# Tensor Rank Regularization with Bias Compensation for Millimeter Wave Channel Estimation
## 4 The Proposed Algorithm
### Proposed CP Tensor Decomposition with Weighted Bias  
We propose a novel regularized alternating least squares (RALS) with sparsity promoting and weighted bias compensation as follows,  
\[\underset{\mathbf{\hat{A}},\,\mathbf{B},\,\mathbf{C}}{\mathrm{min}}\left\| \mathcal{Y}-\sum_{r=1}^{R}\hat{\mathbf{a}}_{r}\circ\mathbf{\hat{b}}_{r}\circ \mathbf{\hat{c}}_{r}\right\|_{F}^{2}+\lambda\Big{(}\|\mathbf{\hat{A}}-\gamma \mathbf{A}^{(n)}\|_{\mathrm{F}}^{2}+\|\mathbf{\hat{B}}-\gamma\mathbf{B}^{(n)} \|_{\mathrm{F}}^{2}+\|\mathbf{\hat{C}}-\gamma\mathbf{C}^{(n)}\|_{\mathrm{F}}^ {2}\Big{)} \tag{28}\]  
where \(\gamma\) is the weighting parameter. The objective function (28) can be divided into three sub-problems:  
\[\mathbf{A}^{(n+1)} =\underset{\mathbf{\hat{A}}}{\mathrm{argmin}}\|\mathbf{Y}_{(1)} -\mathbf{\hat{A}}\Big{(}\mathbf{C}^{(n)}\odot\mathbf{B}^{(n)}\Big{)}^{ \mathrm{T}}\|_{\mathrm{F}}^{2}+\lambda\|\mathbf{\hat{A}}-\gamma\mathbf{A}^{(n) }\|_{\mathrm{F}}^{2} \tag{29}\] \[\mathbf{B}^{(n+1)} =\underset{\mathbf{\hat{B}}}{\mathrm{argmin}}\|\mathbf{Y}_{(2)} -\mathbf{\hat{B}}\Big{(}\mathbf{C}^{(n)}\odot\mathbf{A}^{(n+1)}\Big{)}^{ \mathrm{T}}\|_{\mathrm{F}}^{2}+\lambda\|\mathbf{\hat{B}}-\gamma\mathbf{B}^{(n) }\|_{\mathrm{F}}^{2}\] (30) \[\mathbf{C}^{(n+1)} =\underset{\mathbf{\hat{C}}}{\mathrm{argmin}}\|\mathbf{Y}_{(3)} -\mathbf{\hat{C}}\Big{(}\mathbf{B}^{(n+1)}\odot\mathbf{A}^{(n+1)}\Big{)}^{ \mathrm{T}}\|_{\mathrm{F}}^{2}+\lambda\|\mathbf{\hat{C}}-\gamma\mathbf{C}^{(n )}\|_{\mathrm{F}}^{2} \tag{31}\]  
The least square solutions of (29)-(31) as follows  
\[\mathbf{A}^{(n+1)^{\mathrm{T}}}=\bigg{(}\Big{(}\mathbf{C}^{(n)}\odot\mathbf{ B}^{(n)}\Big{)}^{\mathrm{H}}\Big{(}\mathbf{C}^{(n)}\odot\mathbf{B}^{(n)} \Big{)}+\lambda\mathbf{I}\bigg{)}^{-1}\bigg{(}\Big{(}\mathbf{C}^{(n)}\odot \mathbf{B}^{(n)}\Big{)}^{\mathrm{H}}\mathbf{Y}_{(1)}^{\mathrm{T}}+\lambda \gamma\mathbf{I}\Big{(}\mathbf{A}^{(n)}\Big{)}^{\mathrm{T}}\bigg{)} \tag{32}\] \[\mathbf{B}^{(n+1)^{\mathrm{T}}}=\bigg{(}\bigg{(}\Big{(}\mathbf{C} ^{(n)}\odot\mathbf{A}^{(n+1)}\Big{)}^{\mathrm{H}}\Big{(}\mathbf{C}^{(n)} \odot\mathbf{A}^{(n+1)}\Big{)}+\lambda\mathbf{I}\bigg{)}\bigg{)}^{-1}\bigg{(} \Big{(}\mathbf{C}^{(n)}\odot\mathbf{A}^{(n+1)}\Big{)}^{\mathrm{H}}\mathbf{Y}_ {(2)}^{\mathrm{T}}+\lambda\gamma\mathbf{I}\Big{(}\mathbf{B}^{(n)}\Big{)}^{ \mathrm{T}}\bigg{)} \tag{33}\]  
\[\mathbf{C}^{(n+1)^{\mathrm{T}}}=\bigg{(}\bigg{(}\Big{(}\mathbf{B}^{(n+1)}\odot \mathbf{A}^{(n+1)}\Big{)}^{\mathrm{H}}\Big{(}\mathbf{B}^{(n+1)}\odot\mathbf{A }^{(n+1)}\Big{)}+\lambda\mathbf{I}\bigg{)}\bigg{)}^{-1}\bigg{(}\Big{(}\mathbf{B} ^{(n+1)}\odot\mathbf{A}^{(n+1)}\Big{)}^{\mathrm{H}}\mathbf{Y}_{(3)}^{\mathrm{T} }+\lambda\gamma\mathbf{I}\Big{(}\mathbf{C}^{(n)}\Big{)}^{\mathrm{T}}\bigg{)} \tag{34}\]  
To explore the role of the weighting parameter \(\gamma\), we will consider the iteration of matrix \(\mathbf{A}\) from (32). First, (32) can be transformed to an iterative sequence as follows  
\[ \begin{split}\mathbf{A}^{(n+1)^{\mathrm{T}}}&=\mathbf{A }^{(n)^{\mathrm{T}}}-\bigg{(}\Big{(}\big{(}\mathbf{C}^{(n)}\odot\mathbf{B}^{(n) }\big{)}^{\mathrm{H}}\Big{(}\mathbf{C}^{(n)}\odot\mathbf{B}^{(n)}\Big{)}+ \lambda\mathbf{I}\big{)}\bigg{)}^{-1}\\ &\quad\quad\cdot\bigg{(}\bigg{(}\Big{(}\mathbf{C}^{(n)}\odot \mathbf{B}^{(n)}\Big{)}^{\mathrm{H}}\Big{(}\mathbf{C}^{(n)}\odot\mathbf{B}^{(n )}\Big{)}+\lambda\mathbf{I}(1-\gamma)\bigg{)}\mathbf{A}^{(n)^{\mathrm{T}}}- \big{(}\mathbf{C}^{(n)}\odot\mathbf{B}^{(n)}\big{)}^{\mathrm{H}}\mathbf{Y}_{(1)} ^{\mathrm{T}}\bigg{)}\end{split}  \tag{35}\]  
When the solution in (35) approaches convergence: \(\mathbf{A}^{(n+1)}\rightarrow\ \mathbf{A}^{(n)}\), we have  
\[\mathbf{A}^{(n+1)^{\mathrm{T}}}=\bigg{(}\Big{(}\mathbf{C}^{(n)}\odot\mathbf{B}^{(n )}\Big{)}^{\mathrm{H}}\Big{(}\mathbf{C}^{(n)}\odot\mathbf{B}^{(n)}\Big{)}+ \lambda\mathbf{I}(1-\gamma)\bigg{)}^{-1}\Big{(}\mathbf{C}^{(n)}\odot\mathbf{B} ^{(n)}\Big{)}^{\mathrm{H}}\mathbf{Y}_{(1)}^{\mathrm{T}} \tag{36}\]  
According to [34], (36) is a modified Levenberg-Marquardt function with variable decay rate when \(0<\gamma<1\).