## 3 Homotopy DCD with Non-Convex Regularization Algorithms
\begin{table}
\begin{tabular}{l l} \hline
**Step** & **Equation** \\ \hline  & **Initialization:**\(\mathbf{x}=\mathbf{0},I=\emptyset,\mathbf{r}=\mathbf{y},\mathbf{b}=\mathbf{A}^{H} \mathbf{r}_{t}\mathbf{R}=\mathbf{A}^{H}\mathbf{A}\) \\ \hline
1 & Choose the first (\(t\)-th) element into the support according to: \\  & \(t=\arg\max_{k}|b_{k}|^{2}\); \(I=\{t\},\tau=|b_{t}|\) \\
2 & Repeat until a termination condition is met: \\
3 & Solve \(\min_{\mathbf{x}_{I}}\left|\mathbf{y}-\mathbf{A}_{I}\mathbf{x}_{I}\right|_{2}^ {2}/2+\tau\sum_{n\in I}|x_{n}|/(1+a|x_{n}|/2)\) on the support \(I\) and \\  & update \(\mathbf{r}\) using DCD iterations \\
4 & Update the regularization parameter: \(\tau\leftarrow\gamma\tau,0<\gamma<1\) \\
5 & Remove the \(t\)-th element from \(I\) according to the rule: \\  & \(t=\arg\min_{k\in I}|x_{k}|^{2}R_{,k\in I}/2+\mathrm{Re}\{x_{k}^{*}b_{k}\}- \tau\,|x_{k}|/(1+a|x_{k}|/2)\) \(s.t.\) \\  & \(|x_{t}|^{2}R_{,t}/2+\mathrm{Re}\{x_{t}^{*}b_{t}\}-\tau\,|x_{t}|/(1+a|x_{t}|/2)<0\) \\  & If the \(t\)-th element is removed, update \(\mathbf{r}\): \(\mathbf{r}\leftarrow\mathbf{r}+x_{t}\mathbf{A}^{(t)}\), \(I\gets I/t\) \\
6 & Add the \(t\)-th element into \(I\) according to the rule: \\  & \(t=\arg\max_{k\in I^{c}}\left(a|b_{k}|-R_{,k\in I}\right)/aR_{,k\in I}^{2}\cdot \left\{\left(R_{,k\in I}+a|b_{k}|\right)^{2}-4 aR_{,k\in I}\tau\right\}\)\(s.t.\)\(\left(R_{,t\in I}+a|b_{t}|\right)^{2}>4 aR_{,t\in I}\tau\) \\  & If the \(t\)-th element is added, update: \(I\gets I\cup t\) \\ \hline \end{tabular}
\end{table} 
Table 2: HR-DCD Algorithm. The HR-DCD algorithm starts with the zero support \(I=\emptyset\), and \(\tau=|b_{t}|\). For each \(\tau\), it updates the support and minimizes the cost function according to the corresponding propositions. Besides, the algorithm will terminate if the iteration times reach the preset parameter \(L\) or \(\tau\leq\tau_{\min}\) (\(\tau_{\min}\) is a predefined parameter). **Remark 1:** When the penalty is logarithmic function, we give two similar propositions as follows:  
**Proposition 3:** Add the \(t\)-th (\(t\in I^{c}\)) element into the support according to the rule:  
\[t=\arg\max_{k\in r^{c}}|b_{k}|\quad s.t.\ \big{|}b_{t}|>\tau \tag{16}\]  
**Proposition 4:** Remove the \(t\)-th (\(t\in I\)) element from the support according to the rule:  
\[t=\arg\min_{k\in r^{c}}\frac{1}{2}\big{|}x_{k}\big{|}^{2}\ R_{k,k}+\Re\big{\{}x _{k}^{\prime}b_{k}\big{\}}-\tau\frac{1}{a}\log\big{(}1+a\big{|}x_{k}\big{|} \big{)}\quad s.t.\ \frac{1}{2}\big{|}x_{r}\big{|}^{2}\ R_{k,s}+\Re\big{\{}x_{r}^{ \prime}b_{t}\big{\}}-\tau\frac{1}{a}\log\big{(}1+a\big{|}x_{i}\big{|}\big{)}<0 \tag{17}\]  
**Remark 2:** When the penalty is \(l_{p}\) norm, we can also give two similar propositions as follows:  
**Proposition 5:** Add the \(t\)-th (\(t\in I^{c}\)) element into the support according to the rule:  
\[t=\arg\max_{k\in r^{c}}|b_{k}|\quad s.t.\ \big{|}b_{t}|^{2-p}>2\tau R_{k,s}^{ 1-p} \tag{18}\]  
**Proposition 6:** Remove the \(t\)-th (\(t\in I\)) element from the support according to the rule:  
\[t=\arg\min_{k\in I}\frac{1}{2}\big{|}x_{k}\big{|}^{2}\ R_{k,k}+\Re\big{\{}x_{ k}^{\prime}b_{k}\big{\}}-\tau\big{|}x_{k}\big{|}^{p}\quad s.t.\ \frac{1}{2}\big{|}x_{i}\big{|}^{2}\ R_{k,s}+\Re\big{\{}x_{i}^{ \prime}b_{t}\big{\}}-\tau\big{|}x_{i}\big{|}^{p}<0 \tag{19}\]  
The proofs for Equations (18)-(21) will be shown in the Appendix. By combining the corresponding propositions with related DCD algorithms, it is easy to obtain Hlog-DCD algorithm and H\(l_{p}\)-DCD algorithm, respectively, and we omit the tabulated expressions here for simplicity.