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arxiv: 2605.18325 · v1 · pith:4EFX5XAYnew · submitted 2026-05-18 · 📡 eess.SP

Mixture-of-Experts Diffusion Models for Adaptive Massive MIMO Channel Estimation via Variational Bayesian Inference

Zhuorui Jiang , Jun Fang , Boyu Ning , Hongbin Li , Ying-Chang Liang This is my paper

Pith reviewed 2026-05-20 00:16 UTC · model grok-4.3

classification 📡 eess.SP
keywords mixture-of-expertsdiffusion modelschannel estimationmassive MIMOvariational Bayesian inference3GPP CDL channelsadaptive estimation
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The pith

A mixture of specialized diffusion models adapts to different wireless propagation environments for improved massive MIMO channel estimation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that multiple pre-trained diffusion models, each tuned to a distinct channel type, can be combined via variational Bayesian inference to jointly recover the channel realization and select the active expert. A sympathetic reader cares because real-world massive MIMO systems encounter varying propagation conditions, and a single model trained on pooled data loses accuracy when samples from different environments are uneven. The framework treats the channel as drawn from one of several candidate generative priors with an unknown discrete probability, then infers both the continuous channel values and the expert indicator from noisy observations.

Core claim

The central claim is that embedding a mixture-of-experts structure inside a variational inference loop lets the estimator automatically activate the diffusion prior that best matches the current propagation environment, yielding lower estimation error than a single diffusion model trained on aggregated data, with the advantage growing when channel samples from different 3GPP CDL types are imbalanced.

What carries the argument

A probabilistic graphical model in which the channel is a latent variable drawn from one of several pre-trained diffusion-model priors according to a discrete expert indicator, with both variables recovered jointly by variational Bayesian inference.

If this is right

  • The estimator automatically selects the appropriate prior without explicit environment labels at test time.
  • Estimation accuracy improves most when the training data across environments is unbalanced.
  • The same joint-inference structure can be applied to other generative priors beyond diffusion models.
  • The method supports deployment in scenarios where the channel distribution shifts over time or location.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach could be extended by periodically retraining or adding new experts as new propagation environments are encountered.
  • It may generalize to joint channel and parameter estimation tasks where multiple generative models compete to explain the same observations.
  • The variational step provides a natural way to quantify uncertainty over both the channel values and the active expert choice.

Load-bearing premise

A collection of pre-trained diffusion models for separate propagation environments already exists, and variational inference can correctly identify which expert generated the observed noisy channel samples.

What would settle it

Run the estimator on test channels drawn from one known 3GPP CDL environment and measure whether the inferred expert indicator selects the matching pre-trained model at a rate significantly above chance.

Figures

Figures reproduced from arXiv: 2605.18325 by Boyu Ning, Hongbin Li, Jun Fang, Ying-Chang Liang, Zhuorui Jiang.

Figure 1
Figure 1. Figure 1: Illustration of the proposed VB-based automatic model [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: Illustration of the proposed probabilistic graphical [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: NMSE results among different channel estimation methods under balanced CDL datasets when [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: NMSE results among different channel estimation methods under imbalanced CDL datasets when [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: NMSE results among different channel estimation methods with varying pilot density [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Convergence behavior of the proposed method: (a) [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
read the original abstract

Channel estimation is essential to massive multiple-input multiple-output (MIMO) systems. While recent generative model-based approaches using lightweight diffusion models (DMs) have achieved superior performance, they typically rely on a single data-driven prior, which limits their adaptability to varying channel distributions in real-world scenarios. To address this deficiency, we propose a mixture-of-experts (MoE) diffusion model (DM) framework combined with variational Bayesian inference. Specifically, our approach employs multiple pre-trained DMs, with each trained on a specific type of propagation channels. We then propose a probabilistic graphical model in which the channel is modeled as a latent variable drawn from one of these candidate generative priors with a certain probability. By integrating variational Bayesian inference with DM-based data priors, the underlying channel along with the expert indicator variable are jointly inferred, thus enabling automatic model adaptation for channel estimation. The effectiveness of our approach is evaluated on 3GPP CDL channels. Simulation results demonstrate that our proposed approach achieves a clear performance improvement over the standard DM-based method that employs a single prior trained on aggregated data from all channel types, particularly when the channel samples from different propagation environments are imbalanced.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a mixture-of-experts (MoE) diffusion model (DM) framework for adaptive massive MIMO channel estimation. Multiple pre-trained DMs, each specialized to a distinct propagation environment, are combined via a probabilistic graphical model where the channel is a latent variable drawn from one of the priors according to a categorical distribution. Variational Bayesian inference is used to jointly recover the channel realization and the discrete expert indicator variable from noisy observations, enabling automatic adaptation. Evaluations on 3GPP CDL channels show performance gains over a single aggregated prior, especially under imbalanced channel sample distributions.

