# Tensor Rank Regularization with Bias Compensation for Millimeter Wave Channel Estimation

^{*}

## Abstract

**:**

## 1. Introduction

- First, we propose a novel CP decomposition-based method to jointly estimate both the tensor rank and component matrices of the received signal tensor. We formulate the received signals into a third-order tensor in the form of the CP structure of a hybrid MIMO-OFDM system. Unlike the conventional tensor signal analysis assumed a-priori knowledge of the rank, we focus on determining the tensor rank which is often unknown in practice. We also develop a novel sparsity-promoting prior to determine tensor rank, and then estimate channel information from low rank component matrix representations.
- Third, we discuss the trade-off between convergence and rank estimation accuracy for our proposed rank regularization method. Through numerical experiments, we find that our method significantly improves rank estimation success at the expense of slightly more iterations.

**y**is a vector,

**Y**is a matrix, and $\mathcal{Y}$ is a tensor. ${Y}^{\mathrm{T}}$, ${Y}^{*}$, ${Y}^{\mathrm{H}}$, and ${Y}^{\u2020}$ are transpose, conjugate, conjugate transpose, and Moore-Penrose pseudoinverse, respectively. ${\left[Y\right]}_{i,:}$ and ${\left[Y\right]}_{:,j}$ are the i-th row and the j-th column of the matrix

**Y**.

**A**⊗

**B**,

**A**⊛

**B**, and

**A**⊙

**B**denote the Kronecker product, the Hadamard product, and the column-wise Khatri-Rao product.

**a**∘

**b**denotes the outer product, which is also known as the tensor product. Let Rank ($Y$) and ${k}_{Y}$ denote the rank and Kruskal-rank of a matrix $Y$, respectively. Let Re(

**Y**) and Im(

**Y**) denote the real part and the imaginary part of

**Y**.

**d**(

**Y**) denotes a vector of diagonal entries of

**Y,**and

**D**(

**y**) denotes a diagonal matrix constructed from

**y**.

## 2. Tensor Preliminaries

#### 2.1. Tensor Basics

#### 2.2. CP Tensor Decomposition

## 3. Signal Model

_{r}and the DoA—φ

_{r}. ${a}_{MS}\left({\phi}_{r}\right)={\left[1,{e}^{j\pi \mathrm{sin}\left({\phi}_{r}\right)},\cdots ,{e}^{j\pi \left({N}_{ms}-1\right)\mathrm{sin}\left({\phi}_{r}\right)}\right]}^{T}$ and ${a}_{BS}\left({\theta}_{r}\right)={\left[1,{e}^{j\pi \mathrm{sin}\left({\theta}_{r}\right)},\cdots ,{e}^{j\pi \left({N}_{bs}-1\right)\mathrm{sin}\left({\theta}_{r}\right)}\right]}^{T}$ are the steering vectors of the MS and BS, respectively.

## 4. The Proposed Algorithm

#### 4.1. Joint CP Tensor Decomposition

#### 4.2. Proposed CP Tensor Decomposition with Weighted Bias

**A**from (32). First, (32) can be transformed to an iterative sequence as follows

Algorithm 1. The Proposed Joint CP Tensor Decomposition-Based Estimation Method. |

