# Weighted Group Sparse Regularized Tensor Decomposition for Hyperspectral Image Denoising
## 4 Experimental Results
The influence of the rank of the tensor ring \(r=[r_{1},r_{2},r_{3}]\), which is an important parameter to control the correlation of tensors, is discussed. In order to streamline the determination of the tensor ring's rank, we make the assumption that the ranks of the second and third dimensions are identical, denoted as \(r_{2}=r_{3}\). The PSNR values vary under different tensor ring ranks, as shown in Figure 4:1 In order to achieve a balance between the robustness of the tensor ring rank and the effectiveness of the denoising results, we set the tensor ring rank differently in different noise environments. For the previously discussed case, we set the tensor ring to \(r=[7,14,14]\). Figure 4:2 Influence of parameter \(A\). Figure 4:3 Influence of parameter \(\lambda_{2}\). We consider the influence of the adjacent operator parameter \(\sigma\), which is used to limit the step size of a neighboring operator. Here, we select the sets 0.0001, 0.0005, 0.001, 0.005, 0.01 and 0.05 as the test data. Figure 17 shows the data test results. Finally, \(\sigma=0.01\) is selected as the optimal value. Regarding the influence of parameter \(\mu\), Figure 18 shows the recovery results for changes in the sets 0.001, 0.005, 0.006, 0.007, 0.01, 0.05 and 0.1. The results show that \(\mu=0.006\) has the best effect. Regarding the influence of the parameters \(v_{1},v_{2},v_{3},v_{4},v_{5}\) and \(v_{6}\), Figures 19-24 show the data analyzed. Finally, we chose \(v_{1}=1,v_{2}=0.8,v_{3}=0.9,v_{4}=1.1,v_{5}=1\) and \(v_{6}=1\) as the best results. Figure 4:4 Influence of parameter \(r\). Figure 4:5 Influence of parameter \(\sigma\). Figure 4:6 Influence of parameter \(v_{1}\). Figure 4:7 Influence of parameter \(v_{2}\). Figure 4:8 Influence of parameter \(v_{5}\). Figure 4:9 Influence of parameter \(v_{3}\). Figure 3:0 Influence of parameters \(v_{4}\). Because the parameters \(\mathcal{P}\), \(\mathcal{S}\), \(\mathcal{N}\) and others are solved using simple algebraic calculations, our main focus is on the time spent on \(\mathcal{X}\) and \(\mathcal{Q}\). \(h\times w\times b\) is the size of the hyperspectral image affected by noise. A tensor ring representation is used to estimate \(\mathcal{G}\) for updating \(\mathcal{X}\). Assuming that the size is \(\mathcal{X}\in\mathbb{R}^{l\times l\times l}\) and the TR rank is set as \(r_{1}=r_{2}=r_{3}=r\), there is a cost of O(\(r^{6}+l^{2}r^{4}+l^{3}r^{2}\)) to \(\mathcal{X}\) in each iteration. \(O(D_{1}logD_{1})\) refers to the computational complexity of the FFT, where \(D_{1}\) is the size of the data. Consequently, the cost of updating the subproblem of \(\mathcal{Q}\) is denoted as \(O(hwholog(hwh))\). According to the above, the proposed algorithm has a complexity of O(\(r^{6}+l^{2}r^{4}+l^{3}r^{2}+\) h w b log(h w b)). At last, numerical experiments are conducted to verify the convergence of the proposed algorithm. Figure 25 shows the variation in \(R\) with the number of iterations of LRTRDGS for hyperspectral reconstructions, where \(R=\frac{\|\mathcal{X}r-\mathcal{X}r^{+}\|_{F}^{2}}{\|\mathcal{Y}\|_{F}^{2}}\). As the number of iterations increases, \(R\) approaches zero. Figure 3:1 Influence of parameter \(v_{6}\). Figure 3:2 Change in the value \(R\) for the hyperspectral reconstructed images versus the number of iterations.