# Point Cloud Denoising Algorithm via Geometric Metrics on the Statistical Manifold
## 4 Simulations and Results
From Propositions 1-5 as a basis for calculation norms of influence functions, we repeat simulations for each scenario (\(l=10,40,70\), or 100) one hundred times, as shown in Figure 3. It can be seen that the norms of the influence functions corresponding to the geometric metrics (AIRM, LEM, KLD, and SKLD) are not sensitive to changes in outlier numbers; they remain within a range close to one, regardless of if \(l\) is set from 10 up to 100. On the other hand, norms associated with Euclidean metric (FM) fluctuate significantly--decreasing from around seven or eight down towards one when \(l\) increases from 10 up to 100. Therefore, geometric means are almost independent of the number of outliers, and more stable than the arithmetic mean. Figure 4 provides an intuitive representation by illustrating average norms across all simulations for each metric type (Euclidean vs Geometric). As observed in Figure 3, results hold true here as well: FM consistently yields larger norm values compared to those obtained using geometric metrics when considering various values for \(l\) such as 10, 40, and 70, respectively. Furthermore, Figure 4 shows that if the number of outliers with the FM is 100, which is close to the number of symmetry positive-definite matrices used in simulations, influence functions for FM and geometric metrics become increasingly similar. This further supports the notion that geometric means are more robust than arithmetic means, as demonstrated by Figure 4.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Metric**} & \multicolumn{3}{c}{**SNR 4137:1000**} & \multicolumn{3}{c}{**SNR 4137:2000**} \\ \cline{2-7}  & **TPR** & **FPR** & **SNRG** & **TPR** & **FPR** & **SNRG** \\ \hline
**Arith** & 99.59\% & 74.30\% & 34.04\% & 97.56\% & 66.25\% & 47.26\% \\ \hline
**AIRM** & **100.00**\% & 42.50\% & **135.29**\% & **99.30**\% & **33.50**\% & **195.97**\% \\ \hline
**LEM** & **100.00**\% & 42.50\% & **135.29**\% & **99.30**\% & **33.50**\% & **195.97**\% \\ \hline
**KLD** & 77.45\%  
Figure 4: Means of influence functions. Figure 3: Norms of influence functions. C. Complexity Analysis  
Next, the computational complexity of the mean matrices with FM, LEM, AIRM, KLD, and SKLD will be demonstrated, whose formulas can be obtained from (15) and Table 1. Then, the computational complexity of the influence functions according to FM, LEM, AIRM, KLD, and SKLD will be also given, whose expressions are shown in Propositions 1-5. For iterative algorithms, we only calculate the complexity of single-step iterations. This subsection demonstrates the computational complexity of mean matrices for FM, LEM, AIRM, KLD, and SKLD. The formulas can be obtained from (15) and Table 1. Additionally, the computational complexity of influence functions according to these metrics is provided in Properties 1-5. For iterative algorithms, only the complexity of single-step iterations is considered. Let us assume we have m \(n\times n\) symmetry positive-definite matrices and \(l\) outliers. It is assumed that the complexity of a single element is O(1), \(U^{-1}\sim O(n^{3})\) and \(\log U\sim O(n^{4})\). When calculating the matrix exponential of a symmetric matrix and the half power of a symmetric positive-definite matrix, the eigenvalue decomposition is needed to perform. Thus, we have that \(\exp(A)\sim O(n^{3})\) and \(U^{\frac{1}{2}}\sim O(n^{3})\). Table 3 shows that the arithmetic mean has the lowest computation time followed by KLD and SKLD means. As the complexity of each step of the AIRM mean is equivalent to that of the LEM mean, so the LEM mean has a much faster calculation speed than the AIRM mean. From Table 4, it can be seen that calculating the influence functions with the FM takes less time than those with the geometric metrics. Among geometric metrics, the longest calculation time is the influence function corresponding to the AIRM, followed by that corresponding to the LEM, and the shortest calculation time is the influence function induced by the KLD and the SKLD.