## 1 Introduction
Although this approach is proved to be more robust to outliers and more effective than the classical PCA, seldom work is devoted to study how this approach extract information that can distinguish different classes or clusters for the classification or clustering problems. Another popular approach is the rank-based PCA method. Its idea is to decompose the data matrix into a single low-rank matrix and a sparse component [15; 16; 17]. The low-rank matrix approximates the common features that are linearly correlated in the data while the sparse component approximates less frequently appeared features and these are assumed to be the corrupted and noisy information. For face recognition or clustering problem, the low-rank matrix represents the common facial features of different images that are highly correlated to each other. The scarves and sunglasses are less frequently appeared features in the database. They are relatively sparse information and belong to the sparse component. Although this approach can effectively identify the occluded information, it may discard some information that can distinguish the characteristics of different classes or clusters. In the low-rank formulation, the common facial features of different data objects are not strictly linearly correlated to each other. Some people may have larger eyes while some may have more attractive lips. These are key characteristics to distinguish different people. However, these are less frequently appeared features. They may be discarded and treated as sparse information. Figure 1: Example frontal faces and faces with scarves. To alleviate the aforementioned issues, we introduce a modified version of the \(l_{1}\)-PCA, which incorporates clustering with \(l_{1}\)-PCA. We find that the superiority performance of the \(l_{1}\)-PCA over the classical PCA is not only because of the \(l_{1}\) norm formulation but also its ability to extract information that is beneficial to image classification and clustering problems. We mathematically prove that the \(l_{1}\)-PCA can be expressed as a special two-group \(k\)-means clustering problem. The projection vector of the \(l_{1}\)-PCA is the vector difference between the two cluster centers obtained by the \(k\)-means clustering problem. If the two clusters are two different classes of the data, the corresponding vector difference represents the inter-class direction that groups data objects with similar nature together. This is beneficial to distinguish the two classes. Figure 2 illustrates this situation. This figure shows two classes of data. The left cluster forms a class while the right cluster forms another class. The red stars are the cluster centers of the two different classes. The vector difference between these two centers is in the horizontal direction. Apparently, the horizontal direction provides the key information to distinguish the two classes. Although the \(l_{1}\)-PCA possesses this important property, its performance is not as good as the state-of-the-art methods. The reason is that the \(k\)-means clustering algorithm adopts the squared \(l_{2}\) norm in the formulation and is not robust to outliers. In other words, the \(l_{1}\)-PCA is not robust to outlier. To overcome this limitation, we propose a new method, namely, trimmed-clustering based \(l_{1}\)-PCA (TC-PCA) that replaces the squared \(l_{2}\) norm by a trimming function in the special \(k\)-means clustering algorithm. The contributions of this paper are as follows. * We prove that the \(l_{1}\)-PCA is equivalent to a two-group \(k\)-means clustering model. The projection vector estimated by the \(l_{1}\)-PCA is the vector difference between the two cluster centers that are obtained by the \(k\)-means clustering algorithm. In other words, the projection vector of the \(l_{1}\)-PCA represents the inter-cluster direction, that is beneficial to distinguish data objects from different classes.