solution. The solution to \[vsubprob\] is given by $$\begin{aligned}
\label{vsubexplicit}
\begin{split}
\bsv_{n+1/2}&=\left(\bmP_{\MM}+\mu\La^*\La+\beta_1\bsI\right)^{-1}\left(\bmP_{\MM}\bsf+\mu\La^*\bsW_n^*\bc_n+\beta_1\bsv_n\right),\\
\bsv_{n+1}&=\bmP_{\mC}\left(\bsv_{n+1/2}\right),
\end{split}\end{aligned}$$ where $\bmP_{\mC}$ is defined as \[ProjectOntoC\]. It is worth noting that since $\bmP_{\MM}+\mu\La^*\La+\beta_1\bsI$ is a diagonal matrix, no matrix inversion is needed.

To solve \[Wsubprob,csubprob\], we introduce the following substitutions $$\begin{aligned}
\label{PatchReformulation}
\left(\La\bsv,\bc,\bsW\right)\Leftrightarrow\left(\bsH,\bC,\bsD\right),\end{aligned}$$ where $\bsH=\left[\begin{array}{cc}
\bsH_1&\bsH_2
\end{array}\right]$ with each $\bsH_j$ for $j=1$ and $2$ generated by using $K_1\times K_2$ patches of $\La_j\bsv$. Then \[Wsubprob,csubprob\] can be reformulated as $$\begin{aligned}
\bC_{n+1}&={\operatornamewithlimits{argmin}}~\gamma\left\|\bC\right\|_0+\f{\mu}{2}\left\|\bC-\bsD_{n}^T\bsH_{n+1}\right\|_F^2+\f{\beta_2}{2}\left\|\bC-\bC_n\right\|_F^2,\label{Cupdate}\\
\bsD_{n+1}&={\operatornamewithlimits{argmin}}_{\bsD\bsD^*=M_2^{-1}\bsI}~\f{\mu}{2}\left\|\bsD^T\bsH_{n+1}-\bC_{n+1}\right\|_F^2+\f{\beta_3}{2}\left\|\bsD-\bsD_n\right\|_F^2,\label{Dupdate}\end{aligned}$$ respectively. The closed form solution to \[Cupdate\] is expressed as $$\begin{aligned}
\label{csubexplicit}
\bC_{n+1}=\bmT_{\sqrt{2\gamma/(\mu+\beta_2)}}\left(\f{\mu\bsD_{n}^T\bsH_{n+1}+\beta_2\bC_n}{\mu+\beta_2}\right),\end{aligned}$$ where $\bmT_{\gamma}$ is a hard thresholding; $\bmT_{\gamma}\left(\bC\right)[m,n]=\bC[m,n]$ if $|\bC[m,n]|>\gamma$, and $0$ otherwise. To solve \[Dupdate\], we use the following closed form formula $$\begin{aligned}
\label{Wsubexplicit}
\bsD_{n+1}=M_2^{-1/2}\bX_n\bY_n^*~~~\text{where}~~\bX_n\Sig_n\bY_n^*=\overline{\bsH}_{n+1}\bC_{n+1}^T+\f{\beta_3}{\mu}\bsD_n~~\text{is the SVD}.\end{aligned}$$ Hence, we only compute the SVD of an $M_2\times M_2$ matrix $\overline{\bsH}_{n+1}\bC_{n+1}^T+\left(\beta_3/\mu\right)\bsD_n$, leading to the computational efficiency over directly minimizing the rank of a $2M_1\times M_2$ matrix $\bmH\left(\La\bsv\right)$.

We further mention that it is not necessary to explicitly perform \[PatchReformulation\]. Noting that $\bsH$ is the transpose of $\bmH\left(\La\bsv\right)$, we can compute $\bsD_{n}^T\bsH_{n+1}$ in \[csubexplicit\] and $\overline{\bsH}_{n+1}\bC_{n+1}^T$ in \[Wsubexplicit\] by using $3M_2$ ($M_2$ for $\bsD_{n}^T\bsH_{n+1}$ and $2M_2$ for $\overline{\bsH}_{n+1}\bC_{n+1}^T$) fast Hankel matrix-vector multiplications/two dimensional convolutions directly from $\La\bsv_{n+1}$, requiring $O(M_2N^2\log N)$ floating point operations. In addition, 