## 2 Inverse Solution Algorithms
### Underdetermined Systems
#### 2.2.3 Iterative Approaches: Promote Sparsity by Pruning  
CVX algorithms can solve CS problems, but these algorithms are known to be very computationally expensive, which prevents their use in real-time processing. To address this challenge, several iterative approaches are proposed as follows. In these iterative techniques, the quadratic residual function  
\[J(\mathbf{s})=\frac{1}{2}\|\mathbf{A}\mathbf{s}-\mathbf{p}\|_{2}^{2}=\frac{ 1}{2}\Big{[}\mathbf{s}^{H}\mathbf{A}^{H}\mathbf{A}\mathbf{s}-\mathbf{p}^{H} \mathbf{A}\mathbf{s}-\mathbf{s}^{H}\mathbf{A}^{H}\mathbf{p}+\mathbf{p}^{H} \mathbf{p}\Big{]} \tag{26}\]  
is to be minimized. The key step that executes the \"compressive sampling\" is a pruning process that must be incorporated into each iteration to promote sparsity, as illustrated in Figure 2. First, several main peaks as well as sidelobes may appear in the source diagram. We reset all elements in the source vector \(\mathbf{s}\) below a prespecified threshold (\(s_{\mathrm{max}}-D\)) to zero. \(D\) is initially set to a very small value \(D_{0}\); it is then increased incrementally by \(\Delta D\) in each iteration, typically by the same \(\Delta D\) in every step. The iterative pruning process is summarized with a flowchart in Figure 3. The stopping condition is  
\[D>D_{\text{max}}\text{ or }\|\
abla J(\mathbf{s})\|_{2}^{2}\leq 0.001 \tag{27}\]  
in which \(\
abla J(\mathbf{s})\) is the gradient of the quadratic residual function \(J(\mathbf{s})\). Three approaches were employed in this study to update the solutions in the iterative CS algorithms such that each quadratic residual function \(J(\mathbf{s})\) is minimized. Figure 3: Flowchart of iterative compressive sampling algorithms (adapted from reference [5]). Figure 2: Pruning process to promote sparsity of inverse solution. Conjugate Gradient (CG) Method  
The CG method is an iterative algorithm well suited for the numerical solution of systems of linear equations, associated with symmetric and positive-definite matrices. Instead of the negative gradient used in the SD method, which occasionally causes zigzag convergence, the search direction of the CG method is a linear combination of the current negative gradient and the previous search direction. Development of the CG algorithm is based on nested Krylov subspaces. For details, we refer the interested readers to Ref. [32; 33; 34]. For brevity, we only summarize the CG algorithm with the following pseudocode:  
\[\mathbf{r}_{0}=\mathbf{p}-\mathbf{A}\mathbf{s}_{0}\] \[\mathbf{p}_{0}=\mathbf{r}_{0}\] \[k=0\] \[for\] Figure 2.0  
Figure 2.1 Source localization results obtained using the focal underdetermined system solver (FOCUSS) algorithm and a uniform linear array. (**a**) Tikhonov regularization (TIKR) spectrum (**b**) TIKR averaged spectrum (**c**) FOCUSS spectrum (**d**) FOCUSS frequency-averaged spectrum. (**e**) Compressive sampling-convex optimization (CS-CVX) spectrum (**f**) CS-CVX frequency-averaged spectrum. The segSNR correlates with the effect of noise reduction. The FOCUSS-PINV algorithm was observed to achieve the highest score in PESQ and segSNR (Table 1), although it required more computation time than TIKR and FOCUSS-TIKR. In our previous simulation, we simulated microphones that had no noise; therefore, our regularization parameter was very close to zero. In the current simulation, our microphones did have white noise with a magnitude equal to the magnitude of the microphone signals divided by 100. Therefore, the potential loss of SNR was 40 dB. The regularization parameter is chosen by the maximal singular \(\sigma_{1}\) value dividing 100 in 100 Hz. In our case, the maximal singular value was equal to 5.32. Next, a coarse search was performed by varying \(\beta\) in orders of 10 and using the GSS algorithm to find the optimal regularization parameter. In this case, the optimal regularization parameter was 0.0174, and the FOCUSS-PINV of the robustness to the noise was very ineffective because PINV artifacts caused discontinuities in regularization.