Input$:\mathrm{Observation}\mathrm{signal}\mathrm{tensor}\mathcal{Y}\in {\u2102}^{{M}_{ms}\times {M}_{bs}\times K}$, an initial selection $\tilde{R}$ for rank($\mathcal{Y}$),and the regularization parameters $\lambda $, weighting parameter $\gamma $, threshold $\beta $, number of Iterations iter = 0, Output: Estimated Rank $\widehat{R}$, Estimated Component Matrices
$\widehat{A}$,
$\widehat{B}$, $\widehat{C}$, iter1. Derive unfolding matrices: ${Y}_{\left(1\right)}$, ${Y}_{\left(2\right)}$, ${Y}_{\left(3\right)}$ 2. Generate normally distributed pseudorandom initial matrices ${A}^{\left(0\right)}\in {\u2102}^{{M}_{ms}\times \tilde{R}}$, ${B}^{\left(0\right)}\in {\u2102}^{{M}_{bs}\times \tilde{R}}$, ${C}^{\left(0\right)}\in {\u2102}^{K\times \tilde{R}}$ 3. The initialization of the cost function is $cos{t}^{0}$: $\Vert \mathcal{Y}-\u27e6{A}^{\left(0\right)},{B}^{\left(0\right)},{C}^{\left(0\right)}\u27e7{\Vert}_{\mathrm{F}}^{2}$ 4. while $\left|cos{t}^{iter+1}-cos{t}^{iter}\right|>\epsilon $ do5. iter = iter + 1 6. Calculate ${\widehat{A}}^{\mathrm{T}}\leftarrow {\left({\left({C}^{\left(\mathrm{n}\right)}\odot {B}^{\left(\mathrm{n}\right)}\right)}^{\mathrm{H}}\left({C}^{\left(\mathrm{n}\right)}\odot {B}^{\left(\mathrm{n}\right)}\right)+\lambda I\right)}^{-1}$ $\xb7\left({\left({C}^{\left(\mathrm{n}\right)}\odot {B}^{\left(\mathrm{n}\right)}\right)}^{\mathrm{H}}{Y}_{\left(1\right)}^{T}+\lambda \gamma I{\left({A}^{\left(n\right)}\right)}^{\mathrm{T}}\right)$ 7. Calculate ${\widehat{B}}^{\mathrm{T}}\leftarrow {\left(\left({\left({C}^{\left(\mathrm{n}\right)}\odot {A}^{\left(\mathrm{n}+1\right)}\right)}^{\mathrm{H}}\left({C}^{\left(\mathrm{n}\right)}\odot {A}^{\left(\mathrm{n}+1\right)}\right)+\lambda I\right)\right)}^{-1}$ $\xb7\left({\left({C}^{\left(\mathrm{n}\right)}\odot {A}^{\left(\mathrm{n}+1\right)}\right)}^{\mathrm{H}}{Y}_{\left(2\right)}^{\mathrm{T}}+\lambda \gamma I{\left({B}^{\left(\mathrm{n}\right)}\right)}^{\mathrm{T}}\right)$ 8. Calculate ${\widehat{C}}^{\mathrm{T}}\leftarrow {\left(\left({\left({B}^{\left(\mathrm{n}+1\right)}\odot {A}^{\left(\mathrm{n}+1\right)}\right)}^{\mathrm{H}}\left({B}^{\left(\mathrm{n}+1\right)}\odot {A}^{\left(\mathrm{n}+1\right)}\right)+\lambda I\right)\right)}^{-1}$ $\xb7\left({\left({B}^{\left(\mathrm{n}+1\right)}\odot {A}^{\left(\mathrm{n}+1\right)}\right)}^{\mathrm{H}}{Y}_{\left(3\right)}^{\mathrm{T}}+\lambda \gamma I{\left({C}^{\left(\mathrm{n}\right)}\right)}^{\mathrm{T}}\right)$ 9. Recalculate cost in Equation (10) 10. End while11. Calculate column power of $\widehat{C}\in {\u2102}^{K\times \tilde{R}}$ 12. Set the number of columns whose power > $\beta $ as $\widehat{R}$ and construct the new $\widehat{C}$ by using these columns 13. Based on the index number obtained from 11, we select the columns from $\widehat{A}$ and $\widehat{B}$ to construct new $\widehat{A}$ and new $\widehat{B}$ 14. Return $\widehat{A}\in {\u2102}^{{M}_{ms}\times \widehat{R}}$, $\widehat{B}\in {\u2102}^{{M}_{bs}\times \widehat{R}}$, $\widehat{C}\in {\u2102}^{K\times \widehat{R}}$ |

