Learning a Measurement Matrix in Compressed CSI Feedback for Millimeter Wave Massive MIMO Pengxia Wu * , Zichuan Liu † and Julian Cheng * * School of Engineering, The University of British Columbia, Kelowna, BC, Canada † School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, 639798 Singapore. Email: pengxia.wu@ubc.ca, zliu016@e.ntu.edu.sg, julian.cheng@ubc.ca Abstract—A major challenge to implement the compressed sensing technique for channel state information (CSI) feedback reduction lies in the design of a well-performed measurement matrix to compress linearly the dimension of sparse channel vectors. The widely adopted randomized measurement matrices drawn from Gaussian or Bernoulli distribution are not optimal for all channel realizations. To tackle this problem, a fully data- driven approach is proposed to design the measurement matrix for beamspace channel vectors. This method adopts a model- driven autoencoder which is constructed according to an iterative solution of sparse reconstruction. The constructed autoencoder is parameterized by measurement matrix such that the mea- surement matrix can be optimized by training with beamspace channel vectors to minimize the reconstruction error. Compared with random matrices, the acquired data-driven measurement matrix can achieve accurate CSI reconstructions using fewer measurements, thus the feedback overhead can be substantially reduced by applying this data-driven measurement matrix to compressed sensing based CSI feedback schemes. Index Terms—Compressed sensing, deep learning, massive MIMO, measurement matrix, mmWave I. I NTRODUCTION Compressed sensing technique [1] provides a promising alternative for channel state information (CSI) acquisition in millimetre wave (mmWave) massive multiple-input multiple- output (MIMO) systems. In the proposed compressed sensing based channel estimation schemes [2]–[5], the beamspace channel sparsity is exploited, and the channel estimation problem is formulated as a sparse recovery task. In the proposed compressed sensing based downlink CSI feedback schemes [6]–[12], the user equipment (UE) first compresses the estimated CSI into lower-dimensional measurements, then sends back the compressed measurements to the base station (BS); the downlink CSI is finally recovered at the BS from the received compressed measurements. In these aforementioned compressed sensing based CSI ac- quisition schemes, measurement matrices play essential roles in successful recoveries [3], [13]. Since compressed sensing theory states that some random measurement matrices can achieve accurate recoveries for high probability when the dimension of compressed measurements is sufficiently large, most of the existing literature adopt random matrices as their default choice for measurement matrix. However, it has been found that the random matrices often perform unsatisfactorily in practical applications especially when the dimension of compressed measurements is insufficient [1]. Even though the recovery accuracy can be improved through increasing the dimension of compressed measurements, the larger dimension of randomized compressed measurements means the larger size of training pilot and heavier feedback overhead, which are undesired. Therefore, it is meaningful to optimize the random measurement matrices such that the least number of compressed measurements required for accurate recoveries can be reduced. Besides random matrices, an alternative is to construct a deterministic matrix as the measurement matrix, but the design of a deterministic measurement matrix lacks explicit guidelines. Moreover, the deterministic measurement matrices designed in an ad hoc manner do not perform well for different channel realizations [14]. Therefore, our goal is to seek an effective method to generate a well-performed measurement matrix that can be used for all channel realizations. Motivated by the popularity of deep learning techniques, one promising approach is to employ the data-driven measure- ment matrix. It has been shown that many real-world datasets have structural features that can be exploited to perform data- driven dimensional reductions [15]. However, it is yet known whether additional features beyond sparsity exist in mmWave massive MIMO beamspace channels. Therefore, our goal is to develop an approach that can exploit the underlying dataset structures to perform data-driven linear-dimensional-reduction operations for mmWave massive MIMO beamspace channels. To achieve this goal, we adopt a model-driven autoencoder named ℓ 1 -minimization autoencoder (ℓ 1 -AE) [16], which is constructed by mimicking the linear compression and ℓ 1 -minimization reconstruction of compressed sensing. In spe- cific, we regard the ℓ 1 -minimization reconstruction iterations as a set of stacked neural networks parameterized with the measurement matrix. By backpropagating the reconstruction error through the neural network during training, the measure- ment matrix can be optimized based on the training dataset. We train the ℓ 1 -AE using the dataset of beamspace channel vectors to acquire a data-driven measurement matrix; then the learned measurement matrix is directly applied to classical compressed sensing reconstruction algorithms to perform CSI compression and recovery. Different from other deep learning based schemes aiming to develop end-to-end models for CSI feedback [17]–[22], we de- sign a data-driven measurement matrix adaptive to beamspace 1 arXiv:1903.02127v3 [cs.IT] 21 Mar 2020