  In this manuscript, we propose to use a variational autoencoder-based
framework for parameterizing a conditional linear minimum mean squared error
estimator. The variational autoencoder models the underlying unknown data
distribution as conditionally Gaussian, yielding the conditional first and
second moments of the estimand, given a noisy observation. The derived
estimator is shown to approximate the minimum mean squared error estimator by
utilizing the variational autoencoder as a generative prior for the estimation
problem. We propose three estimator variants that differ in their access to
ground-truth data during the training and estimation phases. The proposed
estimator variant trained solely on noisy observations is particularly
noteworthy as it does not require access to ground-truth data during training
or estimation. We conduct a rigorous analysis by bounding the difference
between the proposed and the minimum mean squared error estimator, connecting
the training objective and the resulting estimation performance. Furthermore,
the resulting bound reveals that the proposed estimator entails a bias-variance
tradeoff, which is well-known in the estimation literature. As an example
application, we portray channel estimation, allowing for a structured
covariance matrix parameterization and low-complexity implementation.
Nevertheless, the proposed framework is not limited to channel estimation but
can be applied to a broad class of estimation problems. Extensive numerical
simulations first validate the theoretical analysis of the proposed variational
autoencoder-based estimators and then demonstrate excellent estimation
performance compared to related classical and machine learning-based
state-of-the-art estimators.
