# Low-Rank Approximation of Difference between Correlation Matrices Using Inner Product
###### Abstract  
In the domain of functional magnetic resonance imaging (fMRI) data analysis, given two correlation matrices between regions of interest (ROIs) for the same subject, it is important to reveal relatively large differences to ensure accurate interpretation. However, clustering results based only on differences tend to be unsatisfactory and interpreting the features tends to be difficult because the differences likely suffer from noise. Therefore, to overcome these problems, we propose a new approach for dimensional reduction clustering. **Methods:** Our proposed dimensional reduction clustering approach consists of low-rank approximation and a clustering algorithm. The low-rank matrix, which reflects the difference, is estimated from the inner product of the difference matrix, not only from the difference. In addition, the low-rank matrix is calculated based on the majorize-minimization (MM) algorithm such that the difference is bounded within the range \(-1\) to \(1\). For the clustering process, ordinal \(k\)-means is applied to the estimated low-rank matrix, which emphasizes the clustering structure. **Results:** Numerical simulations show that, compared with other approaches that are based only on differences, the proposed method provides superior performance in recovering the true clustering structure. Moreover, as demonstrated through a real-data example of brain activity measured via fMRI during the performance of a working memory task, the proposed method can visually provide interpretable community structures consisting of well-known brain functional networks, which can be associated with the human working memory system. **Conclusions:** The proposed dimensional reduction clustering approach is a very useful tool for revealing and interpreting the differences between correlation matrices, even when the true differences tend to be relatively small. \(k\)-means; fMRI data analysis; MM algorithm +
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subnetworks of ROIs such that the differences between conditions are relatively large. However, it is difficult to interpret the features of distinctive clusters because the range of correlations is bounded, i.e., \([-1,1]\), and the differences are affected by observational error. In [10], through the use of benchmark data sets, it was reported that increasing the number of dimensions does not affect the clustering results. On the other hand, in [11], it was demonstrated that clustering results tend to be worse when the data include noise, irrespective of the clustering method used, such as \(k\)-means. Concretely, we assume a situation such that some meaningful variables include a clustering structure, e.g., [12], whereas the other variables do not and are considered as noise. In this situation, it is difficult to detect the clustering structure using a clustering algorithm. One very useful approach for addressing this problem is low-rank approximation for the difference matrix. Here, dimensional reduction signifies estimating a low-rank correlation matrix from the full-rank difference matrix; that is, the estimated low-rank approximation can emphasize the clustering structure. However, such estimated low-rank matrices occasionally lose clustering structure information because the original difference matrix is affected by noise, as demonstrated in upper Figure 1, and mask the true clustering structure. To overcome this problem, we focus on the inner product matrix of the difference matrix. Even if the original difference matrix includes noise, the inner product matrix can emphasize the clustering structure of variables, as demonstrated in the middle of Figure 1. Such an approach, which uses inner products for clustering, was proposed in [13] for processing high-dimensional data. For estimating low-rank matrices, singular value decomposition is very popular. However, the range of the estimated low-rank matrix is not bounded, whereas that of the correlation is bounded within the range \(-1\) to \(1\).