# Alternating Direction Method of Multipliers for Generalized Low-Rank Tensor Recovery
## 4 Generalized Low-Rank Tensor Recovery
### Optimization Algorithm to GLRTR  
As a special splitting method, the ADMM is very efficient to solve a distributed optimization problem with linear equality constraints. It takes the form of the decomposition-coordination procedure and blends the merits of dual decomposition and augmented Lagrangian methods. In this section, we will propose the method of ADMM to solve Equation (9). We first construct the augmented Lagrangian function of the aforementioned convex optimization problem without considering the constraint \(\mathrm{P}_{\boldsymbol{\Omega}}(\boldsymbol{\mathcal{X}})=\mathrm{P}_{ \boldsymbol{\Omega}}(\boldsymbol{\mathcal{D}})\):  
\[ \begin{array}{ll}\mathrm{L}_{\boldsymbol{\mu}}(\boldsymbol{\mathcal{X}},& \boldsymbol{\mathcal{M}},\boldsymbol{\mathcal{E}},\boldsymbol{\mathcal{G}}, \boldsymbol{\mathcal{Y}})=\sum\limits_{\mathrm{n}=\boldsymbol{1}}^{\mathrm{N}} \mathrm{w}_{\mathrm{n}}\,|\,|\,\mathbf{M}_{\mathrm{n}(\mathrm{n})}\,|\,|\,_{ \mathrm{n}}+\boldsymbol{\lambda}\,|\,|\,\boldsymbol{\mathcal{E}}\,|\,_{ \mathrm{1}}+\boldsymbol{\tau}\,|\,|\,\boldsymbol{\mathcal{G}}\,|\,_{\mathrm{F }}^{2}+\\ &\sum\limits_{\mathrm{n}=\boldsymbol{1}}^{\mathrm{N}}\left(\left\langle \mathcal{Y}_{\mathrm{n}},\boldsymbol{\mathcal{X}}-\boldsymbol{\mathcal{M}}_{ \mathrm{n}}-\boldsymbol{\mathcal{G}}-\boldsymbol{\mathcal{E}}\right\rangle+ \frac{\boldsymbol{\mu}}{2}\,|\,|\,\boldsymbol{\mathcal{X}}-\boldsymbol{ \mathcal{M}}_{\mathrm{n}}-\boldsymbol{\mathcal{G}}-\boldsymbol{\mathcal{E}}\,| \,_{\mathrm{F}}^{2}\right),\end{array}  \tag{10}\]  
where \(\boldsymbol{\mu}\) is a positive numerical constant, \(\boldsymbol{\mathcal{Y}}_{\mathrm{n}}\) are Lagrangian multiplier tensors and \(\boldsymbol{\mathcal{Y}}=\left\{\boldsymbol{\mathcal{Y}}_{\mathrm{1}}, \boldsymbol{\mathcal{Y}}_{\mathrm{2}},\ldots,\boldsymbol{\mathcal{Y}}_{\mathrm{ N}}\right\}\). There are totally five blocks of variables in \(\mathrm{L}_{\boldsymbol{\mu}}(\boldsymbol{\mathcal{X}},\boldsymbol{\mathcal{M}}, \boldsymbol{\mathcal{E}},\boldsymbol{\mathcal{G}},\boldsymbol{\mathcal{Y}})\). The ADMM algorithm updates alternatively each block of variables while letting the other blocks of variables be fixed. If \(\boldsymbol{\mathcal{Y}}\) is given, one block of variables in \(\left\{\boldsymbol{\mathcal{X}},\boldsymbol{\mathcal{M}},\boldsymbol{\mathcal{G }},\boldsymbol{\mathcal{E}}\right\}\) can be updated by minimizing \(\mathrm{L}_{\boldsymbol{\mu}}(\boldsymbol{\mathcal{X}},\boldsymbol{\mathcal{M }},\boldsymbol{\mathcal{E}},\boldsymbol{\mathcal{G}},\boldsymbol{\mathcal{Y}})\) with respect to one argument. On the contrary, if \(\left\{\boldsymbol{\mathcal{X}},\boldsymbol{\mathcal{M}},\boldsymbol{\mathcal{G }},\boldsymbol{\mathcal{E}}\right\}\) are given, \(\boldsymbol{\mathcal{Y}}\) is updated by maximizing \(\mathrm{L}_{\boldsymbol{\mu}}(\boldsymbol{\mathcal{X}},\boldsymbol{\mathcal{M }},\boldsymbol{\mathcal{E}},\boldsymbol{\mathcal{G}},\boldsymbol{\mathcal{Y}})\) with respect to \(\boldsymbol{\mathcal{Y}}\). The detailed iterative procedure is outlined as follows:  
**Computing \(\boldsymbol{\mathcal{X}}\)**. Fix \(\left\{\boldsymbol{\mathcal{M}},\boldsymbol{\mathcal{E}},\boldsymbol{ \mathcal{G}},\boldsymbol{\mathcal{Y}}\right\}\) and minimize \(\mathrm{L}_{\boldsymbol{\mu}}(\boldsymbol{\mathcal{X}},\boldsymbol{\mathcal{M }},\boldsymbol{\mathcal{E}},\boldsymbol{\mathcal{G}},\boldsymbol{\mathcal{Y}})\) with respect to \(\boldsymbol{\mathcal{X}}\):  
\[ \begin{array}{ll}\boldsymbol{\mathcal{X}}:&=\arg\min\limits_{\boldsymbol{ \mathcal{X}}}\mathrm{L}_{\boldsymbol{\mu}}(\boldsymbol{\mathcal{X}}, \boldsymbol{\mathcal{M}},\boldsymbol{\mathcal{E}},\boldsymbol{\mathcal{G}}, \boldsymbol{\mathcal{Y}})\\ &=\arg\min\limits_{\boldsymbol{\mathcal{X}}}\sum\
olimits_{\mathrm{n}= \boldsymbol{1}}^{\mathrm{N}}\,|\,|\,\boldsymbol{\mathcal{X}}-\boldsymbol{ \mathcal{M}}_{\mathrm{n}}-\boldsymbol{\mathcal{G}}-\boldsymbol{\mathcal{E}}+ \boldsymbol{\mathcal{Y}}_{\mathrm{n}}/\boldsymbol{\mu}\,|\,_{\mathrm{F}}^{2}\\ &=\widetilde{\boldsymbol{\mathcal{X}}},\end{array}