# Semi-Supervised Manifold Alignment Using Parallel Deep Autoencoders
## 5 Experiments and Results
### Experiments on 2-, 3- and 4-Manifold Alignment
#### 5.2.5 Results of 4-Manifold Alignment  
The 11-dimensional data of the two 3D double pendulums were described in Section 5.2.1. Each of the two datasets \(X\) and \(Y\) was represented by a 20,736 \(\times\) 11 data matrix. In the 3D pendulum motion, the rotation of limb \(u_{2}\) described a sphere \(S^{2}\), and the motion of the other limb \(u_{1}\) described another sphere \(S^{2}\), so that the 3D pendulum motion resembled \(S^{2}\times S^{2}\), which is a 4-manifold. As this was too complex to visualise in full, we took snapshots of the motion around \(J_{1}\) at 90\({}^{\circ}\) steps and for the motion around \(J_{2}\) at 30\({}^{\circ}\) steps. This way, the rotation of \(u_{2}\) resulted in six spherical shapes that were uniformly distributed on a bigger sphere, which represented the motion of \(u_{1}\). The visualisations in Figure 7 show that the instance-level methods were not successful in aligning the high-dimensional nonlinear motion data. The manifolds of the instance-level alignments collapsed even without any noise. Only MAGG-feat and PDAE produced the expected visualisation, comprising six spheres that represented the snapshots we selected in the motion data on \(S^{2}\times S^{2}\). The outcome of MAGG-feat was also supported by our case study [47].  
For the remaining experiments, random noise was added to the data in several stages as described in Section 5.2.1. The noisy datasets were aligned using MAPA-feat, MALG-feat and PDAE. All manifolds collapsed after noise addition, and therefore, only visualisations for zero noise are included in Figure 7. \(\Delta\) and \(\sigma\) were calculated as described in Section 4. The numerical results in Table 4 together with the qualitative visual evaluations of Figure 7 showed that PDAE performed better than MAGG-feat and the other methods.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Noise**} & \multicolumn{2}{c}{**MAPA**} & \multicolumn{2}{c}{**MALG**} & \multicolumn{2}{c}{**MAGG**} & \multicolumn{2}{c}{**PDAE**} \\ \cline{2-7}  & **feat** & **inst** & **feat** & **inst** & **feat** & **inst** & \\ \hline
0 & 0.079 (0.032) & 0.157 (0.073) & 0.025 (0.012) & 0.124 (0.055) & 0.023 (0.012) & 0.062 (0.025) & **0.007 (0.004)** \\ \hline \multicolumn{7}{c}{Actuator noise} \\ \hline \(\pm 2^{\circ}\) & 0.065 (0.027) & 0.427 (0.17) & 0.025 (0.012) & 0.092 (0.063) & 0.026 (0.013) & 0.037 (0.018) & **0.012 (0.006)** \\ \(\pm 4^{\circ}\) & 0.073 (0.031) & 0.379 (0.153) & 0.035 (0.017) & 0.109 (0.06) & 0.029 (0.016) & 0.048 (0.025) & **0.026 (0.012)** \\ \(\pm 6^{\circ}\) & 0.065 (0.3) & 0.115 (0.049) & **0.039 (0.021)** & 0.273 (0.228) & 0.048 (0.023) & 0.074 (0.038) & 0.051 (0.026) \\ \(\pm 8^{\circ}\) & 0.150 (0.071) & 0.082 (0.04) & 0.052 (0.031) & 0.207 (0.244) & 0.047 (0.027) & 0.086 (0.047) & **0.062 (0.026)** \\ \(\pm 10^{\circ}\) & 0.120 (0.056) & 0.169 (0.08) & 0.089 (0.045) & 0.361 (0.164) & 0.078 (0.042) & 0.117 (0.062) & **0.049 (0.023)** \\ \hline \multicolumn{7}{c}{Coordinate noise} \\ \hline \(\pm 0.2\) & 0.073 (0.031) & 0.143 (0.075) & **0.066 (0.025)** & 0.131 (0.066) & 0.08 (0.031) & 0.092 (0.044) & 0.075 (0.033) \\ \(\pm 0.4\) & 0.113 (0.047) & 0.428 (0.169) & 0.101 (0.041) & 0.163 (0.095) & 0.122 (0.05) & 0.141 (0.071) & **0.082 (0.035)** \\ \(\pm 0.6\) & 0.149 (0.0 The experiments were executed on the university's high performance computing grid with 60 GB RAM using two parallel k80 GPUs. The huge speed advantage of inference with the autoencoder was representative of all our data and all comparative simulations we conducted (Table 6). However, these speed results can only be indicative for precise benchmarking of a standalone high-performance machine or a specialised setup would be required.