# Alternating Direction Method of Multipliers for Generalized Low-Rank Tensor Recovery
## 5 Convergence Analysis
### Influence of Noise and Sampling Rate on the Relative Error  
This subsection will evaluate the influence of noise and sampling rate on the relative error. For this purpose, we design four groups of experiments and use the synthetic data generated in the same manner as in the previous subsection. For the sake of convenience, we only carry out experiments on 50 \(\times\) 50 \(\times\) 50 tensors.  
The first group of experiments aims to investigate the influence of noise on the recovery performance. In the data generation process, we only change the manner for generating \(\mathcal{G}\), that is, each entry of \(\mathcal{G}\) is drawn independently from the normal distribution with mean zero and standard deviation \(b\). Let r = 5,p\({}^{\prime}\) = 10,p = 100,a = 50i and b = 0.2j, where i,j \(\in\) {0,1,..,10}. For different combinations of a and b, the relative errors of GLRTR are shown in Figure 3. We can draw two conclusions from this figure. For given b, the relative error is relatively stable with the increasing of a, which means the relative error is not very sensitive to the magnitude of sparse noise. The relative error monotonically increases with the increasing of Gaussian noise level, which validates that large Gaussian noise is disadvantageous for recovering the low-rank component.  
Next, we study the influence of sampling rate \(p\%\) on the relative error for different r. Set p\({}^{\prime}\) = 10,a = 500,b = 1,r = 1 + 2i,i = 1,2,...,14 and vary the value of p\({}^{\prime}\) from 30 to 100 in steps of size 10. For fixed r and p, we obtain the low-rank component according to GLRTR and then plot the relative errors in Figure 4. From the 3-D colored surface in Figure 4, we can see that both r and p have significant influence on the relative error. This observation indicates that small r or large p is conducive to the recovery of low-rank term.  
Figure 2: Relative errors of Generalized Low-Rank Tensor Recovery (GLRTR) with varying \(\uptau\).  
The third group of experiments will validate the robustness of GLRTR to sparse noise. Concretely speaking, we investigate the recovery performance under different ISNR on sparse noise without consideration of Gaussian noise. Set r \(\in\) {3,5 },p\({}^{\prime}\) = 10,p = 100,b = 0, a = 2\({}^{\text{i}}\)\(\times\) 500, i = \(-\)15, \(-\)14,...,10. The experimental results are shown in Figure 5, where the horizontal and the vertical coordinates represent the ISNR on sparse noise and the relative error, respectively. This figure illustrates that the relative error is less than 4.5% for synthetic 3 rd-order tensors, which verifies experimentally that our method is very robust to sparse noise.  
In the last group of experiments, we discuss the performance of GLRTR in removing the small dense noise for given large sparse noise. We also propose a combination strategy: GLRTR + HOSVD, that is, GLRTR is followed by HOSVD. The goal of this new method is to improve the denoising performance of GLRTR. Let r \(\in\) {3,5 },p\({}^{\prime}\) = 10,p = 100,a = 500 and b = 0.1i,i = 0,1,...,50. Different values for b lead to different ISNR on Gaussian noise. We draw four curves to reflect the relationship between the relative error and ISNR on Gaussian noise, as shown in Figure 6, where the black dashed line is a reference line. We have two observations from this figure. When ISNR on Gaussian noise is larger than 3.5%, GLRTR not only successfully separates the sparse noise to some extent but also effectively removes the Gaussian noise. The GLRTR+HOSVD method has better denoising performance than GLRTR in the presence of large Gaussian noise.  
Figure 4: Relative errors for different r and p.  
Figure 3: Relative errors for different a and b.