Significance. If the variational procedure can reliably identify the correct expert under realistic conditions, the approach offers a principled way to handle heterogeneous channel distributions without retraining a single model on aggregated data. This could be particularly valuable for practical deployments where propagation environments vary and data from different types is imbalanced. The integration of generative priors with variational inference for discrete model selection is a potentially useful technical contribution.

major comments (2)
  1. [Probabilistic graphical model and variational inference (Section 3)] The central claim that the MoE framework enables automatic adaptation rests on the variational Bayesian inference jointly recovering both the channel and the expert indicator variable. No analysis or experiments are provided on posterior identifiability when the pre-trained diffusion priors overlap (common for 3GPP CDL variants) or when the variational family is applied under high noise; without this, the reported gains in the imbalanced-data regime cannot be attributed to successful expert selection rather than other factors.
  2. [Numerical results and comparisons (Section 4)] Simulation results claim clear performance improvement over the single-prior baseline, but the manuscript supplies neither error-bar reporting across Monte Carlo trials nor ablations isolating the contribution of the expert-indicator inference. This leaves the quantitative advantage unevaluated and undermines the assertion that the method is particularly effective when channel samples from different environments are imbalanced.
minor comments (2)
  1. [Notation and model description] Explicitly state the mean-field factorization assumed for the variational posterior q(h, z) over the continuous channel h and discrete expert indicator z; this is required to assess the quality of the approximation.
  2. [Figure captions and results presentation] Add confidence intervals or standard deviations to all plotted NMSE curves so that the claimed gains can be assessed for statistical significance.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which highlight important aspects for strengthening the manuscript. We address each major comment below and describe the revisions we will incorporate.

read point-by-point responses

  1. Referee: [Probabilistic graphical model and variational inference (Section 3)] The central claim that the MoE framework enables automatic adaptation rests on the variational Bayesian inference jointly recovering both the channel and the expert indicator variable. No analysis or experiments are provided on posterior identifiability when the pre-trained diffusion priors overlap (common for 3GPP CDL variants) or when the variational family is applied under high noise; without this, the reported gains in the imbalanced-data regime cannot be attributed to successful expert selection rather than other factors.

    Authors: We agree that explicit analysis of posterior identifiability under overlapping priors and high noise would strengthen the attribution of gains to expert selection. In the revised manuscript, we will add a new subsection in Section 3 discussing identifiability conditions drawing on variational inference theory for mixture models, along with empirical results on expert selection accuracy (e.g., posterior probability of correct indicator) across noise levels and CDL variant similarities. These additions will directly address whether the imbalanced-data improvements arise from successful adaptation. revision: yes

  2. Referee: [Numerical results and comparisons (Section 4)] Simulation results claim clear performance improvement over the single-prior baseline, but the manuscript supplies neither error-bar reporting across Monte Carlo trials nor ablations isolating the contribution of the expert-indicator inference. This leaves the quantitative advantage unevaluated and undermines the assertion that the method is particularly effective when channel samples from different environments are imbalanced.

