# Low-Rank Approximation of Difference between Correlation Matrices Using Inner Product
## 2 Methods
### Model of Proposed Method  
Here, the model of the proposed method is explained. Let \(\mathbf{R}_{o}^{(1)}=(r_{ijo}^{(1)})\ r_{ijo}^{(1)}\in[-1,1],\ (i,j=1,2,\cdots,p;\ o=1,2,\cdots,n)\), and \(\mathbf{R}_{o}^{(2)}=(r_{ijo}^{(2)})\ r_{ijo}^{(2)}\in[-1,1],\ (i,j=1,2,\cdots,p;\ o=1,2,\cdots,n)\), be the correlation matrices between variables under condition 1 and condition 2, respectively, where \(n\) is the number of subjects and \(p\) is the number of variables. From \(\mathbf{R}_{o}^{(1)}\) and \(\mathbf{R}_{o}^{(2)}\ (o=1,2,\cdots,n)\), the difference is calculated as follows:  
\[\mathbf{D}=\frac{1}{n}\sum_{i=1}^{n}\Big{(}\mathbf{R}_{o}^{(1)}-\mathbf{R}_{o}^{(2)}\Big{)},\]  
where \(\mathbf{D}=(\mathbf{d}_{1},\mathbf{d}_{2},\cdots,\mathbf{d}_{p})^{T}=(d_{ij}),\ d_{ij}\in[-2,2] \ (i,j=1,2,\cdots,p)\). The clustering structure of \(\mathbf{D}\) is assumed to be in the following form:  
\[\mathbf{D}=\left[ \begin{array}{cccc}\mathbf{D}_{1}&\mathbf{O}_{12}&\cdots&\mathbf{O}_{1k}\\ \mathbf{O}_{21}&\mathbf{D}_{2}&\cdots&\mathbf{O}_{2k}\\ \vdots&\vdots&\ddots&\vdots\\ \mathbf{O}_{k1}&\mathbf{O}_{k2}&\cdots&\mathbf{D}_{k}\end{array} \right], \tag{1}\]  
where \(\mathbf{D}_{\ell}\in[-2,2]^{p_{\ell}\times p_{\ell}}\ (\ell=1,2,\cdots,k)\) is a block matrix such that the absolute values of the elements tend to be higher than those of the elements of \(\mathbf{O}_{\ell_{S}}\ (\ell\
eq s)\), where each  
Figure 1: Image of proposed approach. element of \(\mathbf{O}_{ts}\) is 0. That is, \(\mathbf{D}\) is assumed to be changed to Equation (1) through the permutation of these rows and columns. However, even if \(\mathbf{D}\) has such a clustering structure, the structure is masked because the \(d_{ij}\) values are observed with noise, and the corresponding correlations are bounded within the range \(-1\) to 1. Therefore, it is difficult to capture the clustering structure using low-rank approximation based only on \(\mathbf{D}\). To overcome this problem, the inner product of \(\mathbf{D}\) is focused upon because the clustering structure of the inner product is emphasized, rather than using only \(\mathbf{D}\). Here, the inner product of \(\mathbf{D}\) is defined as follows:  
\[\mathbf{\Delta}=\mathbf{D}^{T}\mathbf{D}, \tag{2}\]  
where the elements of \(\mathbf{\Delta}\) are defined as \(\delta_{ij}\in\mathbb{R}\ (i,j=1,2,\cdots,p)\), and \({}^{T}\) indicates translocation. We focus on the inner product and construct the model based on low-rank approximation from \(\delta_{ij}\). First, from the property of the inner product, for an arbitrary \(\delta_{ij}\), there exists \(\theta_{ij}\ (0<\theta_{ij}\leq 2\pi)\) such that  
\[\delta_{ij}=\|\delta_{i}\|\cdot\|\delta_{j}\|\cos\theta_{ij}, \tag{3}\]  
where \(\delta_{j}=(\delta_{1j},\delta_{2j},\cdots,\delta_{pj})^{T}\ (j=1,2,\cdots,p)\), and \(\|\cdot\|\) is the Euclidean norm. Here, \(\cos\theta_{ij}\in[-1,1]\) can be considered a correlation, and the matrix representation is  
\[\mathbf{\Theta}=\left[ \begin{array}{cccc}1&\cos\theta_{12}&\cdots&\cos\theta_{1 p}\\ \cos\theta_{21}&1&\cdots&\cos\theta_{2p}\\ \vdots&\vdots&\ddots&\vdots\\ \cos\theta_{p1}&\cos\theta_{p2}&\cdots&1\end{array} \right]. \tag{4}\]  
Based on Equations (3) and (4), Equation (2) is expressed as follows:  
\[\mathbf{\Delta}=\mathbf{D}^{T}\mathbf{D}=\mathbf{d}\mathbf{d}^{T}\odot\mathbf{\Theta}, \tag{5}\]  
where \(\mathbf{d}=(\|\mathbf{d}_{1}\|,\ \|\mathbf{d}_{2}\|,\cdots,\|\mathbf{d}_{p}\|)^{T}\), and \(\odot\) is a Hadamard product. From Equation (5), we then construct the following model:  
\[\mathbf{d}\mathbf{d}^{T}\odot\mathbf{\Theta}=\mathbf{d}\mathbf{d}^{T}\odot\mathbf{X}\mathbf{X}^{T}+\mathbf{E}, \tag{6}\]  
where \(\mathbf{X}=(x_{ij})\ x_{iq}\in\mathbb{R}(i=1,2,\cdots,p;\ q=1,2,\cdots,d)\) is a coordinate matrix with rank \(d\ (d\leq p)\), \(d\) is determined by the researchers, and \(\mathbf{E}=(e_{ij})\ e_{ij}\in\mathbb{R}\ (i,j=1,2,\cdots,n)\) is an error matrix.