# Low-Rank Approximation of Difference between Correlation Matrices Using Inner Product
## 4 Conclusions and Remarks  
We proposed a new dimensional reduction clustering approach based on differences between correlation matrices. The estimated low-rank correlation represents the difference, and it is easy to interpret the relations because the estimated values are bounded within the range \(-1\) to \(1\) and because the clustering structure is emphasized. In addition, based on the idea of using the inner product of the difference and not only the difference, we demonstrated that the clustering recovery tends to be accurate. The effectiveness of the proposed method is demonstrated through numerical simulations of the recovery of a true clustering structure. According to these results, the proposed method can detect informative features related to clustering structure and can eliminate noise, and using the low-rank approximation model for the inner product and not only the difference degrades the effect of noise not related to the clustering structure. Certainly, based on the result of numerical simulation, the clustering approach based on eigen decomposition of \(\mathbf{D}\) performs well. However, it is difficult to interpret the estimated low-rank matrix because these values are not bounded. In addition, we show the results of applying the proposed method to real fMRI data related to WM. Through dimensionality reduction, the clustering structures of the ROIs are emphasized, as shown in the left side of Figure 6, because the variances of these estimated low-rank correlation coefficients tend to be larger. Specifically, our proposed method results in clusters that are interpreted based on knowledge regarding WM. Finally, we discuss the future direction of this research topic. There are four things to consider. First, although the proposed method certainly provides us with interpretable results from the perspective of WM, the proposed method requires further evaluation through real-data analyses and comparison with other methods. The properties of the clustering results, in particular, should be evaluated through the use of benchmark data. Second, although we show that the proposed method is based on [PERSON]'s correlation coefficient, other kinds of similarity measures, such as [PERSON]'s correlation coefficient, should be considered. Third, in the numerical simulations, we evaluate the clustering results using ARI. However, there are several other measures for comparing clustering results [44; 45]; these external validation measures are classified into three kinds of measures, i.e., pair counting, information-theory-based approaches, and set-matching measures. In the pair-counting approach, a pair of subjects is regarded as agreeing if the pair belongs to the same cluster in both clustering results or belongs to different clusters in both clustering results; otherwise, these subjects are regarded as disagreeing. In the information-theory-based approach, mutual information measures between two clustering results are used [46; 47; 48]. On the other hand, in set-matching measures, pairs of clusters are used instead of pairs of subjects; for this approach, several indices have already been proposed [49; 50]. Examples of related indices include centroid index [51] and centroid ratio [52]. Therefore, other evaluation indices should be considered for use with our proposed method. Fourth, an ideal number of clusters must be determined. In the real-data example, we used the silhouettes coefficient. However, various methods for determining the number of clusters have also been developed [53; 54; 55], where the method proposed in [55] is also used for determining the number of dimensions. Further simulations must be performed to better evaluate the effects of these approaches on the results of our proposed method. [PERSON] constructed the proposed statistical method, conducted numerical simulations, applied the method to real fMRI data, and drafted the initial manuscript. [PERSON] designed and conducted the experiment, proposed the framework of fMRI data analysis, and reviewed the manuscript.