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In a multiple-input multiple-output frequency-division duplexing (MIMO-FDD) system, the user equipment (UE) sends the downlink channel state information (CSI) to the base station to report link status. Due to the complexity of MIMO systems, the overhead incurred in sending this information negatively affects the system bandwidth. Although this problem has been widely considered in the literature, prior work generally assumes an ideal feedback channel. In this paper, we introduce PRVNet, a neural network architecture inspired by variational autoencoders (VAE) to compress the CSI matrix before sending it back to the base station under noisy channel conditions. Moreover, we propose a customized loss function that best suits the special characteristics of the problem being addressed. We also introduce an additional regularization hyperparameter for the learning objective, which is crucial for achieving competitive performance. In addition, we provide an efficient way to tune this hyperparameter using KL-annealing. Experimental results show the proposed model outperforms the benchmark models including two deep learning-based models in a noise-free feedback channel assumption. In addition, the proposed model achieves an outstanding performance under different noise levels for additive white Gaussian noise feedback channels.