    Authors: We concur that error bars and targeted ablations are necessary to rigorously evaluate the quantitative advantage and isolate the role of expert-indicator inference. In the revision, we will augment Section 4 with error bars (standard deviations over 100 Monte Carlo trials) for all NMSE curves. We will also include an ablation study comparing the full joint VBI approach against a fixed-expert baseline (using only the aggregated prior or random selection), with particular emphasis on the imbalanced sampling regime to demonstrate the contribution of automatic expert inference. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on explicit variational inference over an introduced expert indicator rather than reducing to fitted inputs or self-citations

full rationale

The paper introduces a probabilistic graphical model with an explicit discrete expert indicator variable that selects among pre-trained diffusion priors; variational Bayesian inference is then applied to jointly recover both the channel and the indicator. This structure is presented as an independent modeling choice whose performance benefit is evaluated empirically on 3GPP CDL channels under imbalanced data, not derived by construction from the final metric or from a self-citation chain. No equations or steps in the described framework equate the claimed adaptation gain to a pre-fitted parameter or rename an existing result; the method remains self-contained against external simulation benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The framework rests on the existence of multiple independently trained diffusion models and on the validity of the variational approximation for the joint posterior over channel and expert indicator; no free parameters are explicitly named in the abstract.

axioms (2)
  • domain assumption Variational Bayesian inference can accurately approximate the joint posterior over the continuous channel and discrete expert indicator variables.
    Invoked when the paper states that the channel and expert indicator are jointly inferred via variational Bayesian inference.
  • domain assumption Each pre-trained diffusion model provides a valid generative prior for its corresponding propagation environment.
    Stated when the approach employs multiple pre-trained DMs each trained on a specific type of propagation channels.
invented entities (1)
  • Expert indicator variable no independent evidence
    purpose: Discrete latent variable that selects which diffusion prior generated the observed channel.
    Introduced in the probabilistic graphical model to enable automatic model adaptation.

pith-pipeline@v0.9.0 · 5748 in / 1428 out tokens · 37638 ms · 2026-05-20T00:16:45.797685+00:00 · methodology

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Works this paper leans on

53 extracted references · 53 canonical work pages · 1 internal anchor

  1. [1]

    Intelligent massive MIMO systems for beyond 5G networks: An overview and future trends,

    O. Elijah, S. K. A. Rahim, W. K. New, C. Y . Leow, K. Cumanan, and T. K. Geok, “Intelligent massive MIMO systems for beyond 5G networks: An overview and future trends,”IEEE Access, vol. 10, pp. 102 532–102 563, 2022

    work page 2022

  2. [2]

    A tutorial on extremely large-scale MIMO for 6g: Fundamentals, signal processing, and applications,

    Z. Wang, J. Zhang, H. Du, D. Niyato, S. Cui, B. Ai, M. D ´ebbah, K. B. Letaief, and H. V . Poor, “A tutorial on extremely large-scale MIMO for 6g: Fundamentals, signal processing, and applications,”IEEE Commun. Surveys & Tutorials, vol. 26, no. 3, pp. 1560–1605, 2024

    work page 2024

  3. [3]

    Beamforming technologies for ultra-massive MIMO in terahertz communications,

    B. Ning, Z. Tian, W. Mei, Z. Chen, C. Han, S. Li, J. Yuan, and R. Zhang, “Beamforming technologies for ultra-massive MIMO in terahertz communications,”IEEE Open J. Commun. Soc., vol. 4, pp. 614–658, 2023

    work page 2023

  4. [4]

    Massive MIMO performance comparison of beamforming and multiplexing in the terahertz band,

    S. A. Hoseini, M. Ding, and M. Hassan, “Massive MIMO performance comparison of beamforming and multiplexing in the terahertz band,” in 2017 IEEE Globecom Workshops (GC Wkshps). IEEE, 2017, pp. 1–6

    work page 2017

  5. [5]

    MIMO for millimeter-wave wireless communications: Beamforming, spatial multiplexing, or both?

