# Tensor Rank Regularization with Bias Compensation for Millimeter Wave Channel Estimation
## 4 The Proposed Algorithm  
### Joint CP Tensor Decomposition  
If the number of active paths, or rank \(R\), is known, the CP decomposition of \(\mathcal{Y}\) can be accomplished by a standard alternating least squares (ALS) method [16]. It is likely that the rank \(R\) is unknown in practice. In this case, the CP decomposition of \(\mathcal{Y}\) can be accomplished by minimizing the cost function  
\[\min_{\hat{\mathbf{A}},\ \hat{\mathbf{B}},\ \hat{\mathbf{C}}}\left\|\mathcal{Y}- \sum_{r=1}^{\hat{\mathbf{A}}}\hat{\mathbf{r}}\circ\hat{\mathbf{b}}_{r}\circ \hat{\mathbf{c}}_{r}\right\|_{F}^{2} \tag{10}\]  
where \(\hat{R}\) is an estimation of the true rank \(R\). Instead of estimating tensor rank individually, we seek a more comprehensive CP decomposition-based technique to estimate component matrices and rank jointly. A method of combining CP decomposition and sparsity-promoting prior is presented as an \(l1\) regularization LS method [21] by solving  
\[\min_{\hat{\mathbf{A}},\ \hat{\mathbf{B}},\ \hat{\mathbf{C}}} \|\mathcal{Y}-\mathcal{X}\|_{F}^{2}+\lambda\Big{(}\|\hat{ \mathbf{A}}\|_{F}^{2}+\|\hat{\mathbf{B}}\|_{F}^{2}+\|\hat{\mathbf{C}}\|_{F}^{2 }\Big{)}\] (11) s.t. \[\ \ \mathcal{X}=\sum_{r=1}^{\hat{R}}\hat{\mathbf{a}}_{r}\circ\hat{ \mathbf{b}}_{r}\circ\hat{\mathbf{c}}_{r}\]  
where \(\lambda\) is a tuning parameter for the rank regularization term. The regularizer relies on an alternative characterization of the nuclear norm based on a low-rank factorization of its matrix argument. The optimization problem (11) can also be solved by ALS. However, because the rank \(R\) is unknown, an overestimated CP rank, \(\bar{R}\), must be chosen initially. The objective function (11) can be divided into three sub-problems:  
\[\mathbf{A}^{(n+1)}=\underset{\hat{\mathbf{A}}}{\mathrm{argmin}}\|\mathbf{Y}_{( 1)}-\hat{\mathbf{A}}\Big{(}\mathbf{C}^{(n)}\odot\mathbf{B}^{(n)}\Big{)}^{ \mathrm{T}}\|_{\mathrm{F}}^{2}+\lambda\|\hat{\mathbf{A}}\|_{\mathbf{F}}^{2} \tag{12}\]  
\[\mathbf{B}^{(n+1)}=\underset{\hat{\mathbf{B}}}{\mathrm{argmin}}\|\mathbf{Y}_{( 2)}-\hat{\mathbf{B}}\Big{(}\mathbf{C}^{(n)}\odot\mathbf{A}^{(n+1)}\Big{)}^{ \mathrm{T}}\|_{\mathrm{F}}^{2}+\lambda\|\hat{\mathbf{B}}\|_{\mathbf{F}}^{2} \tag{13}\]  
\[\mathbf{C}^{(n+1)}=\underset{\hat{\mathbf{C}}}{\mathrm{argmin}}\|\mathbf{Y}_{( 3)}-\hat{\mathbf{C}}\Big{(}\mathbf{B}^{(n+1)}\odot\mathbf{A}^{(n+1)}\Big{)}^{ \mathrm{T}}\|_{\mathrm{F}}^{2}+\lambda\|\hat{\mathbf{C}}\|_{\mathbf{F}}^{2} \tag{14}\]  
Sub-problems (12)-(14) do not contain sparsity-promoting prior. However, the entire ALS procedure promotes sparse results because the restraint in (11) is incorporated in the three sub-problems. \(\hat{R}\) can be estimated by removing all negligible rank-one tensor components after convergence. The least squares solutions are derived as follows:  
\[\mathbf{A}^{(n+1)^{\mathrm{T}}}=\bigg{(}\bigg{(}\Big{(}\mathbf{C}^{(n)}\odot \mathbf{B}^{(n)}\Big{)}^{\mathrm{H}}\Big{(}\mathbf{C}^{(n)}\odot\mathbf{B}^{ (n)}\Big{)}+\lambda\mathbf{I}\bigg{)}\bigg{)}^{-1}\bigg{(}\Big{(}\mathbf{C}^{ (n)}\odot\mathbf{B}^{(n)}\Big{)}^{\mathrm{H}}\mathbf{Y}_{(1)}^{\mathrm{T}}\bigg{)} \tag{15}\]  
\[\mathbf{B}^{(n+1)^{\mathrm{T}}}=\bigg{(}\bigg{(}\Big{(}\mathbf{C}^{(n)}\odot \mathbf{A}^{(n+1)}\Big{)}^{\mathrm{H}}\Big{(}\mathbf{C}^{(n)}\odot\mathbf{A}^ {(n+1)}\Big{)}+\lambda\mathbf{I}\bigg{)}\bigg{)}^{-1}\big{(}\big{(}\mathbf{C}^{ (n)}\odot\mathbf{A}^{(n+1)}\big{)}^{\mathrm{H}}\mathbf{Y}_{(2)}^{\mathrm{T}} \tag{16}\]  
\[\mathbf{C}^{(n+1)^{\mathrm{T}}}=\bigg{(}\Big{(}\big{(}\mathbf{B}^{(n+1)}\odot \mathbf{A}^{(n+1)}\big{)}^{\mathrm{H}}\Big{(}\mathbf{B}^{(n+1)}\odot\mathbf{A} ^{(n+1)}\Big{)}+\lambda\mathbf{I}\Big{)}\bigg{)}^{-1}\Big{(}\big{(}\mathbf{B}^ {(n+1)}\odot\mathbf{A}^{(n+1)}\big{)}^{\mathrm{H}}\mathbf{Y}_{(3)}^{\mathrm{T}}\bigg{)} \tag{17}\]  
According to [27], (15)-(17) are also special cases of the iteratively regularized Gauss-Newton method with a linear bounded operator and initial guess set as 0. Equations (15)-(17) have corresponding iterative sequences.