## 2 Background
### Proposed Objective Function  
Given a noisy \(M\)-layer multiplex network, an adjacency tensor, \(\mathcal{A}\in\mathbb{R}^{n\times n\times M}\), is constructed from the network by stacking the normalized adjacencies, \(\{\mathbf{A}_{N}^{(m)}\}_{m=1}^{M}\in\mathbb{R}^{n\times n}\), of the individual layers as the frontal slices of the tensor. The proposed **Mx-CRTSA** aims to detect the community structure of the multiplex network by first extracting a clean self-representation tensor from the noisy adjacency tensor. The extracted clean tensor is then used to compute the low-dimensional individual spectral embeddings or subspaces, \(\{\mathbf{U}_{m}\}_{m=1}^{M}\in\mathbb{R}^{n\times c}\), and a common centroid spectral embedding across all layers, \(\mathbf{U}^{*}\in\mathbb{R}^{n\times c}\) where \(\mathbf{U}^{*}\) is enforced to be close to all \(\{\mathbf{U}_{m}\}_{m=1}^{M}\). The proposed objective function is formulated as follows:  
\[\begin{split}\min_{\mathbf{U}^{*}\mathbf{U}_{\mathbf{E},\mathbf{w }}^{(m)},\mathcal{Z},}\|\mathcal{Z}\|_{\otimes}+\gamma\|\mathbf{E}\|_{2,1}- \lambda\sum_{m=1}^{M}\left[tr\bigg{(}\mathbf{U}^{(m)}{}^{\top}\mathbf{Z}_{N}^{ (m)}\mathbf{U}^{(m)}\bigg{)}+w_{m}tr\bigg{(}\mathbf{U}^{*}{}^{\top}\mathbf{U}^ {(m)}{}^{\top}\mathbf{U}^{*}\bigg{)}\right]\\ \text{s.t.}\quad\mathbf{A}^{(m)}=\mathbf{A}^{(m)}\mathbf{Z}^{(m) }+\mathbf{E}^{(m)},\mathbf{E}=[\mathbf{E}^{(1)};\mathbf{E}^{(2)};\ldots; \mathbf{E}^{(M)}],\mathbf{Z}^{(m)}=\mathbf{Z}^{(m)}{}^{\top},\\ \mathbf{U}^{(m)}{}^{\top}\mathbf{U}^{(m)}=\mathbf{I},m=\{1,\ldots, M\},\mathcal{Z}=\Phi(\mathcal{Z}^{(1)},\mathcal{Z}^{(2)},\ldots,\mathcal{Z}^{(M)}), \\ \|\mathbf{w}\|_{2}=1,\mathbf{w}\geq 0,\mathbf{U}^{*}{}^{\top} \mathbf{U}^{*}=\mathbf{I},\end{split} \tag{12}\]  
where* \(\|\mathcal{Z}\|_{\oplus}\) is considered to recover the low-rank approximation tensor, \(\mathcal{Z}\), from the adjacency tensor, \(\mathcal{A}\). \(\Phi(\cdot)\) is adopted to construct the clean tensor from the recovered self-representation matrices, \(\{\mathbf{Z}^{(m)}\}_{m=1}^{M}\), and rotate it so that the computational complexity is reduced while the tensor's nuclear norm is minimized.
* \(\|\mathbf{E}\|_{2,1}\) represents the \(\mathrm{L}_{2,1}\) of the error matrix \(\mathbf{E}\in\mathbb{R}^{m\times n}\), which is included to remove the noise from the adjacency tensor, \(\mathcal{A}\). The matrix \(\mathbf{E}\) is formed by arranging the individual error matrices, \(\{\mathbf{E}^{(m)}\}_{m=1}^{M}\), vertically.
* The last term consists of two sub-terms. The first sub-term, \(\sum_{m=1}^{M}tr(\mathbf{U}^{(m)}{}^{\top}\mathbf{Z}_{N}^{(m)}\mathbf{U}^{(m)})\), refers to the normalized cut of each layer which is included to find the individual spectral embedding of the separate layers, \(\{\mathbf{U}^{(m)}\}_{m=1}^{M}\), where \(\mathbf{Z}_{N}^{(m)}\) is the normalized clean adjacency matrix of the \(m^{th}\) layer. The second sub-term, \(w_{m}tr(\mathbf{U}^{*\top}\mathbf{U}^{(m)}\mathbf{U}^{(m)}{}^{\top}\mathbf{U} ^{*})\), represents the co-regularization across layers and corresponds to the projection distance between the individual subspaces, \(\{\mathbf{U}^{(m)}\}_{m=1}^{M}\), and the centroid or common subspace, \(\mathbf{U}^{*}\). By minimizing the projection distance, the common subspace is constrained to be close to all \(\{\mathbf{U}^{(m)}\}_{m=1}^{M}\), while the node connectivity in each layer is still preserved. The degree of the consistency between the individual subspaces and the common subspace is controlled by the weighting vector, \(\mathbf{w}=[w_{1},w_{2},\ldots,w_{m}]^{T}\), which also will be optimized.
* \(\gamma\) and \(\lambda\) are regularization parameters to balance the trade-off between the different terms in the proposed objective function.