# Multi-View Cluster Structure Guided One-Class BLS-Autoencoder for Intrusion Detection
## 3 Proposed Method
### Multi-View Co-Association Matrix Optimization  
Although all training samples are labeled as normal traffic data, they contain latent subcategories stemming from diverse network applications. Based on this intuition, a multi-view co-association matrix is first constructed from multi-view data, which preserves the cluster structure of samples. Considering the efficiency and robustness of clustering structure analysis, an ensemble clustering strategy is adopted, which applies K-means in each view to obtain multiple base partitions \(\{P_{1}^{v},\ldots,P_{B}^{v}\}\), where \(B\) is the number of base partition in the \(v\)-th view. Each base partition is represented as a binary partition matrix \(P_{b}^{v}\in\mathbb{R}^{n\times c_{b}}\)[29]; the \(ij\)-th entry of \(P_{b}^{v}\) is defined as follows:  
\[P_{b(ij)}^{v}= \begin{cases}1,&if\ x_{i}\in C_{j},\\ 0,&otherwise,\end{cases}  \tag{1}\]  
where \(C_{j}\) is the \(j\)-th cluster, and \(c_{b}\) is the number of clusters randomly set within a certain range. By horizontally concatenating the binary partition matrices, the ensemble binary partition matrix \(P^{v}\) is obtained for the \(v\)-th view, and the co-association matrix \(G^{v}\in\mathbb{R}^{n\times n}\) can be further computed as follows:  
\[G^{v}=\frac{1}{B}P^{v}(P^{v})^{T}, \tag{2}\]  
\[P^{v}=[P_{1}^{v}\ P_{2}^{v}\ \ldots\ P_{B}^{v}]. \tag{3}\]  
It can be observed that \(G^{v}\) effectively integrates diverse base partitions, representing the probability of sample pairs being assigned to the same cluster. This enables \(G^{v}\) to comprehensively capture the intrinsic cluster structure relationship among samples within the \(v\)-th view. Network traffic data collected from multiple sources reflect inter-sample relationships from different perspectives. Therefore, it is necessary to further integrate cross-view co-association matrices to obtain more robust cluster structure information. However, simply averaging multiple view-specific co-association matrices fails to effectively integrate cluster structure information across different views. To address this problem, a multi-view co-association matrix optimization objective function with doubly-stochastic constraints is proposed as follows:  
\[ \begin{split}&\min_{G}\sum_{v=1}^{V}||G-G^{v}||_{F}^{2},\\ & s.t.G\geq 0,\ G\mathbf{1}=\mathbf{1},\ G^{T}=G.\end{split}  \tag{4}\]  
In the absence of prior knowledge, all views are treated equally to learn a multi-view co-association matrix that approximates the view-specific co-association matrices. By imposing doubly-stochastic constraints, the spectral relationships between samples are further refined, yielding a globally optimized co-association matrix that better reflects the underlying cluster structure across different views. By setting \(Q=\frac{1}{V}\sum_{v=1}^{V}G^{v}\), problem (4) can be reformulated into a doubly-stochastic normalization problem under the [PERSON] norm [30] as follows:  
\[ \begin{split}&\min_{G}||G-Q||_{F}^{2},\\ & s.t.G\geq 0,\ G\mathbf{1}=\mathbf{1},\ G^{T}=G.\end{split}  \tag{5}\]  
Then, according to the von [PERSON] successive projection lemma, the global optimal solution can be derived by iteratively solving the following two problems:  
\[\min_{G}||G-Q||_{F}^{2},\ s.t.\ G\mathbf{1}=\mathbf{1},\ G^{T}=G. \tag{6}\]  
\[\min_{G}||G-Q||_{F}^{2},\ s.t.G\geq 0. \tag{7}\]  
The closed form solution of the problem (6) can be optimized by the Lagrange method as follows:  
\[G=Q+(\frac{I-Q}{n}+\frac{\mathbf{1}\mathbf{1}^{T}Q}{n^{2}})\mathbf{1}\mathbf{ 1}^{T}-\frac{1}{n}\mathbf{1}\mathbf{1}^{T}Q. \tag{8}\]  
As for problem (7), it can be solved by simply projecting it onto a non-zero space as follows:  
\[G=Q^{+}. \tag{9}\]  
In general, the initial \(G\) is \(G=Q=\frac{1}{V}\sum_{v=1}^{V}G^{v}\). Then, it is iteratively updated by computing \(Q=G\) and Equations (8) and (9) until convergence to obtain the multi-view coassociation matrix. The pseudo code of the multi-view co-association matrix optimization is shown in Algorithm 1.