## 4 Experiments
### Experimental Setup  
As mentioned earlier in this paper, the MTS is modeled with the LDS as shown in (1) with order numbers \(d=5\) and \(m=d\). For performance measures, both clustering accuracy (%) and normalized mutual information (NMI) are adopted following [56]. The higher accuracy or NMI (range from 0 to 1) indicates better performance in MTS clustering. As discussed in Section 3, we evaluated 11 Grassmann clustering methods in our experiments, which are summarized in Table 2. The 11 Grassmann clustering methods are comprised of four intrinsic methods, four extrinsic methods, and three semi-intrinsic methods. For the intrinsic approach, spectral clustering methods were evaluated using a fully connected graph and \(m\)NN graph, with principal angle and Procrustes distance. The extrinsic approach consists of two spectral clustering, which use tangent space as the embedding space, and two kernel spectral clustering methods, utilizing the projection kernel and canonical correlation kernel. Similar to the intrinsic approach, two graph types and dissimilarity measures were used for spectral clustering as well in the extrinsic approach. To evaluate the semi-intrinsic approach, the \(k\)-means clustering algorithm was selected with tangent norm, principal angle, and Procrustes distance. \begin{table}
\begin{tabular}{c c c c} \hline
**Cluster No.** & **Gestures** & **Num. of Samples** & **Taken from Subjects** \\ \hline
1 & Forward walking & 46 & \#7, \#8, \#35 \\
2 & Forward running & 22 & \#9, \#35 \\
3 & Jumping & 30 & \#118 \\
4 & Salsa dancing & 30 & \#60, \#61 \\
5 & Picking up something & 25 & \#64, \#115 \\
6 & Golf swing & 11 & \#63, \#64 \\
7 & Story & 30 & \#138 \\
8 & Marching & 10 & \#138 \\
9 & Walking with arm out & 12 & \#132 \\ \hline \multicolumn{2}{c}{Total} & 216 & - \\ \hline \end{tabular}
\end{table} 
Table 1: Summary of the MOCAP subset utilized in our experimental study. \begin{table}
\begin{tabular}{c c c c c} \hline
**Methods** & **Clustering** & \
\begin{tabular}{c} **(Dis)similarity** \\ **Measure/Kernel** \\ **Function** \\ \end{tabular} & **Graph Types** & **Parameters** \\ \hline Intrinsic approach & Spectral & Principal angle, & Fully connected & spreading factor 1 \\ (4 methods) & clustering & Procrustes distance & \(m\)NN & spreading factor, \\ \hline Extrinsic & Tangent space & Spectral & & Fully connected & spreading factor \\  & (2 methods) & clustering & Euclidean distance 3 & \(m\)NN & spreading factor, \\  & & & & \(m\)-neighbor 2 \\ \cline{2-5} approach & Grassmann kernel & Kernel spectral & Projection kernel, & \\  & (2 methods) & clustering & Canonical & Fully connected & - \\ \hline \multicolumn{2}{c}{} & & & \\ Semi-intrinsic approach & \(k\)-means 4 &
\begin{tabular}{c} Tangent norm, \\ Principal angle, \\ Procrustes distance \\ \end{tabular} & - & - \\ \hline \end{tabular} \({}^{1}\) Spreading factors \(\sigma\in[0.1,0.5,1:1:5,10,20,50,100,1000]\) considered for affinity matrix **A** in spectral clustering \({}^{2}\) Number of nearest neighbors \(m\in(3:1:9)\) considered for the **A**. 3 Euclidean distance is chosen as it is the commonly used distance metric in vector space. 4 A total of 100 iterations are used for Karcher mean computation. A total of 30 iterations are used for all clustering methods.  
\end{table} 
Table 2: Eleven combinations of Grassmann clustering techniques used in evaluation. To observe the impact of the spreading factors \(\sigma\) and the number of neighbors \(m\) on clustering performance, the spectral clustering algorithm is evaluated over various settings: \(\sigma\in[0.1,0.5,1:1:5,10,20,50,100,1000]\) and \(m\in\{3:1:9\}\). The maximum number of iterations for the \(k\)-means algorithm and Karcher mean algorithm is set to 100 and 30, respectively. In our experimental study, the number of clusters \(K\) is assumed to be known beforehand. If the \(K\) is unknown, one can use a search algorithm such as an elbow method [57] to determine its optimal value.