# Alternating Direction Method of Multipliers for Generalized Low-Rank Tensor Recovery
## 4 Generalized Low-Rank Tensor Recovery
### Model of GLRTR  
The datasets contaminated by Gaussian noise are very universal in practical engineering applications. In view of this, we assume the data tensor \(\mathbf{\mathcal{D}}\) to be the superposition of the low-rank component \(\mathbf{\mathcal{A}}\), the large sparse corruption \(\mathbf{\mathcal{E}}\) and the Gaussian noise \(\mathbf{\mathcal{G}}\). We also consider the case that some entries of \(\mathbf{\mathcal{D}}\) are missing. To recover simultaneously the above three terms, we establish a convex GLRTR model as follows:  
\[\min_{\mathbf{\mathcal{A}}\mathbf{\mathcal{E}},\mathbf{\mathcal{G}}}\,||\,\mathbf{\mathcal{A}} \,||_{*}+\mathbf{\lambda}\,||\,\mathbf{\mathcal{E}}\,||_{\mathbf{\mathsf{1}}}+\mathbf{\tau}\,|| \,\mathbf{\mathcal{G}}\,||\,\mathrm{F}_{\mathbf{\mathsf{P}}}^{2},\text{ s.t.}\,\mathrm{P}_{\mathbf{\Omega}}(\mathbf{ \mathcal{D}})=\mathrm{P}_{\mathbf{\Omega}}(\mathbf{\mathcal{A}}+\mathbf{\mathcal{E}}+ \mathbf{\mathcal{G}}), \tag{8}\]  
where the regularization coefficients \(\mathbf{\lambda}\) and \(\mathbf{\tau}\) are nonnegative. If we reinforce the constraints \(\mathbf{\mathcal{G}}=\mathbf{\mathcal{O}}\)(or equivalently \(\mathbf{\tau}\to+\infty\)) and \(\mathbf{\mathcal{E}}=\mathbf{\mathcal{O}}\) (or equivalently \(\mathbf{\lambda}\to+\infty\) ), then the GLRTR is transformed into the tensor completion model (4), where the zero tensor \(\mathbf{\mathcal{O}}\) has the same dimensionality as \(\mathbf{\mathcal{D}}\). If only the constraint \(\mathbf{\mathcal{E}}=\mathbf{\mathcal{O}}\) is considered, then the GLRTR is equivalent to Equation (6). Furthermore, if we take \(\mathbf{\mathcal{G}}=\mathbf{\mathcal{O}}\) and \(\mathbf{\Omega}=[\mathrm{I}_{\mathbf{\mathsf{1}}}]\times[\mathrm{I}_{\mathbf{\mathsf{2}}}] \times\cdots\times[\mathrm{I}_{\mathbf{\mathsf{N}}}]\), then the model of GLRTR becomes the model of MRPCA. In summary, the proposed model is the generalization of the existing LRTR. For the convenience of using the splitting method, we discard \(\mathbf{\mathcal{A}}\) and introduce \(N+1\) auxiliary tensor variables \(\mathbf{\mathcal{M}}_{\mathbf{\mathsf{1}}}\), \(\mathbf{\mathcal{M}}_{\mathbf{\mathsf{2}}}\), \(\ldots\), \(\mathbf{\mathcal{M}}_{\mathbf{\mathsf{N}}}\), \(\mathbf{\mathcal{X}}\), where these auxiliary variables have the same dimensionality as \(\mathbf{\mathcal{D}}\). Let \(\mathbf{M}_{\mathbf{\mathsf{n}}(\mathbf{\mathsf{n}})}\) be the \(n\)-mode matricization of \(\mathbf{\mathcal{M}}_{\mathbf{\mathsf{n}}}\) for each \(\mathbf{\mathsf{n}}\in[\mathrm{N}]\) and \(\mathbf{\mathcal{M}}=\{\mathbf{\mathcal{M}}_{\mathbf{\mathsf{1}}},\mathbf{\mathcal{M}}_{\mathbf{ \mathsf{2}}},\ldots,\mathbf{\mathcal{M}}_{\mathrm{N}}\}\). Hence, we have the equivalent formulation of Equation (8):  
\[\min_{\mathbf{\mathcal{X}},\mathbf{\mathcal{E}},\mathbf{\mathcal{G}},\mathbf{ \mathcal{M}}}\,\sum_{\mathbf{\mathsf{N}}=\mathbf{\mathsf{1}}}^{\mathbf{\mathsf{n}}}\,\mathrm{ w}_{\mathbf{\mathsf{n}}}\,||\,\mathbf{M}_{\mathbf{\mathsf{n}}(\mathbf{\mathsf{n}})}\,||_{*}+\mathbf{ \lambda}\,||\,\mathbf{\mathcal{E}}\,||_{\mathbf{\mathsf{1}}}+\mathbf{\tau}\,||\,\mathbf{ \mathcal{G}}\,||\,\mathrm{F}_{\mathbf{\mathsf{F}}}^{2}, \tag{9}\] \[\text{s.t.}\,\,\mathbf{\mathcal{X}}=\mathbf{\mathcal{M}}_{\mathbf{\mathsf{n}} }+\mathbf{\mathcal{G}}+\mathbf{\mathcal{E}},\mathrm{P}_{\mathbf{\Omega}}(\mathbf{\mathcal{X}})= \mathrm{P}_{\mathbf{\Omega}}(\mathbf{\mathcal{D}}),\mathbf{\mathsf{n}}=\mathbf{\mathsf{1}},\mathbf{ \mathsf{2}},\ldots,\mathrm{N}.\]  
As a matter of fact, the discarded low-rank tensor \(\mathbf{\mathcal{A}}\) can be represented by \(\mathbf{\mathcal{A}}=\sum_{\mathbf{\mathsf{N}}=\mathbf{\mathsf{1}}}^{\mathbf{\mathsf{n}}}\mathbf{ \mathcal{M}}_{\mathbf{\mathsf{n}}}/\mathrm{N}\). Now, we explain the low-rankness of \(\mathbf{\mathcal{A}}\) from two viewpoints.