## 4 Experiments
### Image Clustering  
In addition to face image classification, experimental results [60; 61] showed that PCA can be used as a pre-processing step to enhance the accuracy of \(k\)-means clustering. In this experiment, we shall show that the clustering performance of the TC-PCA's subspace is better than that of the other PCA's subspaces. The experimental setting is as follows: the database is projected onto \(U\) and the clustering performance is evaluated on the respective subspace. The clustering rate is obtained based on the known class labels via the following  
\[\text{Clustering rate}=\frac{\sum\textit{best}(C_{i}\cap T_{j(i)})}{\# \textit{elements}}, \tag{21}\]  
where \(C_{i}\) and \(T_{j(i)}\) are \(i\)th cluster and corresponding set of training labels, respectively. In the above calculation, each cluster \(C_{i}\) is assigned to one training label \(j(i)\) only. No two or more clusters are assigned to the same training label. Since estimating the optimal number dimension for unsupervised learning problem is still a challenging research problem, we use the normalized area under the curve (NAUC) of clustering rates over the number of orthonormal projection vectors, \(D\), to quantify the results. Note that for each value of \(D\), we repeat the \(k\)-means clustering 30 times with the same starting vectors and the clustering result with the smallest objective value is obtained.  
Figure 6: The 1 st and 2 nd projection vectors obtained by \(I_{1}\)-PCA and TC-PCA for AR Face Database with (**a**) Normal + Lighting, (**b**) Normal + Sunglasses and (**c**) Normal + Scarves.  
To evaluate the robustness of PCA methods for data clustering problem with the presence of outliers and noisy dimensions, we consider the following synthetic dataset. The dataset has 103 dimensions. The first three dimensions have 10 clusters with 100 outliers around them. Each cluster follows a normal distribution with 100 samples. Thus, the dataset has totally 1100 samples. The remaining dimensions are generated by a mixture of normal and uniform distributions. They do not carry any information about the 10 clusters. They are generated by a mixture of normal and uniform distributions. To compute the clustering rate as shown in Equation (21) of different PCA methods, we only consider the class labels of the 10 clusters and ignore the outliers in the calculation. In other words, we applied the \(k\)-means clustering algorithm to 1100 samples and only evaluate on 1000 samples, which are the 10 clusters. Table 7 shows the clustering results after applying different PCA methods. The TC-PCA method performs the best. It is 40% better than the clustering result without applying any PCA method. Besides, it performs better than other PCA methods such as RPCA nearly 35%. This implies that the proposed method is not confused by the noisy dimensions and is able to cluster the data more effectively.  
Table 7 shows the comparative clustering performance for the databases used in the face image classification experiments. Generally speaking, the proposed method performs better than HQ-PCA and RPCA by around 0.5%-4.80% and the rest by around 1.78%-5.03%. The results indicate that TC-PCA's subspace is better than the other PCA's subspaces for clustering. Moreover, the NAUC of the proposed method is higher than the clustering results with all dimensions in all five experiments, which may not be achieved by other PCA methods.