## 3 Homotopy DCD with Non-Convex Regularization Algorithms  
We consider minimization of the following cost function to solve the problem Equation (2):  
\[J\left[\mathbf{x}\right]=\frac{1}{2}\left\|\mathbf{y}-\mathbf{Ax}\right\|_{2}^{2 }+\tau\rho(\mathbf{x}) \tag{1}\]  
where the sparsity measurement function \(\rho(\mathbf{x})\) is generally separable, or \(\rho(\mathbf{x})=\sum_{i}\phi\left(x_{i}\right)\), where \(\phi(\cdot)\) denotes the parameterized form and satisfies such conditions that \(\rho(\mathbf{x})\) is sparsity-preserving, and \(\tau\geq 0\) is a regularization parameter. As mentioned in previous researches, we can use DCD iterations for minimizing the cost function, which is represented in a fixed point. The DCD algorithm is appealing for practical designs as it operates at the bit level, resulting in stable hardware implementations, which is shown in Table 1. Different from [PERSON]'s studies [28, 37, 38, 39, 40], this paper utilizes the non-convex regularizations instead of convex regularizations embedding with the DCD algorithm, and chooses three typical penalty functions as follows:  
Figure 3: Illustration of grid refinement approach. Figure 2: Off grid and on grid scatterers. 1. The first order rational function penalty: \[\phi(x_{i})=\frac{\left|x_{i}\right|}{1+\frac{a}{2}\left|x_{i}\right|}\] (2)
2. The logarithmic penalty: \[\phi(x_{i})=\frac{1}{a}\log\left(1+a\left|x_{i}\right|\right)\] (3)
3. The \(l_{p}\) penalty: \[\phi\big{(}x_{i}\big{)}=\left|x_{i}\right|^{p}\] (4)  
In order to reduce the complexity of the DCD algorithm, we develop a greedy algorithm that is based on homotopy method with respect to a set of the parameter \(\tau\). Besides, the updates of our DCD algorithm are only done within the support instead of all elements. As the support is usually much smaller than \(N\), therefore, the complexity is further reduced. We consider the first order rational function penalty at first, and develop HR-DCD algorithm. Then Hlog-DCD algorithm and H\(l_{p}\)-DCD algorithm can be similarly obtained. The following two propositions define the rules for adding/removing elements into/from the support of HR-DCD algorithm. Let \(I\) be the support at some homotopy iteration, and \(\Gamma\) be its complementary set. Denote \(\mathbf{r}=\mathbf{y}-\mathbf{A}\mathbf{x}\), \(\mathbf{R}=\mathbf{A}^{H}\mathbf{A}\) and \(\mathbf{b}=\mathbf{A}^{H}\mathbf{r}\). **Proposition 1**: Add the \(t\)-th (\(t\in I^{c}\)) element into the support \(I\) according to the rule:  
\[t=\arg\max_{k\leq\tau}\frac{a\left|b_{k}\right|-R_{k,k}}{aR_{k,k}^{2}}\left\{ \left(R_{k,k}+a\left|b_{k}\right|\right)^{2}-4 aR_{k,k}\tau\right\}\quad s.t. \left(R_{k,t}+a\left|b_{k}\right|\right)^{2}>4 aR_{k,t}\tau \tag{5}\] \begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{2}{c}{**Step**} & \multicolumn{1}{c}{**Equation**} \\ \hline \multicolumn{2}{c}{**Initialization:**\(\mathbf{x}=\mathbf{0},\mathbf{r}=\mathbf{y},\mathbf{R}=\mathbf{A}^{H}\mathbf{A}\), \(m=0,\delta=H\)} \\ \hline
1 & for \(m=1\) to \(M_{\text{b}}\), repeat: \\
2 & \(\delta=\delta/2,\alpha=\left[\delta,-\delta,j\delta,-j\delta\right]\) \\
3 & for \(i=1\) to \(N\), repeat: \\
4 & for \(k=1\) to \(4\), repeat: \\
5 & \(\Delta J=\delta^{2}R_{i,i}/2-\operatorname{Re}\left\{\alpha_{k}^{*}\big{(} \mathbf{A}^{(i)}\big{)}^{H}\mathbf{r}\right\}+\tau\big{(}\phi(x_{i}+\alpha_{k} )-\phi(x_{i})\big{)}\) \\
6 & if \(\Delta J<0\), do: \\
7 & \(x_{i}\gets x_{i}+\alpha_{k},\mathbf{r}=\mathbf{r}-\alpha_{k}\mathbf{A}^{( i)}\) \\
8 & end if \\
9 & end for \\
10 & end for \\
11 & end for \\ \hline \hline \end{tabular}
\end{table} 
Table 1: DCD Algorithm.