    S. Sun, T. S. Rappaport, R. W. Heath, A. Nix, and S. Rangan, “MIMO for millimeter-wave wireless communications: Beamforming, spatial multiplexing, or both?”IEEE Commun. Mag., vol. 52, no. 12, pp. 110– 121, 2014

    work page 2014

  6. [6]

    Precoding and beamforming techniques in mmwave-massive MIMO: Performance assessment,

    T. Kebede, Y . Wondie, J. Steinbrunn, H. B. Kassa, and K. T. Kornegay, “Precoding and beamforming techniques in mmwave-massive MIMO: Performance assessment,”IEEE access, vol. 10, pp. 16 365–16 387, 2022

    work page 2022

  7. [7]

    Precoding matrix indicator in the 5G NR protocol: A tutorial on 3GPP beamforming codebooks,

    B. Ning, H. Yin, S. Liu, H. Deng, S. Yang, Y . Zhang, W. Mei, D. Gesbert, J. Park, R. W. H. Jr., and E. Bj ¨ornson, “Precoding matrix indicator in the 5G NR protocol: A tutorial on 3GPP beamforming codebooks,”IEEE Commun. Surv. Tutorials, vol. 28, pp. 4581–4623, 2026

    work page 2026

  8. [8]

    Overview of deep learning- based CSI feedback in massive MIMO systems,

    J. Guo, C.-K. Wen, S. Jin, and G. Y . Li, “Overview of deep learning- based CSI feedback in massive MIMO systems,”IEEE Trans. Commun., vol. 70, no. 12, pp. 8017–8045, 2022

    work page 2022

  9. [9]

    Deep learning based one bit-ADCs efficient channel estimation using fewer pilots overhead for massive MIMO system,

    M. H. Rahman, M. A. S. Sejan, M. A. Aziz, R. Tabassum, J.-I. Baik, and H.-K. Song, “Deep learning based one bit-ADCs efficient channel estimation using fewer pilots overhead for massive MIMO system,” IEEE Access, vol. 12, pp. 64 823–64 836, 2024

    work page 2024

  10. [10]

    Improved spectral efficiency in massive MIMO ultra-dense networks through optimal pilot-based vector perturbation precoding,

    R. A. Raja and B. Vijayalakshmi, “Improved spectral efficiency in massive MIMO ultra-dense networks through optimal pilot-based vector perturbation precoding,”Optik, vol. 273, p. 170370, 2023. 13

    work page 2023

  11. [11]

    An effective algorithm in uplink massive MIMO systems for pilot decontamination,

    R. Khan, L. Jan, S. Khan, M. H. Zafar, W. Ahmad, and G. Husnain, “An effective algorithm in uplink massive MIMO systems for pilot decontamination,”Results Eng., vol. 21, p. 101873, 2024

    work page 2024

  12. [12]

    Hierarchical BEM based channel estimation with very low pilot overhead for high mobility MIMO-OFDM systems,

    Y . Zhang, X. Zhu, Y . Liu, Y . Jiang, Y . L. Guan, and V . K. N. Lau, “Hierarchical BEM based channel estimation with very low pilot overhead for high mobility MIMO-OFDM systems,”IEEE Trans. V eh. Technol., vol. 71, no. 10, pp. 10 543–10 558, 2022

    work page 2022

  13. [13]

    Regression shrinkage and selection via the Lasso,

    R. Tibshirani, “Regression shrinkage and selection via the Lasso,”J. R. Stat. Soc. B, vol. 58, no. 1, pp. 267–288, 1996

    work page 1996

  14. [14]

    Channel estimation based on compressed sensing for massive MIMO systems with lens antenna array,

    E. Sharifi, M. M. Feghhi, G. Azarnia, S. Nouri, D. Lee, and M. J. Piran, “Channel estimation based on compressed sensing for massive MIMO systems with lens antenna array,”IEEE Access, vol. 11, pp. 79 016– 79 032, 2023

    work page 2023

  15. [15]

    Message-passing algo- rithms for compressed sensing,

    D. L. Donoho, A. Maleki, and A. Montanari, “Message-passing algo- rithms for compressed sensing,”Proceedings of the National Academy of Sciences, vol. 106, no. 45, pp. 18 914–18 919, 2009

    work page 2009

  16. [16]

    Triple-structured compressive sensing- based channel estimation for RIS-aided MU-MIMO systems,

