# Tensor Rank Regularization with Bias Compensation for Millimeter Wave Channel Estimation
## 4 The Proposed Algorithm
### Proposed CP Tensor Decomposition with Weighted Bias
We will therefore restrict the range of \(\gamma\) to (0, 1). Sub-problems(29)-(31) can be considered modified Levenberg-Marquardt variational characterizations. Compared to \(l1\) LS solution (15) which is a Levenberg-Marquardt function, we know that \(\lambda\mathbf{I}(1-\gamma)\) will affect both the direction and step size of each iteration [35]. The scaled identity matrix \(\lambda\mathbf{I}(1-\gamma)\) in the modified Levenberg-Marquardt method in (36) also controls the direction of each iteration to favor low-rank solutions through the modified pseudoinverse calculation (\(\Big{(}\Big{(}\mathbf{C}^{(n)}\odot\mathbf{B}^{(n)}\Big{)}^{\mathrm{H}}\Big{(} \mathbf{C}^{(n)}\odot\mathbf{B}^{(n)}\Big{)}+\lambda\mathbf{I}(1-\gamma)\Big{)} ^{-1}\cdot\Big{(}\mathbf{C}^{(n)}\odot\mathbf{B}^{(n)}\Big{)}^{\mathrm{H}}\)) of (36). Compared to that of the \(l1\) LS solution (15), the added term \(1-\gamma\) will slightly change the direction of (15) and still favor low rank solutions because we restrict \(0<\gamma<1\). Moreover, compared to the \(l1\) LS solution (15) and RALS solution (21), the weighting parameter \(\gamma\) in (32) controls the bias compensation. The analyses to \(\mathbf{B}^{(n+1)}\) and \(\mathbf{C}^{(n+1)}\)T are similar. The estimated component matrices from the novel joint CP tensor decomposition-based method are associated with the weighting parameter. The novel joint CP tensor decomposition-based method is summarized in Algorithm 1. ```
0: Observation signal tensor \(\mathcal{Y}\in\mathbb{C}^{M_{\mathrm{m}}\times M_{\mathrm{m}}\times K}\), an initial selection \(\bar{k}\) for rank(\(\mathcal{Y}\)), and the regularization parameters \(\lambda\), weighting parameter \(\gamma\), threshold \(\beta\), number of Iterations iter = 0, Estimated Rank \(\bar{R}\), Estimated Component Matrices \(\hat{\mathbf{A}}\), \(\hat{\mathbf{B}}\), \(\hat{\mathbf{C}}\), iter
1. Derive unfolding matrices: \(\mathbf{Y}_{(1)}\), \(\mathbf{Y}_{(2)}\), \(\mathbf{Y}_{(3)}\)
2. Generate normally distributed pseudorandom initial matrices \(\mathbf{A}^{(0)}\in\mathbb{C}^{M_{\mathrm{m}}\times\bar{R}}\), \(\mathbf{B}^{(0)}\in\mathbb{C}^{M_{\mathrm{m}}\times\bar{R}}\), \(\mathbf{C}^{(0)}\in\mathbb{C}^{K\times\bar{R}}\)
3. The initialization of the cost function is \(cost^{0}\): \(\|\mathcal{Y}-\big{[}\mathbf{A}^{(0)},\mathbf{B}^{(0)},\mathbf{C}^{(0)}\| \big{]}_{\mathrm{F}}^{2}\)
4.  while\(\left|cost^{iter+1}-cost^{iter}\right|>\varepsilon\)do
5. \(iter\) = iter + 1
6. Calculate \(\hat{\mathbf{A}}^{\mathrm{T}}\leftarrow\Big{(}\Big{(}\mathbf{C}^{(n)}\odot \mathbf{B}^{(n)}\Big{)}^{\mathrm{H}}\Big{(}\mathbf{C}^{(n)}\odot\mathbf{B}^{ (n)}\Big{)}+\lambda\mathbf{I}\Big{)}^{-1}\)
7. Calculate \(\hat{\mathbf{B}}^{\mathrm{T}}\leftarrow\Big{(}\Big{(}\Big{(}\mathbf{C}^{(n)} \odot\mathbf{A}^{(n+1)}\Big{)}^{\mathrm{H}}\Big{(}\mathbf{C}^{(n)}\odot \mathbf{A}^{(n+1)}\Big{)}+\lambda\mathbf{I}\Big{)}\Big{)}^{-1}\)
8. Calculate \(\hat{\mathbf{C}}^{\mathrm{T}}\leftarrow\Big{(}\Big{(}\Big{(}\mathbf{B}^{(n+1 )}\odot\mathbf{A}^{(n+1)}\Big{)}^{\mathrm{H}}\Big{(}\mathbf{B}^{(n+1)}\odot \mathbf{A}^{(n+1)}\Big{)}+\lambda\mathbf{I}\Big{)}\Big{)}^{-1}\)
9. Recalculate cost in Equation (10)
10. End while
11. Calculate column power of \(\hat{\mathbf{C}}\in\mathbb{C}^{K\times\bar{R}}\)
12. Set the number of columns whose power \(>\beta\) as \(\bar{R}\) and construct the new \(\hat{\mathbf{C}}\) by using these columns
13. Based on the index number obtained from \(11\), we select the columns from \(\hat{\mathbf{A}}\) and \(\hat{\mathbf{B}}\) to construct new \(\hat{\mathbf{A}}\) and new \(\hat{\mathbf{B}}\)
14. Return \(\hat{\mathbf{A}}\in\mathbb{C}^{M_{\mathrm{m}}\times\bar{R}}\), \(\hat{\mathbf{B}}\in\mathbb{C}^{M_{\mathrm{m}}\times\bar{R}}\), \(\hat{\mathbf{C}}\in\mathbb{C}^{K\times\bar{R}}\)
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**Algorithm 1** The Proposed Joint CP Tensor Decomposition-Based Estimation Method.