# Alternating Direction Method of Multipliers for Generalized Low-Rank Tensor Recovery
## 5 Convergence Analysis
### Synthetic Data
Meanwhile, large values of ISNR on Gaussian noise are disadvantageous for recovering the low-rank components; (II) GLKTR has better recovery performance than MRPCA. In an average sense, the relative error of GLKTR is 2.68% smaller than that of MRPCA for \(50\times 50\times 50\) tensors, and 14.17% for \begin{table}
\begin{tabular}{c c c c c} \hline \hline (\(\mathbf{r}\), \(\mathbf{p}^{\prime}\)) & \(11\mathbf{\mathcal{E}}11_{\mathrm{F}}/11\mathbf{\mathcal{A}}11_{\mathrm{F}}\) (\%) & \(11\mathbf{\mathcal{G}}11_{\mathrm{F}}/11\mathbf{\mathcal{A}}11_{\mathrm{F}}\) (\%) & **RE of MRPCA (\%)** & **RE of GLKTR (\%)** \\ \hline \((3,5)\) & 1253 & 18.33 & \(9.71\pm 0.98\) & \(6.60\pm 0.69\) \\ \((3,10)\) & 1769 & 19.20 & \(8.04\pm 0.84\) & \(7.99\pm 0.84\) \\ \((3,15)\) & 2218 & 20.05 & \(9.84\pm 1.97\) & \(8.75\pm 1.87\) \\ \((5,5)\) & 569 & 8.79 & \(7.39\pm 0.79\) & \(3.74\pm 0.42\) \\ \((5,10)\) & 855 & 9.40 & \(8.55\pm 0.97\) & \(4.47\pm 0.52\) \\ \((5,15)\) & 1040 & 9.16 & \(9.13\pm 0.52\) & \(5.05\pm 0.31\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of experimental results on \(50\times 50\times 50\) tensors. \begin{table}
\begin{tabular}{c c c c c} \hline \hline (\(\mathbf{r}\), \(\mathbf{p}^{\prime}\)) & \(11\mathbf{\mathcal{E}}11_{\mathrm{F}}/11\mathbf{\mathcal{A}}11_{\mathrm{F}}\) (\%) & \(11\mathbf{\mathcal{G}}11_{\mathrm{F}}/11\mathbf{\mathcal{A}}11_{\mathrm{F}}\) (\%) & **RE of MRPCA (\%)** & **RE of GLKTR (\%)** \\ \hline \((2,5)\) & 1722 & 24.05 & \(29.24\pm 7.95\) & \(13.90\pm 4.68\) \\ \((2,10)\) & 2823 & 37.72 & \(48.07\pm 18.47\) & \(23.98\pm 9.66\) \\ \((2,15)\) & 3052 & 27.17 & \(44.71\pm 13.66\) & \(23.10\pm 8.52\) \\ \((4,5)\) & 448 & 6.97 & \(12.00\pm 1.60\) & \(8.96\pm 1.15\) \\ \((4,10)\) & 563 & 6.04 & \(15.12\pm 1.98\) & \(6.26\pm 0.87\) \\ \((4,15)\) & 758 & 6.58 & \(24.22\pm 2.61\) & \(12.16\pm 2.90\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of experimental results on \(20\times 20\times 20\times 20\) tensors. 20 \(\times\) 20 \(\times\) 20 \(\times\) 20 tensors; (III) For 3 rd-order tensors, GLRTR removes effectively Gaussian noise, and, on average, its relative error is 5.96% smaller than the value of ISNR on Gaussian noise; for 4 th-order tensors, GLRTR effectively removes Gaussian noise only in the case \(r=2\). In summary, GLRTR is more effective than MRPCA in recovering the low-rank components. The second group of experiments considers four different sampling rates for \(\mathbf{\Omega}\) and one fixed degree of sparsity for \(\mathbf{\Omega}^{\prime}\), that is, \(\mathrm{p}\in\{\mathbf{30},\mathbf{50},\mathbf{70},\mathbf{90}\}\) and \(\mathrm{p}^{\prime}=5\). We set \(\boldsymbol{\tau}=0.02\) for both 3 rd-order and 4 th-order tensors, and choose the superior tradeoff parameter \(\boldsymbol{\lambda}\) for each \(p\). The comparisons of experimental results between MRPCA and GLRTR are shown in Tables 3 and 4 respectively. We can see from these two tables that MRPCA is very sensitive to the sampling rate \(p^{\circ}\)%, and it hardly recovers the low-rank components. In contrast, GLRTR achieves better recovery performance for 3 rd-order tensors. As for 4 th-order tensors, it also has smaller relative error when the sampling rate \(p^{\circ}\)% is relatively large. These observations show that GLRTR is more robust to missing values than MRPCA. We will evaluate the sensitivity of GLRTR to the choice of \(\boldsymbol{\lambda}\) and \(\boldsymbol{\tau}\) in the last group of experiments. For convenience of designing experiments, we only perform experiments on \(50\times 50\times 50\) tensors and consider the case that \(\mathrm{p}=100\). The values of \(\boldsymbol{\lambda}\) and \(\boldsymbol{\tau}\) are set according to the following manner: we vary the value of one parameter while letting the other be fixed. In the first case, the parameter \(\boldsymbol{\tau}\) is chosen as 0.01.