# An Asymptotically MSE-Optimal Estimator based on Gaussian Mixture Models

arXiv: https://arxiv.org/abs/2112.12499 · HTML: https://mlsanity.com/paper/2112.12499/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Dec 2021.

## Built on by (most cited first, 14 of 14)

- [Generative Diffusion Models for High Dimensional Channel Estimation](https://mlsanity.com/paper/2408.10501/) (2024)
- [Diffusion-based Generative Prior for Low-Complexity MIMO Channel Estimation](https://mlsanity.com/paper/2403.03545/) (2024)
- [Generative Diffusion Models for Wireless Networks: Fundamental, Architecture, and State-of-the-Art](https://mlsanity.com/paper/2507.16733/) (2025)
- [On the Asymptotic Mean Square Error Optimality of Diffusion Models](https://mlsanity.com/paper/2403.02957/) (2024)
- [Variational Autoencoder Leveraged MMSE Channel Estimation](https://mlsanity.com/paper/2205.05345/) (2022)
- [Leveraging Variational Autoencoders for Parameterized MMSE Estimation](https://mlsanity.com/paper/2307.05352/) (2023)
- [Diffusion Models for Future Networks and Communications: A Comprehensive Survey](https://mlsanity.com/paper/2508.01586/) (2025)
- [Channel Estimation for Quantized Systems based on Conditionally Gaussian Latent Models](https://mlsanity.com/paper/2309.04014/) (2023)
- [Variational Inference Aided Estimation of Time Varying Channels](https://mlsanity.com/paper/2210.17177/) (2022)
- [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)

This paper investigates a channel estimator based on Gaussian mixture models (GMMs) in the context of linear inverse problems with additive Gaussian noise. We fit a GMM to given channel samples to obtain an analytic probability density function (PDF) which approximates the true channel PDF. Then, a conditional mean estimator (CME) corresponding to this approximating PDF is computed in closed form and used as an approximation of the optimal CME based on the true channel PDF. This optimal CME cannot be calculated analytically because the true channel PDF is generally unknown. We present mild conditions which allow us to prove the convergence of the GMM-based CME to the optimal CME as the number of GMM components is increased. Additionally, we investigate the estimator's computational complexity and present simplifications based on common model-based insights. Further, we study the estimator's behavior in numerical experiments including multiple-input multiple-output (MIMO) and wideband systems.