# Low-Rank Approximation of Difference between Correlation Matrices Using Inner Product
## 2 Methods
### Simulation Study  
In this subsection, the superiority of the proposed method is shown via the results of a numerical simulation. In particular, the recovery of clustering results is evaluated in this simulation. First, we reveal the simulation design. To evaluate the clustering results, artificial data with a true clustering structure are generated and correlation matrices between variables are calculated from the data. Dimensional reduction clustering approaches are then applied to the difference between the two correlation matrices, and the clustering results are obtained. In this numerical simulation, the true number of clusters is assumed to be known beforehand. Finally, an adjusted Rand index (ARI) [28] between the true clustering structure and the estimated clustering structure is calculated, and the effectiveness of the proposed method is compared with that of existing approaches. Here, ARI is the similarity index between two clustering results. When two clustering results are completely equivalent to each other, ARI becomes 1; otherwise, ARI becomes close to 0. Afterward, we explain the method for generating the artificial data. Multivariate data representing condition 1 and condition 2 are generated as follows:  
\[X\sim N(\mathbf{0}_{p},\mathbf{\Sigma}_{X})\quad\text{and}\quad Y\sim N(\mathbf{0}_{p},\mathbf{ \Sigma}_{Y}), \tag{16}\]  
where \(X\) and \(Y\) are random vectors of conditions 1 and 2, respectively; \(\mathbf{0}_{p}\) is a vector with length \(p\), for which the elements are all zero; and \(\mathbf{\Sigma}_{X}\in[-1,1]^{p\times p}\) and \(\mathbf{\Sigma}_{X}\in[-1,1]^{p\times p}\) are true correlation matrices of condition 1 and condition 2, respectively. Here, in this simulation, \(p\) is set to 120. From Equation (16), \(p\) dimensional vectors are generated 60 times for each condition, and sample correlation matrices of condition 1 and condition 2 are calculated as \(\mathbf{R}(X)\in[-1,1]^{p\times p}\) and \(\mathbf{R}(Y)\in[-1,1]^{p\times p}\), respectively. The input data are then calculated as \(\mathbf{D}=\mathbf{R}(X)-\mathbf{R}(Y)\). In this simulation, four factors are utilized; a summary of the simulation is shown in Table 1. As a result, there are \(4\times 3\times 3\times 2=72\) patterns in this simulation. For each pattern, the corresponding artificial data are generated 100 times and evaluated using the ARI. The descriptions of the factors are presented as follows. For this factor, we evaluate four methods. The purpose of setting this factor is to evaluate the effect of using an inner product and to estimate a bounded low-rank matrix. Here, the proposed method is referred to as method 1. By contrast, method 2 is designed to have an approach similar to that of the proposed approach, except method 2 is based only on difference and not on the inner product. Through a comparison between the proposed method and method 2, we can evaluate the effect of using the inner product. On the other hand, in method 3, the low-rank matrix is estimated from only the difference and calculated via Cholesky decomposition, where these estimated values are characterized by no constraint. Based on a comparison between the proposed method and method 3, the effect of both the inner product and bounded constraint is evaluated. Meanwhile, in method 4, the low-rank matrix is estimated from the inner product; however, the estimated values are characterized by no constraint. Therefore, based on a comparison between the proposed method and method 4, the effect of the bounded constraints is estimated. These four methods are then explained from the perspective of calculation. The first method is the proposed approach based on the inner product of \(\mathbf{D}\). For the second method, an approach based on difference, and not on the inner product model, is adopted. The second method consists of two steps, as does the proposed method.