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.

中文翻译：

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利用变分自动编码器进行参数化 MMSE 估计


在本手稿中，我们建议使用基于变分自动编码器的框架来参数化条件线性最小均方误差估计器。变分自动编码器将底层未知数据分布建模为条件高斯分布，在给定噪声观测的情况下产生估计量的条件一阶矩和二阶矩。导出的估计器通过利用变分自动编码器作为估计问题的生成先验来逼近最小均方误差估计器。我们提出了三种估计器变体，它们在训练和估计阶段对地面实况数据的访问有所不同。所提出的仅针对噪声观测进行训练的估计器变体特别值得注意，因为它不需要在训练或估计期间访问地面实况数据。我们通过限制所提出的和最小均方误差估计器之间的差异，将训练目标和最终的估计性能联系起来，进行严格的分析。此外，所得的界限表明，所提出的估计量需要偏差-方差权衡，这在估计文献中是众所周知的。作为示例应用程序，我们描述了信道估计，允许结构化协方差矩阵参数化和低复杂度实现。然而，所提出的框架并不限于信道估计，而是可以应用于广泛的估计问题。广泛的数值模拟首先验证了所提出的基于变分自动编码器的估计器的理论分析，然后与相关的经典和基于机器学习的最先进估计器相比，展示了出色的估计性能。