1}{2\tau}||A||^2_F+\frac{1}{2\tau}||E||^2_F\quad\text{s.t. }D=A+E$$ which can be reformulate as Lagrange function, $$\mathcal{L}(Y,A,E)= ||A||_*+\lambda||E||_1+\frac{1}{2\tau}||A||^2_F+\frac{1}{2\tau}||E||^2_F+\frac{1}{\tau}\langle Y,D-A-E \rangle$$ I will focus on the $A$ component (the low rank), which a well-known analysis shows that, $$U\mathcal{S}_\epsilon[S]V^H=\text{arg}\min_{X}||X||_*+||X-W||^2_F$$ where, $USV^H$ is the SVD of $W$. My understanding is that I have to reform the Lagrange function into the well-known form, which leads to, $$||L-Y||^2_F =||A||^2+\langle Y,D-A-E \rangle$$ I'm still unable to prove this, is it the correct way to implement it? Thank you in advance!