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.

翻译：本文提出了一种基于变分自编码器的框架，用于参数化条件线性最小均方误差估计器。该变分自编码器将底层未知数据分布建模为条件高斯分布，从而在给定噪声观测的情况下导出估计量的条件一阶矩和二阶矩。通过将变分自编码器作为估计问题的生成先验，所推导的估计器被证明可近似实现最小均方误差估计。我们提出了三种估计器变体，它们在训练和估计阶段对真实数据的访问权限有所不同。其中仅基于噪声观测进行训练的估计器变体尤其值得关注，因为其在训练和估计过程中均不需要真实数据。我们通过严格的理论分析，界定了所提估计器与最小均方误差估计器之间的差异，从而建立了训练目标与估计性能之间的理论联系。进一步地，所得界揭示了所提估计器存在偏差-方差权衡特性，这一特性在估计理论中广为人知。作为示例应用，我们展示了信道估计场景，该框架支持结构化协方差矩阵参数化与低复杂度实现。然而，所提框架不仅限于信道估计，可广泛应用于各类估计问题。大量数值仿真首先验证了所提基于变分自编码器的估计器的理论分析，随后通过与相关经典估计器及基于机器学习的最先进估计器进行比较，证明了其优异的估计性能。