## 5. Computational Complexity Analysis

- (1)
- To calculate ${\left({\left({C}^{\left(\mathrm{n}\right)}\odot {B}^{\left(\mathrm{n}\right)}\right)}^{\mathrm{H}}\left({C}^{\left(\mathrm{n}\right)}\odot {B}^{\left(\mathrm{n}\right)}\right)+\lambda I\right)}^{-1}$, the complexity is $\mathcal{O}\left({M}_{bs}K{\tilde{R}}^{2}+{\tilde{R}}^{3}\right)$,
- (2)
- To calculate $\left({\left({C}^{\left(\mathrm{n}\right)}\odot {B}^{\left(\mathrm{n}\right)}\right)}^{\mathrm{H}}{Y}_{\left(1\right)}^{T}+\lambda \gamma I{A}^{(n{)}^{\mathrm{T}}}\right)$, the complexity is $\mathcal{O}\left({M}_{ms}{M}_{bs}K\tilde{R}\right)$,
- (3)
- To calculate ${\left({\left({C}^{\left(\mathrm{n}\right)}\odot {B}^{\left(\mathrm{n}\right)}\right)}^{\mathrm{H}}\left({C}^{\left(\mathrm{n}\right)}\odot {B}^{\left(\mathrm{n}\right)}\right)+\lambda I\right)}^{-1}\left({\left({C}^{\left(\mathrm{n}\right)}\odot {B}^{\left(\mathrm{n}\right)}\right)}^{\mathrm{H}}{Y}_{\left(1\right)}^{\mathrm{T}}+\lambda \gamma I{A}^{(n{)}^{\mathrm{T}}}\right)$, the complexity is $\mathcal{O}\left({M}_{ms}{\tilde{R}}^{2}\right)$.

## 6. Numerical Experiments

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Rank estimation performance of the l1 Regularization LS method as a function of $\lambda $ with real rank $R$ = 4.

**Figure 3.**Comparison of the rank estimation performance of the proposed method as a function of $\gamma $ and the l1 regularization LS versus SNR with $\lambda $ =${10}^{-4}$ and real rank $R$ = 4.

**Figure 4.**Robustness of the proposed method regrading different initial rank values with $R$ = 4, SNR = 20 dB, $\lambda $ =${10}^{-4}$ and $\gamma $ = 0.3.

**Figure 5.**Robustness of the proposed method in success rate regrading different values of initial rank with $R$ = 4, SNR = 20 dB, $\lambda $ =${10}^{-4}$ and $\gamma $ = 0.3.

**Figure 6.**Success rate regrading different values of $\gamma $ with $R$ = 4, SNR = 20 dB, $\lambda $ =${10}^{-4}$.

**Figure 7.**Relative error vs. Iteration using the proposed method for different values of $\gamma $ with $R$ = 4, SNR = 20 dB, $\lambda $ =${10}^{-4}$. (

**a**) $\gamma =0$, (

**b**) $\gamma =0.3$, (

**c**) $\gamma =0.5$, (

**d**) $\gamma =0.7$.

**Figure 8.**NMSEs of channel estimation schemes versus the system SNR and values of $\gamma $ with $R$ = 4, $\lambda $ =${10}^{-4}$.

**Figure 9.**Rank estimation performance of the proposed method as a function of weighting parameter $\gamma $ with real rank $R$ = 4, 5, and 6, SNR = 20 dB, $\lambda $ =${10}^{-4}$.

**Figure 10.**Number of iterations for each trail of the proposed method as a function of weighting parameter $\gamma $ with real rank $R$ = 4, 5, and 6, SNR = 20 dB, $\lambda $ =${10}^{-4}$.

Initial Rank | 8 | 16 | 24 | |
---|---|---|---|---|

Real Rank | ||||

$R=$ 4 | 97.4% | 97.96% | 98.36% | |

$R=$ 5 | 98.4% | 98.52% | 98.6% | |

$R=$ 6 | 99.36% | 99.47% | 99.48% |

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## Share and Cite

**MDPI and ACS Style**

He, F.; Harms, A.; Yang, L.Y.
Tensor Rank Regularization with Bias Compensation for Millimeter Wave Channel Estimation. *Signals* **2022**, *3*, 664-681.
https://doi.org/10.3390/signals3040040

**AMA Style**

He F, Harms A, Yang LY.
Tensor Rank Regularization with Bias Compensation for Millimeter Wave Channel Estimation. *Signals*. 2022; 3(4):664-681.
https://doi.org/10.3390/signals3040040

**Chicago/Turabian Style**

He, Fei, Andrew Harms, and Lamar Yaoqing Yang.
2022. "Tensor Rank Regularization with Bias Compensation for Millimeter Wave Channel Estimation" *Signals* 3, no. 4: 664-681.
https://doi.org/10.3390/signals3040040