# Low-Rank Approximation of Difference between Correlation Matrices Using Inner Product
## 3 Results
### Results of fMRI Data Analysis  
In this subsection, we show the results of applying the proposed method to the fMRI data. Concretely, the purpose of this example is to detect clustering structures where the difference between two experimental conditions, i.e., High-WM and Low-WM tasks, is emphasized. In addition, the features of these estimated clusters are interpreted in combination with knowledge on ROIs related to WM, including the task-positive network (TPN), ventral attention network (VAN), salience network (SN), visual network (VN), and default mode network (DMN). The TPN consists of the fronto-parietal network (FPN), dorsal attention network (DAN), and cingulo-opercular network (CON). The FPN and DAN are related to executive function [36] and top-down attention, respectively, whereas the CON is involved in exerting control over the contents of WM [37]. In addition, the VAN is associated with response to stimuli and bottom-up attention, whereas SN activation is said to be observed in situations wherein changes in behavior may be advisable [38; 39]. The VN is literally associated with visual information processing, whereas the DMN relates to internally focused tasks such as retrieving autobiographical memory, envisioning the future, and conceiving the perspectives of others [40]. We then explain how to construct the difference between correlation matrices. For each subject, using a matrix of difference between the correlation matrices of High-WM and Low-WM, we calculate one mean matrix of these differences. As described in the previous section, the input difference is a \(90\times 90\) matrix. In the proposed method, rank and the number of clusters must be selected. For the determination of rank, we set the rank candidates to \(2,\ 3,\ 4,\ 5,\) and \(6\), and select one among them. Generally, when the rank is larger, the values of the objective function tend to be lower. The proposed method is then applied to difference matrices of rank \(=2,\ 3,\ 4,\ 5,\ 6,\) and the rate of change in values of the objective function is calculated as follows:  
\[\frac{L_{d}(\mathbf{X}|\ \mathbf{\Delta})-L_{d+1}(\mathbf{X}|\ \mathbf{\Delta})}{L_{d}(\mathbf{X}|\ \mathbf{\Delta})}\ \ \ (d=2,\ 3,\ 4,\ 5,\ 6),\]  
where \(L_{d}(\mathbf{X}|\ \mathbf{\Delta})\) is the value of the objective function with rank \(d\). From Figure 5, \(d=3\) is selected because, among the candidates, the change in values for the objective function tends not to change even when the rank is larger than 3. For the number of clusters, the silhouettes coefficient [41] is used; the number of clusters that provides the highest silhouettes coefficient is \(k=3\). Figure 5: Change in rate for objective function; vertical and horizontal axes indicate rank and ratio of values for objective function, respectively. The left side of Figure 6 shows the results of applying the proposed method. Through the estimated low-rank correlation matrix, we confirm that the clustering structure is emphasized and that the relations among ROIs belonging to the same cluster are estimated to be higher. The right-hand side of Figure 6 shows the original difference matrix, in which both rows and columns are permutated by the estimated clusters. The right-hand side of Figure 6 also shows that relations among ROIs belonging to cluster 1 tend to be higher during the difficult task than during the easy task, although those among ROIs belonging to cluster 2 or cluster 3 tend to be lower. The features of the estimated cluster are then interpreted in combination with knowledge on WM. Figure 7 represents the difference matrix (High-WM - Low-WM) permutated based on both estimated clusters and WM. We particularly focused on differences related to the TPN and DMN. For the relations of ROIs belonging to the TPN, those belonging to cluster 1 tend to be higher than those belonging to cluster 2 or cluster 3. Therefore, in cluster 1, correlation among ROIs related to the TPN tended to be active under the condition of High-WM.