# Semi-Supervised Manifold Alignment Using Parallel Deep Autoencoders
## 5 Experiments and Results
### Experiments on 2-, 3- and 4-Manifold Alignment  
#### 5.2.1 Double Pendulum Datasets  
We generated high-dimensional datasets by simulating the motion of double pendulums where three different conditions were considered (Figure 4): 2D-2D, 2D-3D and 3D-3D. The two limbs of the double pendulum are denoted as \(u_{1}\) and \(u_{2}\), and the joints are denoted as \(J_{1}\) and \(J_{2}\). One end \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{2}{c}{**Methods**} & \multicolumn{2}{c}{\(\Delta(\sigma)\)} \\ \hline \multirow{2}{*}{**MAPA**} & feat & 0.094 (0.039) \\  & inst & 0.030 (0.021) \\ \hline \multirow{2}{*}{**MALG**} & feat & 0.022 (0.016) \\  & inst & 0.023 (0.021) \\ \hline \multirow{2}{*}{**MAGG**} & feat & 0.033 (0.024) \\  & inst & 0.034 (0.030) \\ \hline \multicolumn{2}{c}{**PDAE**} & **0.017 (0.010)** \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The table shows \(\Delta\) of the alignment of Models 1 and 21 for all methods that were tested and visualised in Figure 3. The standard deviation \(\sigma\) is provided in parenthesis. PDAE performed the best. Figure 3: Protein structure manifold alignment. 3D structure of the glutaredoxin protein PDB-IG7O: The blue graph shows Model 1, and the red graph shows Model 21 scaled by a factor of four. (**a**) shows the manifolds before alignment. The other subfigures show the resulting alignments using (**b**) MAPA-feature (feat), (**c**) MAPA-instance (inst), (**d**) MALG-feat, (**e**) MALG-inst, (**f**) MAGG-feat, (**g**) MAGG-inst and (**h**) PDAE. of \(u_{1}\) is fixed at \(R_{1}\), and the limb can rotate around this joint. The other end of \(u_{1}\) is attached to \(u_{2}\) at joint \(R_{2}\). \(u_{2}\) can rotate freely around joint \(J_{2}\). The free end point \(e\) of the pendulum is referred to as the \"end-effector\" and has coordinates \(\left(e_{x},e_{y}\right)\) in the two-dimensional case and \(\left(e_{x},e_{y},e_{z}\right)\) in the three-dimensional version using a right-handed coordinate system with the origin at joint \(J_{1}\). The data were generated in simulation for the 2D-2D, 2D-3D and 3D-3D cases as follows (Figure 4):  
1. 2D-2D motion: The pendulum has two Degrees-Of-Freedom (DOF), that is, both limbs \(u_{1}\) and \(u_{2}\) rotate in the two-dimensional (\(x\)-\(y\))-plane, each of them describing a circle. In Figure 4a, \(\theta_{1}\) and \(\theta_{2}\) are the rotation angles of limbs \(u_{1}\) and \(u_{2}\) at joints \(J_{1}\) and \(J_{2}\), respectively. Accordingly, the manifold representing the dynamics of the 2D-2D case is the cross-product of two circles, \(S^{1}\times S^{1}\), which is homeomorphic to the two-dimensional torus, that is a 2-manifold. 2. 2D-3D motion: The pendulum has three DOFs, where limb \(u_{2}\) can rotate on a two-dimensional sphere \(S^{2}\) in three-dimensional space, while \(u_{1}\) is restricted to rotate on a circle \(S^{1}\) in a two-dimensional plane. That is, the manifold representing the dynamics of the 2D-3D case is homeomorphic to \(S^{1}\times S^{2}\), which is a 3-manifold. As the pendulum moves in 3D space, the end-effector has the 3D coordinates \(\left(e_{x},e_{y},e_{z}\right)\). In Figure 4b, \(\theta_{y}\) and \(\theta_{z^{\prime}}\) are the angles of \(u_{2}\) with axes \(y^{\prime}\) and \(z^{\prime}\), respectively, and describe the motion on the sphere \(S^{2}\). \(\theta_{1}\) is the angle between the \(x\)-axis and \(u_{1}\) and describes the two-dimensional rotation of the sphere's centre in the (\(x\)-\(y\))-plane. 3. 3D-3D motion: In this case, the pendulum has four DOFs, where both limbs can rotate on two-dimensional spheres in 3D space. In Figure 4c, \(\theta_{y}\) and \(\theta_{z}\) are the angles of \(u_{1}\) with the \(y\) and \(z\) axes, respectively, and \(\theta_{y^{\prime}}\) and \(\theta_{z^{\prime}}\) are the angles of \(u_{2}\) with the \(y^{\prime}\) and \(z^{\prime}\) axes, respectively. Accordingly, we expected that the manifolds representing the dynamics of the 3D-3D case were homeomorphic to \(S^{2}\times S^{2}\), which is a 4-manifold.