## 3 A Trimmed-Clustering Based \(I_{1}\)-PCA Model
### Relating \(l_{1}\)-PCA to Two-Groups K-Means Clustering  
Consider the following \(l_{1}\)-PCA  
\[\underset{\|u\|_{2}=1}{max}\sum\
olimits_{i=1}^{n}\lvert(x_{i}-c)^{T}u\rvert, \tag{1}\]  
where \(u\) is the projection vector and \(c\) is a centroid of the dataset. The centroid \(c\) is usually set as the mean or median of the whole data. In our proposed model, the projection vector \(u\) can be obtained without necessity of data centralization procedure. This will be explained later. Now, we show that the \(l_{1}\)-PCA can be expressed as the two-group \(k\)-means clustering model below  
\[J(\hat{u},c,l)=\sum\
olimits_{i=1}^{n}(l_{i,1}\lVert x_{i}-(\hat{u}+c) \rVert_{2}^{2}+l_{i,2}\lVert x_{i}-(-\hat{u}+c)\rVert_{2}^{2}), \tag{2}\]  
where \(\lVert\ \rVert_{2}\) is the \(l_{2}\) norm distance. \(l=\left\{l_{i,1},l_{i,2}\right\}\) is the indicator variable with \(l_{i,k}=1\), \(k=1,2\), if \(x_{i}\) belongs to the \(k\)th cluster and \(0\) otherwise and \(l_{i,2}=1-l_{i,1}\). This clustering model partitions the data to two disjoint parts. They are represented by the two centroids, \(\hat{u}+c\) and \(-\hat{u}+c\). The data points that are closer to \(\hat{u}+c\) are classified as first cluster and the rest are the second cluster. Now, we examine this property in \(l_{1}\)-PCA model. The absolute term of this model can be expressed as \(\lvert(x_{i}-c)^{T}u\rvert=\mathit{sign}((x_{i}-c)^{T}u)((x_{i}-c)^{T}u)\). This expression introduces another interpretation of the \(l_{1}\)-PCA. The sign function virtually assigns \(\{-1,1\}\) to each data point and each data point belongs to one of the two regions or \(\mathit{sign}((x_{i}-c)^{T}u)=-1\). This assignment means that if the point \(x_{i}-c\) is in the same semi-plane as \(u\), it is assigned as the first cluster (i.e., \(\mathit{sign}((x_{i}-c)^{T}u)=1\)). Otherwise, it is the second cluster (i.e., \(\mathit{sign}((x_{i}-c)^{T}u)=-1\)). In other words, the point \(x_{i}\) that is closer to \(u+c\) belongs to the first cluster. If it is closer to \(-u+c\), it belongs to the second cluster. This is the same as the two-group clustering model. Figure 3 gives a visual illustration to this relationship. Figure 3 shows the 1 st projection vector obtained by the \(l_{1}\)-PCA on the 2-dimensional normal dataset with zero mean and variances 1 and 10, whereas Figure 3 shows the same dataset with the cluster centers estimated by the two-group \(k\)-means clustering model. In Figure 3, the data points \(x_{i}-c\) that are in the same semi-plane (\((x_{i}-c)^{T}u>0\), the upper half of the data) of the vector \(u\) belong to the first cluster (shaded region). Otherwise, they are in the opposite semi-plane (\((x_{i}-c)^{T}u<0\), the lower half of the data) and belong to the second cluster (non-shaded region). In Figure 3, the data are partitioned into two parts. The data points closer to the upper centroid belong to the first cluster (shaded region) while the rest of them belong to the second cluster (non-shaded region). With this connection, we can see that the vector difference between the two clusters (or \(\hat{u}+c-(-\hat{u}+c)\)) shown in Figure 3 is the projection vector \(u\) shown in Figure 3. This will be mathematically justified next. The following gives a mathematical justification about the equivalence between the \(l_{1}\)-PCA model and the special two-group \(k\)-means clustering model. By expressing each absolute term as a binary variable \(s_{i}\), Equation (1) becomes  
\[\underset{\|u\|_{2}=1}{max}\sum_{i=1}^{n}|(x_{i}-c)^{T}u|=\underset{\|u\|_{2}=1 s_{i}\in[-1,1]}{max}\underset{i=1}{max}s_{i}(x_{i}-c)^{T}u=\underset{s_{i}\in[-1,1]}{ max}\underset{i=1}{max}s_{i}(x_{i}-c)\|\sum_{2}. \tag{3}\]  
The first equality holds because \(|x|=\underset{s\in[-1,1]}{max}\underset{s\in[-1,1]}{max}\). The second equality holds because \(\underset{\|u\|_{2}=1}{max}x^{T}u=\|x\|_{2}\). It is noted that the maximum point of the \(l_{2}\) norm function is the same as the squared \(l_{2}\) norm function.