## 4 Experiments
### Results and Discussion
#### 4.3.4 Discussion  
In our experiments, the intrinsic approach demonstrated the best clustering performance among the three approaches. This result is attributed to the fact that the clustering process is conducted directly on the surface of the Grassmann manifold, effectively capturing the inherent nonlinear nature of high-dimensional MTS data [20]. Consequently, both NMI and accuracy achieved the highest results, underscoring the intrinsic approach's ability to handle MTS data efficiently. Furthermore, these intrinsic methods also exhibited the fastest CPU times in our experimental setup, as they do not require additional embedding or averaging processes.  
In contrast, the extrinsic approach relies on the idea of manifold embedding. As a result, the performance of the extrinsic approach was highly dependent on the choice of projection methods, as has been highlighted in previous studies [15; 24; 58]. Additionally, the efficiency of affinity computation was influenced by the projection operator. Despite its drawbacks, the extrinsic approach allows for the use of any Euclidean-based clustering algorithms or distance measures, offering flexibility in method selection. While we have considered commonly adopted embedding techniques from the literature [43; 59] in this work, our results are sufficient to draw conclusions.  
The semi-intrinsic approach, on the other hand, performed comparable or worse in terms of NMI and accuracy compared to the other methods. Specifically, its NMI and accuracy were similar to those of the extrinsic approach but slightly lower than those of the intrinsic approach. However, the clustering performance of the semi-intrinsic methods remained more stable than that of the extrinsic approach across different datasets and evaluation metrics. Due to the iterative nature of the Karcher mean computation, the semi-intrinsic methods exhibited the worst CPU times during the clustering process. Despite these drawbacks, the concept of averaging MTS on manifold space has been shown to improve performance in tasks like action recognition, as reported in [21; 31]. Moreover, the semi-intrinsic approach remains the only method that can leverage various mean-based clustering algorithms.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{4}{c}{**CPU Time on Affinity Calculation**} \\
**Clustering** & \multicolumn{4}{c}{**[CPU Time on Clustering Process]**} \\ \cline{2-5}
**Methods** & **Euclidean** & **Tangent** & **Principal** & **Procrustes** \\  & **Distance** & **Norm** & **Angle** & **Distance** \\ \hline \multirow{2}{*}{\(k\)-means} & \multirow{2}{*}{-} & 29.043 & 1.879 & 1.163 \\  & & [102.612] & [102.488] & [102.361] \\ \hline \multirow{2}{*}{FC-SC} & 6.656 & \multirow{2}{*}{-} & 0.429 & 0.253 \\  & [0.064] & & [0.065] & [0.062] \\ \hline \multirow{2}{*}{\(m\)NN-SC} & 6.749 & \multirow{2}{*}{-} & 0.431 & 0.252 \\  & [0.074] & & [0.074] & [0.068] \\ \hline \multirow{2}{*}{PK-KSC} & \multicolumn{4}{c}{169.069} \\  & & [0.063] & \\ \hline \multirow{2}{*}{CCK-KSC} & \multicolumn{4}{c}{0.380} \\  & & [0.065] & \\ \hline \hline \end{tabular}
\end{table}
Table 5: Clustering (CPU time) performances of the \(11\) methods on C3D data (in seconds).