# Low-Rank Approximation of Difference between Correlation Matrices Using Inner Product
## 2 Methods
### Simulation Study
First, \(\mathbf{D}\) is decomposed as \(\mathbf{D}=\mathbf{B}^{\intercal}\mathbf{B}\) using Cholesky decomposition, where \(\mathbf{B}=(\mathbf{b}_{1},\mathbf{b}_{2},\cdots,\mathbf{b}_{p})\ \mathbf{b}_{i}\in\mathbb{R}^{q}(i=1,2, \cdots,p;\ q\leq p)\). Let \(\mathbf{b}^{\intercal}=(\|\mathbf{b}_{1}\|,\ \|\mathbf{b}_{2}\|,\cdots,\ \|\mathbf{b}_{p}\|)^{T}\); therefore, there exists \(\cos\phi_{ij}\) such that \(d_{ij}=\|\mathbf{b}_{i}\|\ \|\mathbf{b}_{j}\|\ \cos\phi_{ij}\ (0\leq\phi_{ij}<2\pi)\). From the decomposition, the optimization problem of the second method is formulated as follows:  
\[L_{(2)}(\mathbf{X}_{2})=\|\mathbf{D}-\mathbf{b}^{\intercal}\mathbf{b}^{\intercal}\odot\mathbf{X}_ {2}\mathbf{X}_{2}^{T}\|_{F}^{2}\] (17) subject to \[\|\mathbf{x}_{i}^{(2)}\|=1\quad(i=1,2,\cdots,p),\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Names of Factors** & **Levels** & **Descriptions** \\ \hline Methods & 4 & proposal, Method 2, Method 3, and Method 4 \\ Rank & 3 & Rank \(=2,3\) and 4 \\ The number of clusters & 3 & \(k=2,3\), and 4 \\ The difference between true correlation & 2 & 0.1, and 0.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of simulation design. where \(\mathbf{X}_{2}=(\mathbf{x}_{1}^{(2)},\ \mathbf{x}_{2}^{(2)},\ \cdots,\ \mathbf{x}_{p}^{(2)}),\ \mathbf{x}_{i}^{(2)} \in\mathbb{R}^{d}\). The parameters of Equation (17) can be estimated in the same manner as in the proposed method. Subsequently, \(k\)-means is applied to the estimated \(\mathbf{X}_{2}\). The third method also consists of two steps. Eigenvalue decomposition is applied to \(\mathbf{D}\), and the low-rank matrix corresponding to \(\lambda_{1},\ \lambda_{2},\ \cdots,\ \lambda_{d}\) is estimated, where \(\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{q}\ (q\leq p)\) are eigenvalues of \(\mathbf{D}\). Afterward, \(k\)-means is applied to the estimated low-rank matrix. The fourth method is similar to the third method, with the only difference being that eigenvalue decomposition is applied to the inner product matrix of \(\mathbf{D}\) and not only to \(\mathbf{D}\). These methods, i.e., method 2, method 3, and method 4, are referred to as tandem approaches [29], each of which consists of two steps. First, the existing dimensional reduction method is applied to the data and the low-rank data matrix is estimated. Second, a typical clustering algorithm, such as \(k\)-means clustering, is applied to the estimated low-rank data matrix. In a practical situation, these approaches are sometimes used. Both the third and fourth methods provide us with low-rank matrices and clustering results. However, from these results, it is difficult to interpret the degree of the relation because these estimated values are not bounded. On the other hand, both the first and second methods allow us to interpret the results easily because these estimated values are bounded within the range \(-1\) to \(1\). **Factor 2: Rank**  
For the four methods mentioned in the previous subsection, the rank must be determined. In this simulation, ranks are set to \(2,\ 3\), and \(4\). **Factor 3: Number of clusters**  
All four methods mentioned in the Factor 1: Methods subsection adopt \(k\)-means clustering. In the simulation, we assume that the the number of true clusters is known beforehand. The number of clusters is set to \(k=2,\ 3\), and \(4\). **Factor 4: Difference between true correlations**  
At first, the generation of true clustering structures is determined by \(k=2,3\), and \(4\). The true clustering structure is dependent on \(\mathbf{\Sigma}_{X}\) and \(\mathbf{\Sigma}_{Y}\). When \(k=2\), the equations for \(\mathbf{\Sigma}_{X}\) and \(\mathbf{\Sigma}_{Y}\) are  
\[\mathbf{\Sigma}_{X}=\left[ \begin{array}{cc}\mathbf{\Sigma}_{1}&\mathbf{O}_{60}\\ \mathbf{O}_{60}&\mathbf{I}_{60}\end{array} \right]\quad\text{and}\quad\mathbf{\Sigma}_{Y}= \left[ \begin{array}{cc}\mathbf{I}_{60}&\mathbf{O}_{60}\\ \mathbf{O}_{60}&\mathbf{\Sigma}_{1}\end{array}