# Robust Hessian Locally Linear Embedding Techniques for High-Dimensional Data
## 5 Experiments
### High-Dimensional Image Datasets
Figure 9 shows 10 original face images, their corresponding outlier images and noise-corrupted images. Figure 8: 2D embedding results of RLLE, RLTSA, SLE and RHLLE on the original teapot image set without being corrupted. Figure 7: 2D embedding results of RLLE, RLTSA, SLE and RHLLE on the teapot image set with outliers, noise and both outliers and noise respectively. The 2D embedding results of RLLE, RLTSA, SLE and RHLLE are shown in Figure 10. For completeness and to provide a baseline, the results on the original data without being corrupted are shown in Figure 11. In the embedding we randomly mark about 8% points with red circles and attach their corresponding training images. In the experiment, we set the number of nearest neighbors \(k\) to 15. Figure 10 reveals several interesting observations: (1) For the outliers situation, the embedding result of RLLE and SLE show that the orientations of the face change smoothly along the horizontal direction and the orientations of the face change from down to up along the vertical direction (see Figure 10a,c). The 2-D embedding results of RLTSA and RHLLE show that the poses and the light of the face images vary smoothly along the horizontal direction and the vertical direction respectively (see Figure 10b,d). Thus, all methods can approximately discover the intrinsic structures of this dataset with outliers; (2) For the noise-corrupted situation, the embedding results of RLLE and RLTSA are severely compressed and the changes in face rotation are not obvious (see Figure 10e and the right-hand side of Figure 10f). Therefore, RLLE and RLTSA cannot recover the underlying manifold structure well when there exist large amounts of noise-corrupted images; (3) For the outliers  
Figure 10: 2-D embedding results of RLLE, RLTSA, SLE and RHLLE on the ISOFACE image set with outliers, noise and both outliers and noise respectively. Figure 9: Ten ISOFACE images, their corresponding outlier images and noise-corrupt images. plus noise-corrupted situation, RLLE and RLTSA perform even worse (see Figure 10i,j), while SLE and our RHLE successfully reveal the underlying manifold of the ISOFACE dataset with both outliers and noise-corrupted images (see Figure 10k,l). To further quantitatively evaluate the embedding qualities obtained by RLLE, RLTSA, SLE and our RHLLE, we employ the global smoothness and co-directional consistence (GSCD) criteria introduced in [36]. Roughly speaking, the smaller the value of GSCD, the higher the global smoothness and the better co-directional consistence. For details about the GSCD criteria, please refer to [36]. Table 2 shows the GSCD values computed by different algorithms on the ISOFACE image set with outliers, noise and both outliers and noise respectively. As we can observe from Table 2, our RHLLE algorithm obtains the smallest GSCD value, this result reveals that our RHLLE performs the best in recovering the underlying manifold structure of the ISOFACE dataset with either or both outliers and noise-corrupted points. Moreover, these GSCD results further verify the visualization results (as shown in Figure 10) in a quantitative way. The average computation times over 10 independent runs of different methods are shown in Table 3. As we can observe from Table 3, our RHLLE takes significantly less time than RLLE and RLTSA for high-dimensional datasets. This result is because, for high-dimensional data, the computational complexity of RLLE and RLTSA are dominated by the iterative weighted PCA procedure which has the computational complexity of \(O(kNLdD^{2})\), where \(L\) is the number of iterations (usually less than 15); while our RHLLE designs a fast iterative algorithm with low computational complexity \(O(kNLD)\) to approximate the original RPCA and the time-consuming reconstruction-errors procedure only need to be computed in one iteration, as we have discussed in Section 4.2. As we can observe from Table 3, SLE takes much more time than other algorithms when the number of data points \(N\) is relatively large.