## 2 Centralized Stochastic Approximation Algorithm
### Recursive Principal Component Analysis Algorithm  
Before introducing the recursive algorithm, we first make the following assumption. 1. \(\{A,A_{k},k\geq 1\}\) are \(n\times n\) symmetric matrices. \(A\) is unknown and \(\{A_{k}\}_{k\geq 1}\) is the observation sequence of \(A\) satisfying \(A_{k}\underset{k\to\infty}{\longrightarrow}A\). The principal component analysis algorithm considered in this paper aims at estimating the eigenvalues and eigenvectors based on the observation sequence \(\{A_{k}\}_{k\geq 1}\). Since \(A\) is unknown, if we perform SVD or QR decomposition for each \(A_{k},\ k\geq 1\), it would be rather time-consuming. In the following, we will introduce a SAAWET-based recursive algorithm for solving the problem. The eigenvectors of matrix \(A\) are recursively estimated as follows:Choose the stepsize \(a_{k}=\frac{1}{k}\) and any initial values \(u_{0}^{(1)}\in\mathbb{R}^{n}\) with unit modular and define  
\[\tilde{u}_{k+1}^{(1)}=u_{k}^{(1)}-a_{k}A_{k}u_{k}^{(1)}. \tag{37}\]  
If \(\|\tilde{u}_{k+1}^{(1)}\|\
eq 0\), then  
\[u_{k+1}^{(1)}=\tilde{u}_{k+1}^{(1)}/\|\tilde{u}_{k+1}^{(1)}\|. \tag{38}\]  
Otherwise, \(\|\tilde{u}_{k+1}^{(1)}\|=0\) and reset \(u_{k+1}^{(1)}\) as another vector with a unit modular. Assume that \(u_{k}^{(i)},\ i=1,\cdots,j\) have been well-defined. Next, we inductively define the estimation algorithm for \(u_{k}^{(j+1)}\). Define the \(n\times j\)-matrix  
\[V_{k}^{(j)}\triangleq\Big{[}u_{k}^{(1)}\quad P_{k}^{(1)}u_{k}^{(2)}\quad \cdots\quad P_{k}^{(j-1)}u_{k}^{(j)}\Big{]},\]  
where \(P_{k}^{(i)}\triangleq I-V_{k}^{(i)}V_{k}^{(i)+},\ i=1,\cdots,j-1\) and \(V_{k}^{(i)+}\) is the pseudo inverse of \(V_{k}^{(i)}\). With an arbitrary initial vector \(u_{0}^{(j+1)}\) with a unit modular, define  
\[\tilde{u}_{k+1}^{(j+1)}=P_{k}^{(j)}u_{k}^{(j+1)}-a_{k}P_{k}^{(j)}A_{k}P_{k}^{(j )}\cdot u_{k}^{(j+1)}. \tag{39}\]  
If \(\|\tilde{u}_{k+1}^{(j+1)}\|\
eq 0\), then  
\[u_{k+1}^{(j+1)}=\tilde{u}_{k+1}^{(j+1)}/\|\tilde{u}_{k+1}^{(j+1)}\|. \tag{40}\]  
Otherwise, \(\|\tilde{u}_{k+1}^{(j+1)}\|=0\) and reset \(u_{k+1}^{(j+1)}\) as another vector denoted still by \(u_{k+1}^{(j+1)}\) satisfying \(\|u_{k+1}^{(j+1)}\|=1\) and \(V_{k}^{(j)\tau}u_{k+1}^{(j+1)}=0\). \(\{u_{k}^{(1)},\cdots,u_{k}^{(n)}\}\) serve as the estimates for eigenvectors of matrix \(A\). Based on \(\{u_{k}^{(1)},\cdots,u_{k}^{(n)}\}\), we introduce the recursive algorithms for estimating eigenvalues of matrix \(A\). Choose a positive sequence increasingly diverging to infinity, \(M_{k+1}>M_{k}>0\), \(M_{k}\underset{k\rightarrow\infty}{\longrightarrow}\infty\). With arbitrary initial values \(\lambda_{0}^{(j)},\ \{\lambda_{k}^{(j)}\}_{k\geq 1},\ j=1,\cdots,n\) are generated by the following algorithms:  
\[\lambda_{k+1}^{(j)} =\Big{[}\lambda_{k}^{(j)}-a_{k}\Big{(}\lambda_{k}^{(j)}-u_{k}^{(j )\intercal}A_{k}u_{k}^{(j)}\Big{)}\Big{]}\cdot I\big{[}\big{|}_{\lambda_{k}^{( j)}-a_{k}\big{(}\lambda_{k}^{(j)}-u_{k}^{(j)\intercal}A_{k}u_{k}^{(j)}\big{)} \big{|}\leq M_{\tau_{k}(j)}}^{\ast}\big{]}, \tag{41}\] \[\sigma_{k}(j) =\sum_{i=1}^{k-1}I_{\big{[}\big{|}\lambda_{i}^{(j)}-a_{i}\big{(} \lambda_{i}^{(j)}-u_{i}^{(j)\intercal}A_{i}u_{i}^{(j)}\big{)}\big{|}>M_{\tau (j)}\big{]}}^{\ast},\ \ \ \ \sigma_{0}(j)=0, \tag{42}\]  
where \(\lambda_{k}^{(j)}\) is the estimate for the eigenvalue corresponding to the eigenvector which \(u_{k}^{(j)}\) estimates at time \(k\). Denote \(S\) the unit sphere in \(\mathbb{R}^{n}\) and \(J\) the set of all eigenvectors with unit modular of matrix \(A\). Denote the set of all eigenvalues of matrix \(A\) by \(V(J)\), that is, \(V(J)\triangleq\{\lambda^{(1)},\cdots,\lambda^{(n)}\}\). For algorithms (39)-(42), the following result takes place. **Theorem 4**.