## 3 A Trimmed-Clustering Based \(I_{1}\)-PCA Model
### Mathematical Properties  
In this subsection, we prove that the estimator of the proposed TC-PCA model is insensitive to outliers by breakdown point analysis. Here, to simplify the proof, we assume that the outliers are a group of points that are far away from the majority. We also prove that the proposed TC-PCA algorithm converges.  
In robust statistics, the breakdown point measures the ability of a statistic to resist the outliers contained in the dataset [50]. The higher the breakdown point of an estimator, the more robust it is.  
**Theorem 1**: _The estimator of TC-PCA model has a breakdown point of \(1-p\%\), where \(p\) is a parameter of the trimming set (9)._  
Suppose that \(X=\{x_{1},x_{2},\ldots,x_{n}\}\) is a sample of size \(n\). Without loss of generality, we assume that the first \(n\):\(p\%\) samples are fixed and \(x_{i}\rightarrow\infty\) for \(n\):\(p\%\)\(+1\leq i\leq n\), where \(\cdot\) is a floor function. With the definition of \(\Omega(p)\) in (9), the last \(n\):\((1-p\%)\) samples will be discarded and thus \(\Omega(p)=\{x_{i}:i=1,2,\ldots,\)\(n\):\(p\%\) }. Therefore, the breakdown point of the estimator of TC-PCA model is \(n\):\((1-p\%)/n=1-p\%\).  
Theorem 1 reveals that the estimator of TC-PCA model is insensitive to outliers and reliable if less than \(1-p\%\) of data are outliers in the dataset. Using similar techniques as in Theorem 1, it can be shown that the breakdown point of \(l_{1}\)-PCA is zero, which implies that any outlier exists in the dataset will affect the estimation of a projection vector.  
In addition to the breakdown property of the proposed method, the convergence of the TC-PCA algorithm to obtain a collection of orthonormal projection vectors is of paramount importance. Since the major part of the proposed TC-PCA algorithm is SPVEA, which depends on (8), this is equivalent to proving the objective function of TC-PCA is decreasing and bounded.  
**Theorem 2**: _The proposed TC-PCA algorithm converges._  
Let \(v^{l}=\left\{v_{1}^{l},v_{2}^{l}\right\}\), \(l^{l}=\left\{l_{i,1}^{l},l_{i,2}^{l}\right\}\), \(\Omega(p)^{t}\) and \(J_{obj}(v^{l},l^{l},\Omega(p)^{t})\) be the cluster center, indicator function, trimming set and the function value of TC-PCA at the \(t\)th iteration, respectively.  
Consider the above variables at the \((t+1)\)th iteration, the algorithm starts by updating the trimming set which selects samples in the first \(p\)th percentile of \(l_{i,1}^{t}\|x_{i}-v_{1}^{l}\|_{2}^{2}+l_{i,2}^{t}\|x_{i}-v_{2}^{l}\|_{2}^{2}\) such that  
\[J_{obj}(v^{l},l^{l},\Omega(p)^{t})\geq J_{obj}(v^{l},l^{l},\Omega(p)^{t+1}). \tag{14}\]  
Next, we consider the update of cluster center. It is noted that \(\frac{\partial^{2}J_{obj}  
Finally, the indicator variable selects the smallest of the two terms \(\left\|x_{i}-v_{1}^{t+1}\right\|_{2}^{2}\) and \(\left\|x_{i}-v_{2}^{t+}\right\|_{2}^{2}\) and thus, the update of indicator variables leads to a smaller objective function value, as follows:  
\[J_{obj}(v^{t+1},l^{t},\Omega(p)^{t+1})\geq J_{obj}(v^{t+1},l^{t+1},\Omega(p)^{ t+1}). \tag{16}\]  
Combining (14)-(16) shows that the objective function is decreasing and bounded, as shown below:  
\[J_{obj}(v^{t},l^{t},\Omega(p)^{t})\geq J_{obj}(v^{t+1},l^{t+1},\Omega(p)^{t+1} )\geq 0,\]  
and hence the proposed TC-PCA algorithm converges.