# Semi-Supervised Manifold Alignment Using Parallel Deep Autoencoders
## 5 Experiments and Results
### Experiments on 2-, 3- and 4-Manifold Alignment
#### 5.2.4 Results of 3-Manifold Alignment  
In the case of the 2D-3D motion data, limb \(u_{1}\) rotated around joint \(J_{1}\) in a circle and limb \(u_{2}\) rotated on a two-dimensional sphere around joint \(J_{2}\). In 3D space, this motion is represented by a manifold homeomorphic to \(S^{1}\times S^{2}\) and can be visualised by a circle of spheres. For clearer visualisation of the quality of the alignments, we plotted only six of the aligned spheres equally distributed along the circle in Figure 6. The figures indicate visually (best if enlarged) that already for zero noise, the visualisations of the results of all methods had alignment issues except PDAE, which produced a near-perfect alignment. The alignment errors \(\Delta\) and \(\sigma\) in Table 3 corroborate the visual assessment and show that PDAE was the best-performing method in the experiments using the 2D-3D data.  
\begin{table}
\begin{tabular}{c 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{2-9}  & **feat** & **inst** & **feat** & **inst** & **feat** & **inst** & \\ \hline
0 & 0.186 (0.070) & 0.112 (0.041) & **0.003 (0.001)** & 0.013 (0.012) & 0.054 (0.015) & 0.089 (0.038) & 0.007 (0.003) \\ \hline \multicolumn{9}{c}{Actuator noise} \\ \hline \(\pm 2^{\circ}\) & 0.090 (0.030) & 0.090 (0.037) & 0.016 (0.009) & 0.034 (0.026) & 0.053 (0.017) & 0.097 (0.043) & **0.015 (0.009)** \\ \(\pm 6^{\circ}\) & 0.377 (0.157) & 0.485 (0.205) & 0.047 (0.027) & 0.101 (0.092) & 0.072 (0.031) & 0.098 (0.051) & **0.011 (0.028)** \\ \(\pm 10^{\circ}\) & 0.365 (0.154) & 0.713 (0.304) & 0.078 (0.043) & 0.180 (0.135) & 0.119 (0.055) & 0.101 (0.056) & **0.007 (0.039)** \\ \hline \multicolumn{9}{c}{Coordinate noise} \\ \hline \(\pm 0.2\) & 0.105 (0.035) & 0.150 (0.067) & **0.039 (0.018)** & 0.061 (0.042) & 0.078 (0.029) & 0.117 (0.052) & 0.048 (0.022) \\ \(\pm 0.6\) & 0.125 (0.056) & 0.184 (0.093) & 0.124 (0.056) & 0.145 (0.096) & 0.139 (0.060) & 0.161 (0.076) & **0.085 (0.043)** \\ \(\pm 1.0\) & 0.203 (0.078) & 0.271 (0.145) & 0.135 (0.063) & 0.218 (0.144) & 0.204 (0.086) & 0.190 (0.083) & **0.128 (0.048)** \\ \hline \hline \end{tabular}
\end{table}
Table 2: 2D-2D manifold alignment: Shown are the alignment errors \(\Delta\) for the different alignment methods under different levels of noise. The standard deviation \(\sigma\) as defined as a measure of smoothness of the alignment in (5) is provided in parenthesis. The best results are highlighted in bold.  
Figure 6: Manifold alignments of 2D-3D motion data under the influence of noise: The graphs visualise the outcomes of aligning datasets \(X\) (red) and \(Y\) (blue) using the seven methods mentioned at the top. Each row shows the results obtained under the influence of a different level of noise. The expected outcome is \(S^{1}\times S^{2}\), that is a ring of spheres where, for clarity, our visualisations showed a section comprising six spheres \(S^{2}\) at equally-distributed positions on the circle \(S^{1}\). Of all methods tested, the PDAE shows the best performance with almost perfectly-aligned manifolds shown at the right end of the top row. With the addition of noise, the results deteriorate.