# Distributed Consensus for Global Matrix Formation in the Principal Component Pursuit Scenario
## 4 Results  
First, it is shown why it is important for each agent \(i\) to only update the unobserved entries \(\mathcal{X}_{i}\). This step is required since redundant measurements between agents are allowed, which, in the case of surveillance, would mean a shared vision field, as in the example in Figure 3.  
In this case, a showcase model is employed, updating all entries of each agent, with \(M\in\mathcal{R}^{500\times 500}\), with a low-rank component of 5 and a cardinality of the sparse component of 0.2 with random entries. Each agent will observe the same amount of rows, but adjacent agents share 20 rows. The result is shown in Figure 4. The highlighted rows in \(L\) are the overlapped field of view; hence, it is clear that it is required to update only the unobserved entries \(\mathcal{X}_{i}\) to avoid this overestimation, as stated in Algorithm 1. Since it has already been explained why a local update is not performed for every entry of the matrix state, now, some results of the convergence of the consensus process and an example of image recovery are shown. Two cases, \(a\) and \(b\), are tested with four agents. The graphs are shown in Figure 5, the respective adjacency matrices are shown in (6), and the convergence consensus is shown in Figure 6, where exponential convergence can be observed, which is a known property of this type of static graph. Figure 3: Two agents with redundant measurements in the overlapping field of view (gray region). \[A_{a}= \begin{bmatrix}1/3&0&1/3&1/3\\ 0&1/3&1/3&1/3\\ 1/3&1/3&1/3&0\\ 1/3&1/3&0&1/3\end{bmatrix} ,A_{b}= \begin{bmatrix}3/4&1/4&0&0\\ 1/4&1/4&1/4&1/4\\ 0&1/4&5/12&1/3\\ 0&1/4&1/3&5/12\end{bmatrix}  \tag{18}\]  
Figure 4: The agents reach consensus with overestimation (highlighted rows in L) in the overlapping regions in the field of view when the filter using \(\mathcal{X}_{i}\) is not applied. Figure 5: Connected graphs. Figure 6: Convergence in consensus. The first studied case involves decomposing a corrupted image of the moon's surface into a low-rank \(L\) and a sparse \(S\) matrix. Every agent knows part of the global \(M\), and there is some amount of redundant values between the agent states, but every agent knows each of its own observed entries. After reaching a consensus, each agent runs the alternate direction optimization into the convex relaxation of (3), as described in Algorithm 1. The resulting low-rank image \(L\) is employed to replace the corrupted data in the original image \(M\); the results are shown in Figure 7.  
To deal with colored RGB images, the algorithm splits the original image \(M\in\mathcal{R}^{n\times n\times 3}\) into three matrices, \(M_{R}\in\mathcal{R}^{n\times n}\), \(M_{G}\in\mathcal{R}^{n\times n}\), and \(M_{B}\in\mathcal{R}^{n\times n}\), one for each color channel, and a consensus is reached and optimization is carried for each matrix. Finally, each low-rank matrix is employed to replace the corrupted data in each RGB channel, and then they are stacked into an RGB image. Further examples are shown in Figures 8 and 9. Figure 8: Earthimage recovered by every agent. Figure 7: Moon image recovered by every agent, case \(a\). Figure 9: Marsimage recovered by every agent. Now, we investigate the impact of the size of the network. We designed random, strongly connected graphs with different number of nodes, and we ran the toy model of Figure 4. We compared the maximum Frobenius norms of any two agents in the network at step 20. The graphs for 20 and 100 agents are shown in Figure 10. The convergence results are shown in Table 1. The results show that the convergence speed is improved with the number of agents for a small network. In the case of 100 agents, the convergence is still higher than the smallest network of 4 agents, but worse than the other cases, probably caused by sparsity. Finally, we compare the convergence results after 20 iterations for a network with a fixed number of agents (nodes) with a different number of edges, meaning lower-density communication. We tested a network of 50 agents.