# Leveraging Variational Autoencoders for Parameterized MMSE Estimation

arXiv: https://arxiv.org/abs/2307.05352 · HTML: https://mlsanity.com/paper/2307.05352/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 53 references that are in the map.
- First posted on arXiv 11 Jul 2023.

## Builds on (most cited first, 23 of 23)

- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Weight Normalization: A Simple Reparameterization to Accelerate Training of Deep Neural Networks](https://mlsanity.com/paper/1602.07868/) (2016)
- [Tune: A Research Platform for Distributed Model Selection and Training](https://mlsanity.com/paper/1807.05118/) (2018)
- [Compressed Sensing using Generative Models](https://mlsanity.com/paper/1703.03208/) (2017)
- [Deep Learning Techniques for Inverse Problems in Imaging](https://mlsanity.com/paper/2005.06001/) (2020)
- [Robust Compressed Sensing MRI with Deep Generative Priors](https://mlsanity.com/paper/2108.01368/) (2021)
- [Structured Pruning of Deep Convolutional Neural Networks](https://mlsanity.com/paper/1512.08571/) (2015)
- [An Introduction to Deep Generative Modeling](https://mlsanity.com/paper/2103.05180/) (2021)
- [Phase Retrieval Under a Generative Prior](https://mlsanity.com/paper/1807.04261/) (2018)
- [Towards Deeper Understanding of Variational Autoencoding Models](https://mlsanity.com/paper/1702.08658/) (2017)
- [MIMO Channel Estimation using Score-Based Generative Models](https://mlsanity.com/paper/2204.07122/) (2022)
- [Blind Image Deconvolution using Deep Generative Priors](https://mlsanity.com/paper/1802.04073/) (2018)
- [High Dimensional Channel Estimation Using Deep Generative Networks](https://mlsanity.com/paper/2006.13494/) (2020)
- [Wideband Channel Estimation with A Generative Adversarial Network](https://mlsanity.com/paper/2012.12063/) (2020)
- [An Asymptotically MSE-Optimal Estimator based on Gaussian Mixture Models](https://mlsanity.com/paper/2112.12499/) (2021)
- [Learning to Estimate Without Bias](https://mlsanity.com/paper/2110.12403/) (2021)
- [Variational Autoencoder Leveraged MMSE Channel Estimation](https://mlsanity.com/paper/2205.05345/) (2022)
- [Channel Estimation based on Gaussian Mixture Models with Structured Covariances](https://mlsanity.com/paper/2205.03634/) (2022)
- [Over-the-Air Design of GAN Training for mmWave MIMO Channel Estimation](https://mlsanity.com/paper/2205.12445/) (2022)
- [A Low-Complexity MIMO Channel Estimator with Implicit Structure of a Convolutional Neural Network](https://mlsanity.com/paper/2104.12667/) (2021)

## Built on by (most cited first, 6 of 6)

- [Generative Diffusion Models for High Dimensional Channel Estimation](https://mlsanity.com/paper/2408.10501/) (2024)
- [Enhancements in Score-based Channel Estimation for Real-Time Wireless Systems](https://mlsanity.com/paper/2509.07839/) (2025)
- [Achieving Robust Channel Estimation Neural Networks by Designed Training Data](https://mlsanity.com/paper/2507.12630/) (2025)
- [Enhancing Channel Estimation in Quantized Systems with a Generative Prior](https://mlsanity.com/paper/2405.03542/) (2024)
- [Sparse Bayesian Generative Modeling for Compressive Sensing](https://mlsanity.com/paper/2411.09483/) (2024)
- [Null-Space Flow Matching for MIMO Channel Estimation in Latency-Constrained Systems](https://mlsanity.com/paper/2604.22005/) (2026)

## Abstract (arXiv)

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.