    X. Shi, J. Wang, and J. Song, “Triple-structured compressive sensing- based channel estimation for RIS-aided MU-MIMO systems,”IEEE Trans. Wireless Commun., vol. 21, no. 12, pp. 11 095–11 109, 2022

    work page 2022

  17. [17]

    Expectation-maximization Gaussian-mixture approximate message passing,

    J. P. Vila and P. Schniter, “Expectation-maximization Gaussian-mixture approximate message passing,”IEEE Trans. Signal Process., vol. 61, no. 19, pp. 4658–4672, 2013

    work page 2013

  18. [18]

    Two-dimensional pattern-coupled sparse Bayesian learning via generalized approximate message passing,

    J. Fang, L. Zhang, and H. Li, “Two-dimensional pattern-coupled sparse Bayesian learning via generalized approximate message passing,”IEEE Trans. Image Process., vol. 25, no. 6, pp. 2920–2930, 2016

    work page 2016

  19. [19]

    Low- rank tensor decomposition-aided channel estimation for millimeter wave MIMO-OFDM systems,

    Z. Zhou, J. Fang, L. Yang, H. Li, Z. Chen, and R. S. Blum, “Low- rank tensor decomposition-aided channel estimation for millimeter wave MIMO-OFDM systems,”IEEE J. Sel. Areas Commun., vol. 35, no. 7, pp. 1524–1538, 2017

    work page 2017

  20. [20]

    Sparse mmWave OFDM channel estimation using compressed sensing,

    F. Gomez-Cuba and A. J. Goldsmith, “Sparse mmWave OFDM channel estimation using compressed sensing,” inICC 2019–2019 IEEE Inter- national Conference on Communications (ICC). IEEE, 2019, pp. 1–7

    work page 2019

  21. [21]

    Channel estimation for quantized systems based on conditionally Gaussian latent models,

    B. Fesl, N. Turan, B. B ¨ock, and W. Utschick, “Channel estimation for quantized systems based on conditionally Gaussian latent models,”IEEE Trans. Signal Process., vol. 72, pp. 1475–1490, 2024

    work page 2024

  22. [22]

    Deep learning-based channel estimation for beamspace mmWave massive MIMO systems,

    H. He, C.-K. Wen, S. Jin, and G. Y . Li, “Deep learning-based channel estimation for beamspace mmWave massive MIMO systems,”IEEE Wireless Commun. Lett., vol. 7, no. 5, pp. 852–855, 2018

    work page 2018

  23. [23]

    Pruning the pilots: Deep learning- based pilot design and channel estimation for MIMO-OFDM systems,

    M. B. Mashhadi and D. G ¨und¨uz, “Pruning the pilots: Deep learning- based pilot design and channel estimation for MIMO-OFDM systems,” IEEE Trans. Wireless Commun., vol. 20, no. 10, pp. 6315–6328, 2021

    work page 2021

  24. [24]

    Deep CNN-based channel estimation for mmWave massive MIMO systems,

    P. Dong, H. Zhang, G. Y . Li, I. S. Gaspar, and N. NaderiAlizadeh, “Deep CNN-based channel estimation for mmWave massive MIMO systems,” IEEE J. Sel. Topics Signal Process., vol. 13, no. 5, pp. 989–1000, 2019

    work page 2019

  25. [25]

    3GPP 3D MIMO channel model: A holistic implementation guideline for open source simulation tools,

    F. Ademaj, M. Taranetz, and M. Rupp, “3GPP 3D MIMO channel model: A holistic implementation guideline for open source simulation tools,”EURASIP J. Wirel. Commun. Netw., vol. 2016, no. 1, p. 55, 2016

    work page 2016

  26. [26]

    A novel millimeter- wave channel simulator and applications for 5G wireless communi- cations,

    S. Sun, G. R. MacCartney, and T. S. Rappaport, “A novel millimeter- wave channel simulator and applications for 5G wireless communi- cations,” in2017 IEEE International Conference on Communications (ICC). IEEE, 2017, pp. 1–7

    work page 2017

  27. [27]

    Open-source and low-cost test bed for automated 5G channel measurement in mmWave band,

