# Robust Hessian Locally Linear Embedding Techniques for High-Dimensional Data
## 5 Experiments
### High-Dimensional Image Datasets  
In the section, we present two experiments on high-dimensional real datasets to illustrate the effectiveness of our RHLLE algorithm. We compare our RHLLE with RLLE, RLTSA and SLE on these datasets. The first experiment is performed on the teapot image [6] dataset. There are 400 color images of a teapot in this database. They are captured from different angles in a plane. In our experiment, we convert the color images to the 76 \(\times\) 101 gray-level images, so the dimensionality of the vectorized image space is 7676. In order to add some outliers to the original image dataset, we first randomly select 10% of the images and for each selected image we replace a rectangular block of the image (about 20% pixels) with independent and identically-distributed samples from a uniform distribution. The location of the corrupt rectangular block is unknown to the algorithm. In order to generate noise-corrupted images, we first randomly select 90% of the images and for each selected image we corrupt 10% of randomly-chosen pixels by replacing their values with independent and identically-distributed samples from a uniform distribution. The corrupted pixels are randomly-chosen for each image, and the locations are unknown to the algorithm. Figure 6 shows 10 original teapot images, their corresponding outlier images and noise-corrupted images. The low-dimensional embeddings in \(\mathbb{R}^{2}\) of RLLE, RLTSA, SLE and RHLLE are shown in Figure 7. For completeness and to provide a baseline, the results on the original data without being corrupted are shown in Figure 8. In the embeddings we uniformly mark about 2.5% of the points with red circles and attach their corresponding training images. In the experiment, we set the number of nearest neighbors \(k\) to 10. Figure 7 reveals the following points: (1) RLTSA and RHLLE perform better than RLLE and SLE when there are outliers in the image set (see Figure 7a-d); (2) SLE and RHLLE perform better than RLLE and RLTSA when there exist large amounts of noise-corrupted images (see Figure 7e-h); (3) RLLE and RLTSA perform even worse and fail to recover the underlying manifold structure when there exist both outliers and large amounts of noise-corrupted images in the teapot dataset (see Figure 7i,j); (4) The 2D embedding results obtained by our RHLLE are shown in Figure 7d,h and l. Along the circle we can see a clear and smooth change in the orientations of the teapot, which means that our RHLLE successfully uncovers the geometric structure of the image dataset. Therefore, our RHLLE is superior to RLLE, RLTSA and SLE in revealing the underlying manifold when there are either or both outliers and noise-corrupted images in the teapot dataset. Figure 6: Ten teapot images, their corresponding outlier images and noise-corrupted images. Besides the teapot image dataset, we also conduct experiments on the ISOFACE [1] dataset which is another benchmark dataset used by many manifold learning algorithms. This dataset contains 698 images of a 3D human head, which are collected under different poses and lighting directions. The resolution of each image is 64 \(\times\) 64 and each image is represented as a 4096-dimensional vector. The intrinsic degrees of freedom are the horizontal rotation, vertical rotation and lighting direction. In order to add some outliers to the original face images, we first randomly select 10% of the images and for each selected face image we replace a square block of the image (about 20% pixels) with independent and identically-distributed samples from a uniform distribution. The location of the corrupt square block is unknown to the computer. In order to generate noise-corrupted face images, we first randomly select 30% of the face images and for each selected image we corrupt 10% of randomly-chosen pixels by replacing their values with independent and identically-distributed samples from a uniform distribution. The corrupted pixels are randomly-chosen for each face image, and the locations are unknown to the algorithm.