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Leveraging Variational Autoencoders for Parameterized MMSE Estimation

Michael Baur, Benedikt Fesl, Wolfgang Utschick

TL;DR

This paper introduces a variational autoencoder–based framework to parameterize a conditional linear minimum MSE estimator by modeling the unknown data distribution as conditionally Gaussian. By learning conditional moments through a VAE with an encoder for ${m{y}}$ and a decoder for ${m{h}}|{m{z}}$, the authors derive a tractable CME approximation that integrates a generative prior into a closed-form LMMSE step inside an outer expectation over the latent variable ${m{z}}$. They propose three estimator variants (VAE-genie, VAE-noisy, VAE-real), provide a bound linking training objectives to CME performance, and reveal a conditional bias–variance tradeoff controlled by SNR. As an application, they apply the framework to MIMO channel estimation with circulant covariance parameterization, achieving substantial NMSE gains over classical and ML baselines, including a variant that requires no ground-truth data for training or evaluation. The results demonstrate the framework’s robustness across channel models (3GPP and QuaDRiGa) and configurations, while offering a path toward low-complexity, data-efficient inverse problem solvers in communications and beyond.

Abstract

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.

Leveraging Variational Autoencoders for Parameterized MMSE Estimation

TL;DR

This paper introduces a variational autoencoder–based framework to parameterize a conditional linear minimum MSE estimator by modeling the unknown data distribution as conditionally Gaussian. By learning conditional moments through a VAE with an encoder for and a decoder for , the authors derive a tractable CME approximation that integrates a generative prior into a closed-form LMMSE step inside an outer expectation over the latent variable . They propose three estimator variants (VAE-genie, VAE-noisy, VAE-real), provide a bound linking training objectives to CME performance, and reveal a conditional bias–variance tradeoff controlled by SNR. As an application, they apply the framework to MIMO channel estimation with circulant covariance parameterization, achieving substantial NMSE gains over classical and ML baselines, including a variant that requires no ground-truth data for training or evaluation. The results demonstrate the framework’s robustness across channel models (3GPP and QuaDRiGa) and configurations, while offering a path toward low-complexity, data-efficient inverse problem solvers in communications and beyond.

Abstract

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.
Paper Structure (22 sections, 1 theorem, 45 equations, 13 figures, 1 table)

This paper contains 22 sections, 1 theorem, 45 equations, 13 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Signal Model and Problem Formulation
  4. VAE Fundamentals
  5. VAE-Parameterized Estimator
  6. MMSE Estimation with the VAE
  7. Covariance Matrix Parameterization
  8. Variants of VAE-based Estimators
  9. MSE-Optimality and Conditional Bias-Variance Tradeoff
  10. MSE-Optimality
  11. Conditional Bias-Variance Tradeoff
  12. Example Application: Channel Estimation
  13. Training Loss and Network Architecture
  14. Computational Complexity
  15. Channel Models
  16. ...and 7 more sections

Key Result

Theorem 1

Consider a decorrelated observation ${\bm{y}} = {\bm{h}} +~{\bm{n}}$ with ${\bm{n}} \sim \mathcal{N}_{\mathbb{C}}({\bm{0}}, \varsigma^2 \mathop{\mathrm{\mathbf{I}}}\nolimits)$ and let eq:cg-vae and eq:total_exp hold. Further, assume the decoder neural network functions are Lipschitz continuous, i.e. Then, the expected Euclidean distance between the cme eq:total_exp and the map-vae estimator eq:vae

Figures (13)

  • Figure 1: Bayesian network illustrating the VAE decoder distribution $p_{\bm{\theta}}({\bm{h}}{\,|\,}{\bm{z}})$, encoder distribution $q_{\bm{\phi}}({\bm{z}}{\,|\,}{\bm{y}})$, and the known $p({\bm{y}}{\,|\,}{\bm{h}})=\mathcal{N}_{\mathbb{C}}({\bm{A}}{\bm{h}}, \mathbf{\Sigma})$.
  • Figure 2: Structure of a VAE with cg distributions for $q_{{\bm{\phi}}}({\bm{z}}{\,|\,}{\bm{y}})$ and $p_{{\bm{\theta}}}({\bm{h}}{\,|\,}{\bm{z}})$. The encoder and decoder each represent a nn.
  • Figure 3: Detailed illustration of the different layers constituting our VAE implementation. The real and imaginary parts of the input are stacked as cc and processed. The colored arrows represent different layers or layer compositions. Purple stands for a $1\times 1$cl, orange for a block of a cl, bn layer, and ReLU activation function, gray for a rl, green for a ll, and red for a block of a transposed cl, bn layer, and ReLU activation function.
  • Figure 4: Training of the VAE-noisy variant for the 3gpp channel model (simo case) with three propagation clusters at an snr of 10dB. ELBO refers to the complete training loss in \ref{['eq:loss_training']}, REC to the negative of \ref{['eq:dec-like-diag']}, and KL to \ref{['eq:vae-kl']}. REC is plotted including the in \ref{['eq:dec-like-diag']} omitted constants.
  • Figure 5: Normalized mse for different numbers of training samples at an SNR of 10dB for the 3gpp channel model (simo case) with three propagation clusters and $128$ antennas at the receiver. The dotted lines display the achieved result with the complete training dataset of $180{,}000$ samples.
  • ...and 8 more figures

Theorems & Definitions (2)

  • Theorem 1
  • proof