    C. C. Chan, F. G. Kurnia, A. Al-Hournani, K. M. Gomez, S. Kandeepan, and W. Rowe, “Open-source and low-cost test bed for automated 5G channel measurement in mmWave band,”J. Infrared Millim. Terahertz Waves, vol. 40, no. 5, pp. 535–556, 2019

    work page 2019

  28. [28]

    Deep generative models for downlink channel estimation in FDD massive MIMO systems,

    J. Mirzaei, S. S. Panahi, R. S. Adve, and N. K. M. Gopal, “Deep generative models for downlink channel estimation in FDD massive MIMO systems,”IEEE Trans. Signal Process., vol. 70, pp. 2000–2014, 2022

    work page 2000

  29. [29]

    Variational autoencoder leveraged MMSE channel estimation,

    M. Baur, B. Fesl, M. Koller, and W. Utschick, “Variational autoencoder leveraged MMSE channel estimation,” in2022 56th Asilomar Confer- ence on Signals, Systems, and Computers. IEEE, 2022, pp. 527–532

    work page 2022

  30. [30]

    High dimensional channel estimation using deep generative networks,

    E. Balevi, A. Doshi, A. Jalal, A. Dimakis, and J. G. Andrews, “High dimensional channel estimation using deep generative networks,”IEEE J. Sel. Areas Commun., vol. 39, no. 1, pp. 18–30, 2021

    work page 2021

  31. [31]

    Channel estimation in massive MIMO systems using a modified Bayes-GMM method,

    P. Su and Y . Wang, “Channel estimation in massive MIMO systems using a modified Bayes-GMM method,”Wirel. Pers. Commun., vol. 107, no. 4, pp. 1521–1536, 2019

    work page 2019

  32. [32]

    Denoising diffusion probabilistic models,

    J. Ho, A. Jain, and P. Abbeel, “Denoising diffusion probabilistic models,”Adv. Neural Inf. Process. Syst., vol. 33, pp. 6840–6851, 2020

    work page 2020

  33. [33]

    arXiv preprint arXiv:2208.11970 , year=

    C. Luo, “Understanding diffusion models: A unified perspective,”arXiv preprint arXiv:2208.11970, 2022

    work page arXiv 2022

  34. [34]

    Score-Based Generative Modeling through Stochastic Differential Equations

    Y . Song, J. Sohl-Dickstein, D. P. Kingma, A. Kumar, S. Ermon, and B. Poole, “Score-based generative modeling through stochastic differential equations,”arXiv preprint arXiv:2011.13456, 2020

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  35. [35]

    MIMO channel estimation using score- based generative models,

    M. Arvinte and J. I. Tamir, “MIMO channel estimation using score- based generative models,”IEEE Trans. Wireless Commun., vol. 22, no. 6, pp. 3698–3713, 2022

    work page 2022

  36. [36]

    Generative diffusion model- based variational inference for MIMO channel estimation,

    Z. Chen, H. Shin, and A. Nallanathan, “Generative diffusion model- based variational inference for MIMO channel estimation,”IEEE Trans. Commun., vol. 73, no. 10, pp. 9254–9269, 2025

    work page 2025

  37. [37]

    Generative diffusion models for high dimensional channel estimation,

    X. Zhou, L. Liang, J. Zhang, P. Jiang, Y . Li, and S. Jin, “Generative diffusion models for high dimensional channel estimation,”IEEE Trans. Wireless Commun., vol. 24, no. 7, pp. 5840–5854, 2025

    work page 2025

  38. [38]

    Diffusion- based generative prior for low-complexity MIMO channel estimation,

    B. Fesl, M. Baur, F. Strasser, M. Joham, and W. Utschick, “Diffusion- based generative prior for low-complexity MIMO channel estimation,” IEEE Wireless Commun. Lett., vol. 13, no. 12, pp. 3493–3497, 2024

    work page 2024

  39. [39]

    Time-varying channel estimation scheme for uplink MU-MIMO in 6G systems,

    J. Wang, W. Zhang, Y . Chen, Z. Liu, J. Sun, and C.-X. Wang, “Time-varying channel estimation scheme for uplink MU-MIMO in 6G systems,”IEEE Trans. V eh. Technol., vol. 71, no. 11, pp. 11 820–11 831, 2022

    work page 2022

  40. [40]

