  We propose to utilize a variational autoencoder (VAE) for data-driven channel
estimation. The underlying true and unknown channel distribution is modeled by
the VAE as a conditional Gaussian distribution in a novel way, parameterized by
the respective first and second order conditional moments. As a result, it can
be observed that the linear minimum mean square error (LMMSE) estimator in its
variant conditioned on the latent sample of the VAE approximates an optimal MSE
estimator. Furthermore, we argue how a VAE-based channel estimator can
approximate the MMSE channel estimator. We propose three variants of VAE
estimators that differ in the data used during training and estimation. First,
we show that given perfectly known channel state information at the input of
the VAE during estimation, which is impractical, we obtain an estimator that
can serve as a benchmark result for an estimation scenario. We then propose
practically feasible approaches, where perfectly known channel state
information is only necessary in the training phase or is not needed at all.
Simulation results on 3GPP and QuaDRiGa channel data attest a small performance
loss of the practical approaches and the superiority of our VAE approaches in
comparison to other related channel estimation methods.
