# Robust Hessian Locally Linear Embedding Techniques for High-Dimensional Data
## 5 Experiments
### Synthetic Data  
We apply our RHLLE to datasets that have been commonly used by other researchers: Swiss roll (Figure 3), S-curve (Figure 4) and helix manifold (Figure 5). For each manifold, we add uniformly distributed random noise with large amplitude to a small percentage of the data to simulate the dataset with outliers; we add Gaussian noise to a large percentage of the data to simulate the noise-corrupted dataset. Table 1 shows the parameter settings used in these synthetic data experiments. The parameters include the dimensionality of the ambient space D, the dimensionality of the intrinsic dimensionality of the nonlinear manifold d, the number of nearest neighbors \(k\), the number of the data points, the percentage of the outliers, the percentage of the noise-corrupted points, the amplitude of the outliers and the standard deviation of the Gaussian noise.  
Figure 4: The first row shows the S-curve dataset with outliers and the embedding results of different algorithms; The second row shows the S-curve dataset with noise and the embedding results of different algorithms; The third row shows the S-curve dataset with both outliers and noise and the embedding results of different algorithms.  
Figure 3: The first row shows the Swiss roll dataset with outliers and the embedding results of different algorithms;The second row shows the Swiss roll dataset with noise and the embedding results of different algorithms; The third row shows the Swiss roll dataset with both outliers and noise and the embedding results of different algorithms.  
We compare our RHLLE with HLLE [5], RLLE [15], RLTSA [17] and SLE [35] on these datasets. The performance of these algorithms can be seen by comparing the coloring of the data points, the smoothness and the shape of the projection coordinates with their original manifolds. As we can see from Figures 3-5, the traditional HLLE algorithm fails to recover the underlying manifold structure when there exist either or both outliers and noise-corrupted points (see Figures 3(b,h and n)-5(b,h and n). RLLE and RLTSA can recover the manifold structure when there are outliers (see Figures 3(c,d)-5(c,d)). However, RLLE and RLTSA cannot recover the manifold structure well when there exist large amounts of noise-corrupted points or both outliers and noise-corrupted points (see Figures 3(i,j,o and p)-5(i,j,o and p)). SLE can usually preserve the local geometry of the data manifolds in the embedding space when there exist large amounts of noise-corrupted points (see Figures 3k-5k)). However, SLE cannot preserve the local geometry of the data manifolds well in the embedding space when there exist outliers or both outliers and noise-corrupted points (see Figures 3(e,d)-5(e,q)). The results in Figures 3-5 show that only our RHLLE algorithm can accurately unfold and recover the original manifolds in the presence of either or both outliers and noise-corrupted points (see Figures 3(f,l and r)-5(f,l and r)). From the above experiments we can see that the proposed RHLLE algorithm fails to recover the manifold structure when there exist large amounts of noise-corrupted points (see Figures 3(f,l and r)-5(f,l and r)). However, the results in Figures 3(f,l and r)-5 show that the proposed RHLLE algorithm fails to recover the manifold structure when there exist large amounts of noise-corrupted points (see Figures 3(f,l and r)-5(f  
conclude that: (1) our RHLLE algorithm can significantly improve the performance of the Hessian LLE algorithm on the outliers and noise-corrupted data; (2) our RHLLE algorithm performs the best among the five methods on the synthetic datasets.