# Alternating Direction Method of Multipliers for Generalized Low-Rank Tensor Recovery
## 5 Convergence Analysis
### Synthetic Data  
In this subsection, we synthesize data tensors with missing entries. First, we generate an \(N\)th-order low-rank tensor as follows: \(\mathbf{\mathcal{A}}=\mathbf{\mathcal{C}}\times_{n=1}^{N}\mathbf{U}^{(n)}\), with the core tensor \(\mathbf{\mathcal{C}}\in\mathbb{R}^{1_{1}\times_{2}\times\cdots\times_{N}}\) and mode matrices \(\mathbf{U}^{(n)}\in\mathbb{R}^{\mathbb{R}_{n}\times_{2}}(J_{n}<I_{n})\). The entries of \(\mathbf{\mathcal{C}}\) and \(\mathbf{U}^{(n)}\) are independently drawn from the standard normal distribution. Then, we generate randomly a dense noise tensor \(\mathbf{\mathcal{G}}\in\mathbb{R}^{1_{1}\times_{2}\times\cdots\times_{N}}\) whose entries also obey the standard normal distribution. Next, we construct a sparse noise tensor \(\mathbf{P}_{\mathbf{\Omega}^{\prime}}(\mathbf{\mathcal{E}})\), where \(\mathbf{\mathcal{E}}\in\mathbb{R}^{1_{1}\times_{2}\times\cdots\times_{N}}\) is produced by a uniform distribution on the interval \((-\mathbf{a},\mathbf{a})\) and the index set \(\mathbf{\Omega}^{\prime}\) is produced by uniformly sampling on \([I_{1}]\times[I_{2}]\times\cdots\times[I_{N}]\) with probability \(\mathrm{p}^{\prime}\%\). Finally, the generation of the sampling index set \(\mathbf{\Omega}\) is similar to \(\mathbf{\Omega}^{\prime}\) and the corresponding sampling rate is set to be \(\mathrm{p}^{\prime}\%\). Therefore, an incomplete data tensor is synthesized as \(\mathbf{\mathcal{D}}_{\mathbf{\Omega}}=\mathrm{P}_{\mathbf{\Omega}}(\mathbf{\mathcal{A}}+\bm {\mathcal{G}}+\mathbf{\mathcal{E}})\). For given data tensor \(\mathbf{\mathcal{D}}_{\mathbf{\Omega}}\) with missing values, its low-rank component recovered by some method is denoted by \(\mathbf{\mathcal{\hat{A}}}\). The Relative Error (RE) is employed to evaluate the recovery performance of the low-rank structure and its definition is given as follows: \(\mathrm{RE}=\ \left|1\mathbf{\mathcal{\hat{A}}}-\mathbf{\mathcal{A}}\right|_{1}\mathrm{F}/ \left|1\mathbf{\mathcal{\hat{A}}}\right|_{1}\mathrm{F}\). Small relative error means good recovery performance. The experiments are carried out on \(50\times 50\times 50\) tensors and \(20\times 20\times 20\times 20\) tensors, respectively. Furthermore, we set a = 500 and \(\mathrm{J}_{1}=\mathrm{J}_{2}=\cdots=J_{N}=\mathrm{r}\). For convenience of comparison, we design three groups of experiments. In the first group of experiments, we only consider the case that there are no missing entries, that is, \(\mathbf{\Omega}=[I_{1}]\times[I_{2}]\times\cdots\times[I_{N}]\) or \(\mathrm{p}=100\). The values of \(\left|1\mathbf{\mathcal{E}}\right|\mathrm{i}_{F}/1\mathbf{\mathcal{\hat{A}}}\right|_{1} \mathrm{F}\) and \(\left|1\mathbf{\mathcal{\hat{G}}}\right|_{1}\mathrm{F}/\left|1\mathbf{\mathcal{\hat{A}}} \right|_{1}\mathrm{F}\) are adopted to indicate the Inverse Signal-to-Noise Ratio (ISNR) with respect to the sparse and the Gaussian noise respectively. Three different degrees of sparsity are taken into account, that is, \(\mathrm{p}^{\prime}=5\), \(10\) and \(15\). In addition, we take \(\mathbf{\lambda}=\mathbf{0.025},\mathbf{\tau}=\mathbf{0.02},\mathrm{r}\in\{3,5\}\) for 3 rd-order tensors and \(\mathbf{\lambda}=\mathbf{0.03},\mathbf{\tau}=\mathbf{0.01},\mathrm{r}\in\{2,4\}\) for 4 th-order tensors. For given parameters, we repeat the experiments ten times and report the average results. As a low-rank approximation method for tensors, the Higher-Order SVD (HOSVD) truncation [12] to rank-\((r_{1},r_{2},\ldots,r_{N})\) is not suitable for gross corruptions owing to the fact that its relative error reaches up to 97% or even 100%. Hence, we do not compare this method with our GLKTR in subsequent experiments. The experimental results are shown in Tables 1 and 2 respectively. From the above two tables, we have the following observations: (I) Although the values of ISNR on sparse noise are very large, both MRPCA and GLKTR remove efficiently sparse noise to some extent.