Nuclear norm minimization is very popularization and formulation is least squares term with nuclear norm term as following, $$\min\limits_{X} \frac{1}{2}\|X_{3\times3}-Y_{3\times3}\|_F^2+\lambda\|X_{3\times3}\|_{*}.$$
The above minimization can be solved by SVT (singular value thresholding). And the least squares term is based on the hypothesis that the noise is Gausssian noise and i.i.d..
However if the noise is Gaussian noise with different variance, we can get the weighted least squares as following, \begin{equation} \min\limits_{X} \frac{1}{2}\|\big(X_{3\times3}-Y_{3\times3}\big)W_{}\|_F^2+\lambda\|X_{3\times 3}\|_{*} \end{equation}
where $W$ is the diagonal matrix,
$$W=\begin{pmatrix} w_1&0&0 \\0&w_2 &0\\ 0& 0 & w_3\end{pmatrix}$$
My question is how to solve the optimization problem above? Any suggestion or reference paper is appreciated.