# Tensor Rank Regularization with Bias Compensation for Millimeter Wave Channel Estimation
## 1 Introduction
The Vandermonde structure of the component matrix of a third-order low-rank tensor model was proposed in [15], which developed a non-iterative structured CP decomposition-aided channel estimation algorithm and showed the advantages of avoiding random initialization and iterations. The authors in [15] also proposed a tensor rank estimation method by applying singular value decomposition (SVD) to a tensor unfolding matrix, which is suitable in a noiseless channel. A CP decomposition-based method for mmWave channel parameter estimation with frequency selectivity in a hybrid orthogonal frequency division multiplexing (OFDM)-MIMO scenario was proposed in [16], which showed advantages over a compressed sensing (CS)-based method. A novel tensor-decomposition channel estimation method for IRS-assisted mmWave OFDM systems was proposed in [17]. By exploiting the inherent sparse structure of the cascade channel, the authors formulate the received signal as a low-rank third-order tensor. Only a very small amount of training overhead is used to obtain a reliable estimate of the cascade channel because of the low rank structure. A sparse Bayes tensor and DOA tracking inspired channel estimation for V2X millimeter wave massive MIMO system was proposed in [18]. This paper dealt with accurate channel estimation in highly mobile scenarios. The sparse Bayes tensor is used to calculate the angle offset. Then a direction of arrival (DOA) tracking method is developed to acquire the angle value at the next moment by exploiting the sparse Bayes tensor. These methods rely on knowing, or estimating, the rank of the tensors. Tensor rank is analogous to matrix rank, but the properties of matrix and tensor ranks are quite different. It is not clear how to define a nuclear norm surrogate for tensors because singular values defined by the Tucker decomposition are not generally related to the rank of a tensor. Usually, finding the tensor rank is NP-hard [19], and even very simple tensors stubbornly resist rank determination, such as for the \(3^{2}\times 3^{2}\times 3^{2}\) tensor whose rank is only known to be between 19 and 23 [20]. While the tensor rank is difficult to compute precisely, it is not known whether the rank can be approximated. A regularization term of the CP decomposition factors capturing the tensor's rank was proposed in [21]. The proposed regularization term relies on an alternative characterization of the nuclear norm based on a low-rank factorization of its matrix argument. A method to simultaneously perform CP decomposition and tensor rank approximation was proposed in [22], which divided a third-order low rank CP tensor into 4 blocks, i.e., 3 component matrices and 1 weighting vector. In this paper, we propose a novel CP decomposition based method to jointly estimate the tensor rank and channel component matrices for the hybrid MIMO architecture. We construct the received signal into a third-order tensor and exploit the low rank characteristics of mmWave channel. The main contributions of the paper are summarized as follows:  
* 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. * Second, we analyze the effectiveness of the weighting parameter in our proposed new prior. Based on the numerical results, we show that the proposed scheme outperforms the conventional strategies in [16, 21] in terms of estimation accuracy and robustness. * Third, we discuss the trade-off between convergence and rank estimation accuracy for our proposed rank regularization method.