# Tensor Rank Regularization with Bias Compensation for Millimeter Wave Channel Estimation
## 1 Introduction  
Millimeter wave (mmWave) transmission technology is an attractive candidate technology for next-generation wireless communication systems [1]. The main benefit of utilizing mmWave carrier frequencies is the larger spectral bandwidth available for higher data rates facilitating the ever-increasing communication traffic [2]. To compensate for the significant path-loss at such high frequencies, large antenna arrays are common at both the base station (BS) and the mobile station (MS) to provide sufficient beamforming gain [3]. Compared to conventional multiple-input multiple-output (MIMO) systems, the large antenna arrays in mmWave systems make it unlikely to have a dedicated radio frequency (RF) chain for each antenna without costly hardware [3]. A hybrid analog/digital structure uses fewer RF chains to reduce the number of power-consuming devices, such as analog-to-digital converters (ADCs) or digital-to-analog converters (DACs) [3]. To achieve considerable beamforming gain in the precoding stage, accurate channel state information (CSI) is required through channel estimation. Channel estimation in mmWave hybrid systems is challenging due to the hybrid precoding structure and the large number of antennas. The mmWave channel can be modeled in parametric form based on the path angles, i.e., directions of arrival and departure (DoA/DoD), and the corresponding complex path gains. As a result, mmWave channel estimation is a problem of estimating the path angles and gains rather than estimating the conventional MIMO channel matrix [4, 5]. An attractive approach to directly estimate themmWave channel parameters is to use compressive sensing (CS) based methods [6, 7, 8, 9, 10, 11, 12]. Due to the limited propagation range of mmWave channels, the parametric form of the channel description generally has few active paths. Therefore, these CS-based methods leverage the sparsity of the channel to cast estimation as a sparse recovery problem. The virtual angular representation [6] or angle grids [7] are used to describe the path angles. There is a gain coefficient for each angle grid point that represents a pair of DoA/DoD. Because of the spatial sparsity of the mmWave channel, most of the gain coefficients are negligible. The sparse recovery approach intends to recover the significant gain coefficients providing estimates of the path directions and reducing training overhead. A compressive architecture for estimating and tracking sparse spatial channels in mmWave picocellular networks with large arrays was proposed and investigated in [8]. In [9], it considered multi-user massive MIMO systems and deployed CS-based techniques to reduce the training and feedback overhead in the channel state information at the transmitter estimation. In [10, 11], the authors showed that significant reductions in training overhead can be achieved via CS-based methods in mmWave MIMO communication systems. [12] developed a system that uses CS-based estimation on the uplink to configure precoders and combiners for the downlink in a hybrid mmWave MIMO system. In addition to CS-based methods, tensor-based methods for mmWave MIMO communication have been studied in [13, 14, 15, 16, 17, 18], which formulate a third order tensor with multiple dimensions containing information about the channel parameters. A two-stage tensor decomposition-based method for mmWave time-varying channel estimation was proposed in [13]. The CP decomposition aided method is used to estimate the DOAs/DODs in the first stage. The estimated angles are then used to estimate the path gains and doppler shifts. A spatial channel covariance estimation method, proposed in [14], is based on higher-order tensor decomposition for the hybrid single-input multiple-output (SIMO) architecture over uplink time-varying frequency-selective channels.