# Tensor Rank Regularization with Bias Compensation for Millimeter Wave Channel Estimation
## 4 The Proposed Algorithm
### Joint CP Tensor Decomposition
To compensate the regularization term \(\lambda\mathbf{I}\mathbf{A}^{(n+1)^{\mathrm{T}}}\) we can add to the right-hand side the similar term \(\lambda\mathbf{I}\mathbf{A}^{(n)^{\mathrm{T}}}\) which gradually removes the bias as \(\mathbf{A}^{(n+1)}\rightarrow\ \hat{\mathbf{A}}\), where \(\hat{\mathbf{A}}\) is the estimate of \(\mathbf{A}\)  
\[\min_{\hat{\mathbf{A}},\ \hat{\mathbf{B}},\ \hat{\mathbf{C}}} \|\mathcal{Y}-\mathcal{X}\|_{F}^{2}+\lambda\Big{(}\|\hat{ \mathbf{A}}\|_{F}^{2}+\|\hat{\mathbf{B}}\|_{F}^{2}+\|\hat{\mathbf{C}}\|_{F}^{2 }\Big{)}\]1  
After some manipulation we get  
\[\min_{\hat{\mathbf{A}},\ \hat{\mathbf{B}},\ \hat{\mathbf{C}}} \|\mathcal{Y}-\mathcal{X}\|_{F}^{2}+\lambda\Big{(}\|\hat{ \mathbf{A}}\|_{F}^{2}+\|\hat{\mathbf{B}}\|_{F}^{2}+\|\hat{\mathbf{C}}\|_{F}^{2 }\Big{)}\]2  
Likewise, we can compensate the estimation bias of \(\hat{\mathbf{B}}^{\mathrm{T}}\) and \(\hat{\mathbf{C}}^{\mathrm{T}}\) by using  
\[\min_{\hat{\mathbf{A}},\ \hat{\mathbf{B}},\ \hat{\mathbf{C}}} \|\mathcal{Y}-\mathcal{X}\|_{F}^{2}+\lambda\Big{(}\|\hat{ \mathbf{A}}\|_{F}^{2}+\|\hat{\mathbf{B}}\|_{F}^{2}+\|\hat{\mathbf{C}}\|_{F}^{2 }\Big{)}\]3  
\[\min_{\hat{\mathbf{A}},\ \hat{\mathbf{B}},\ \hat{\mathbf{C}}} \|\mathcal{Y}-\mathcal{X}\|_{F}^{2}+\lambda\Big{(}\|\hat{ \mathbf{A}}\|_{F}^{2}+\|\hat{\mathbf{B}}\|_{F}^{2}+\|\hat{\mathbf{C}}\|_{F}^{2 }\Big{)}\]4  
According to [30,31], (21)-(23) are least square solutions of the regularized alternating least squares (RALS) objective function  
\[\min_{\hat{\mathbf{A}},\ \hat{\mathbf{B}},\ \hat{\mathbf{C}}} \|\mathcal{Y}-\mathcal{X}\|_{F}^{2}+\lambda\Big{(}\|\hat{ \mathbf{A}}\|_{F}^{2}+\|\hat{\mathbf{B}}\|_{F}^{2}+\|\hat{\mathbf{C}}\|_{F}^{2 }\Big{)}\]5  
The second term in (24) penalizes the difference between the current values and previous iterates. However, the limit points of (24) are the critical points of a standard ALS-the first term in (24) without the regularization term. Equation (24) can be divided into three sub-problems:  
\[\min_{\hat{\mathbf{A}},\ \hat{\mathbf{B}},\ \hat{\mathbf{C}}} \|\mathcal{Y}-\mathcal{X}\|_{F}^{2}+\lambda\Big{(}\|\hat{ \mathbf{A}}\|_{F}^{2}+\|\hat{\mathbf{B}}\|_{F}^{2}+\|\hat{\mathbf{C}}\|_{F}^{2 }\Big{)}\]6  
\[\min_{\hat{\mathbf{A}},\ \hat{\mathbf{B}},\ \hat{\mathbf{C}}} \|\mathcal{Y}-\mathcal{X}\|_{F}^{2}+\lambda\Big{(}\|\hat{ \mathbf{A}}\|_{F}^{2}+\|\hat{\mathbf{B}}\|_{F}^{2}+\|\hat{\mathbf{C}}\|_{F}^{2 }\Big{)}\]7\[\min_{\hat{\mathbf{A}},\ \hat{\mathbf{B}},\ \hat{\mathbf{C}}} \|\mathcal{Y}-\mathcal{X}\|_{F}^{2}+\lambda\Big{(}\|\hat{ \mathbf{A}}\|_{F}^{2}+\|\hat{\mathbf{B}}\|_{F}^{2}+\|\hat{\mathbf{C}}\|_{F}^{2 }\Big{)}\]8  
The iteration approaches (25)-(27) work well for CP tensor decomposition with faster convergence if the rank \(R\) is known. If the rank \(R\) is unknown, objective function (24) is incapable of obtaining tensor rank because its solutions are not critical points of the regularized version. According to [32, 33], the sub-problems (25)-(27) are variational characterizations of the Levenberg-Marquardt method with linear bounded operator. We need to seek a compromise between estimation bias and sparsity promotion.