    Tweedie’s formula and selection bias,

    B. Efron, “Tweedie’s formula and selection bias,”J. Am. Stat. Assoc., vol. 106, no. 496, pp. 1602–1614, 2011

    work page 2011

  41. [41]

    Diffusion model based posterior sampling for noisy linear inverse problems.arXiv preprint arXiv:2211.12343,

    X. Meng and Y . Kabashima, “Diffusion model based posterior sampling for noisy linear inverse problems,”arXiv preprint arXiv:2211.12343, 2022

    work page arXiv 2022

  42. [42]

    Generative modeling by estimating gradients of the data distribution,

    Y . Song and S. Ermon, “Generative modeling by estimating gradients of the data distribution,”Adv. Neural Inf. Process. Syst., vol. 32, 2019

    work page 2019

  43. [43]

    Training deep neural networks on imbalanced data sets,

    S. Wang, W. Liu, J. Wu, L. Cao, Q. Meng, and P. J. Kennedy, “Training deep neural networks on imbalanced data sets,” in2016 Int. Joint Conf. Neural Netw. (IJCNN). IEEE, 2016, pp. 4368–4374

    work page 2016

  44. [44]

    Robust Bayesian compressed sensing with outliers,

    Q. Wan, H. Duan, J. Fang, H. Li, and Z. Xing, “Robust Bayesian compressed sensing with outliers,”Signal Process., vol. 140, pp. 104– 109, 2017

    work page 2017

  45. [45]

    Mean-field theory for scale- free random networks,

    A.-L. Barab ´asi, R. Albert, and H. Jeong, “Mean-field theory for scale- free random networks,”Physica A, vol. 272, no. 1-2, pp. 173–187, 1999

    work page 1999

  46. [46]

    The variational approximation for Bayesian inference,

    D. G. Tzikas, A. C. Likas, and N. P. Galatsanos, “The variational approximation for Bayesian inference,”IEEE Signal Process. Mag., vol. 25, no. 6, pp. 131–136, 2008

    work page 2008

  47. [47]

    B. G. Osgood,Lectures on F ourier Transform and Its Applications. American Mathematical Soc., 2019, vol. 33

    work page 2019

  48. [48]

    Diffusion models beat GANs on image synthesis,

    P. Dhariwal and A. Nichol, “Diffusion models beat GANs on image synthesis,”Adv. Neural Inf. Process. Syst., vol. 34, pp. 8780–8794, 2021

    work page 2021

  49. [49]

    Generalization error in deep learning,

    D. Jakubovitz, R. Giryes, and M. R. D. Rodrigues, “Generalization error in deep learning,” inCompressed Sens. Its Appl.Springer, 2019, pp. 153–193

    work page 2019

  50. [50]

    3GPP tr 38.901 channel model,

    Q. Zhu, C.-X. Wang, B. Hua, K. Mao, S. Jiang, and M. Yao, “3GPP tr 38.901 channel model,” inthe Wiley 5G Ref: the essential 5G reference online. Wiley, 2021, pp. 1–35

    work page 2021

  51. [51]

    On the asymptotic mean square error optimality of diffusion models,

    B. Fesl, B. B ¨ock, F. Strasser, M. Baur, M. Joham, and W. Utschick, “On the asymptotic mean square error optimality of diffusion models,” arXiv preprint arXiv:2403.02957, 2024

    work page arXiv 2024

  52. [52]

    Regularized least-squares,

    R. Rifkin, G. Yeo, and T. Poggio, “Regularized least-squares,”Adv. Learn. Theory: Methods, Models Appl., vol. 190, p. 131, 2003

    work page 2003

  53. [53]

    Compressed sensing using generative models,

    A. Bora, A. Jalal, E. Price, and A. G. Dimakis, “Compressed sensing using generative models,” inInt. Conf. Mach. Learn., 2017, pp. 537– 546

    work page 2017