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Deep-Learning-Aided Alternating Least Squares for Tensor CP Decomposition and Its Application to Massive MIMO Channel Estimation

Xiao Gong, Wei Chen, Bo Ai, Geert Leus

TL;DR

A deep-learning-aided CPALS (DL-CPALS) method that uses a deep neural network (DNN) to generate favorable initializations that facilitates fast and accurate CPD and channel estimation is proposed.

Abstract

CANDECOMP/PARAFAC (CP) decomposition is the mostly used model to formulate the received tensor signal in a massive MIMO system, as the receiver generally sums the components from different paths or users. To achieve accurate and low-latency channel estimation, good and fast CP decomposition (CPD) algorithms are desired. The CP alternating least squares (CPALS) is the workhorse algorithm for calculating the CPD. However, its performance depends on the initializations, and good starting values can lead to more efficient solutions. Existing initialization strategies are decoupled from the CPALS and are not necessarily favorable for solving the CPD. This paper proposes a deep-learning-aided CPALS (DL-CPALS) method that uses a deep neural network (DNN) to generate favorable initializations. The proposed DL-CPALS integrates the DNN and CPALS to a model-based deep learning paradigm, where it trains the DNN to generate an initialization that facilitates fast and accurate CPD. Moreover, benefiting from the CP low-rankness, the proposed method is trained using noisy data and does not require paired clean data. The proposed DL-CPALS is applied to millimeter wave MIMO-OFDM channel estimation. Experimental results demonstrate the significant improvements of the proposed method in terms of both speed and accuracy for CPD and channel estimation.

Deep-Learning-Aided Alternating Least Squares for Tensor CP Decomposition and Its Application to Massive MIMO Channel Estimation

TL;DR

A deep-learning-aided CPALS (DL-CPALS) method that uses a deep neural network (DNN) to generate favorable initializations that facilitates fast and accurate CPD and channel estimation is proposed.

Abstract

CANDECOMP/PARAFAC (CP) decomposition is the mostly used model to formulate the received tensor signal in a massive MIMO system, as the receiver generally sums the components from different paths or users. To achieve accurate and low-latency channel estimation, good and fast CP decomposition (CPD) algorithms are desired. The CP alternating least squares (CPALS) is the workhorse algorithm for calculating the CPD. However, its performance depends on the initializations, and good starting values can lead to more efficient solutions. Existing initialization strategies are decoupled from the CPALS and are not necessarily favorable for solving the CPD. This paper proposes a deep-learning-aided CPALS (DL-CPALS) method that uses a deep neural network (DNN) to generate favorable initializations. The proposed DL-CPALS integrates the DNN and CPALS to a model-based deep learning paradigm, where it trains the DNN to generate an initialization that facilitates fast and accurate CPD. Moreover, benefiting from the CP low-rankness, the proposed method is trained using noisy data and does not require paired clean data. The proposed DL-CPALS is applied to millimeter wave MIMO-OFDM channel estimation. Experimental results demonstrate the significant improvements of the proposed method in terms of both speed and accuracy for CPD and channel estimation.
Paper Structure (13 sections, 1 theorem, 25 equations, 8 figures, 3 tables, 2 algorithms)

This paper contains 13 sections, 1 theorem, 25 equations, 8 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Notations
  3. Preliminaries
  4. Deep-learning-aided CP decomposition
  5. Motivations
  6. The proposed deep-learning-aided CPALS (DL-CPALS)
  7. DNN designs
  8. Application to massive MIMO-OFDM Tensor Channel Estimation
  9. Simulations
  10. Results on synthetic data
  11. Results of MIMO-OFDM channel estimation
  12. Complexity analysis
  13. Conclusion

Key Result

Theorem 1

Let an $N$th-order tensor $\boldsymbol{\mathcal{X}}$ satisfy (cp1). Then, the CP decomposition is unique if

Figures (8)

  • Figure 1: Iterative behaviour of ALS using different starting points of $\hat{\mathbf{z}}^0_3=[z_{3,1}, z_{3,2}]^T$ for computing a random tensor $\boldsymbol{\mathcal{Y}}$. (a) The reconstruction NSE of $\frac{\left\|\hat{\boldsymbol{\mathcal{Y}}}-\boldsymbol{\mathcal{Y}}\right\|^2_F}{\left\|\boldsymbol{\mathcal{Y}}\right\|^2_F}$ in dB with $K=50$ using different starting points, where $\hat{\boldsymbol{\mathcal{Y}}}$ is the reconstructed tensor. (b) The reconstruction NSE as a function of iteration number.
  • Figure 2: An illustration of the model forward propagation on a single tensor.
  • Figure 3: An illustration of the pilot pattern.
  • Figure 4: The training loss of DL-CPALS.
  • Figure 5: The performance comparison of CPALS using different initializations on synthetic data. (a) Iterative behavior of the objective value. (b) Iterative behavior of the ANSE. (c) Curves of the cumulative distribution function (CDF) of the ANSE with $K=50$.
  • ...and 3 more figures

Theorems & Definitions (1)

  • Theorem 1: Uniqueness conditionCPuniqueness