Pith. sign in

REVIEW 3 major objections 5 minor 59 references

Radio Map Estimation via Latent Domain Plug-and-Play Denoising

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Without radio-map training data, latent-domain PnP reconstructs full spatio-spectral radio maps from sparse sensor measurements.

desk verdict Latent-domain PnP for radio maps is a solid engineering contribution, but the recoverability proof has an unstated range-space assumption that should be fixed before the theory is trusted. read the letter →

arxiv 2501.13472 v2 pith:J3VI7JDP submitted 2025-01-23 eess.SP cs.LG

Le Xu , Lei Cheng , Junting Chen , Wenqiang Pu , Xiao Fu This is my paper · ORCID
classification eess.SPcs.LG
keywords Radiomapestimationspectrumcartographyplug-and-playdenoisingADMMlatentdomaintensorcompletionrecoverabilityanalysistraining-free
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper claims that a full spatio-spectral radio map can be reconstructed from sparse sensor measurements with no radio-map training data, by running a plug-and-play (PnP) ADMM algorithm that denoises only the latent spatial factors of the map. The key move is to factor the map as a sum of outer products of spatial loss fields and emitter power spectra, $\mathcal{X} = \sum_{r=1}^{R} \mathcal{S}_r \circ \mathbf{c}_r$, and let a grayscale natural-image denoiser act as the implicit regularizer on each $\mathcal{S}_r$. This latent-domain design needs only $R$ denoising calls per iteration instead of one per frequency band, has better noise tolerance than data-domain PnP, and avoids the training cost of deep-learning radio-map estimators. The paper proves that for linear symmetric denoisers every optimal solution is close to the ground truth with a bound that shrinks as the number of sensors grows, and that the ADMM iterates converge to a fixed point, or to a KKT point when the denoiser is linear. If correct, this makes sophisticated image denoisers directly transferable to spectrum cartography without retraining.

What carries the argument

The central object is the latent-factor decomposition $\mathcal{X}(m,n,k) = \sum_{r=1}^{R} \mathcal{S}_r(m,n) \mathbf{c}_r(k)$, where $\mathcal{S}_r$ is the spatial loss field of emitter $r$ and $\mathbf{c}_r$ is its power spectral density; this converts the 3D radio map into a small set of 2D images plus spectra. Around this, the paper builds an ADMM whose $\mathcal{Z}_r$ update is literally a plug-and-play grayscale image denoiser applied to $\mathcal{S}_r + \Psi_r$, so the entire prior is supplied by an off-the-shelf denoiser such as DSG-NLM, BM3D, or DRUnet. The theoretical engine is Lemma 1, which identifies such denoisers with proximal operators of explicit quadratic regularizers when the denoising matrix is symmetric and satisfies a Perron-Frobenius-type spectral condition; that lemma turns the black-box algorithm into a tractable optimization problem and enables both the sample-complexity bound and the KKT characterization.

What would settle it

Generate a radio map whose shadowing fields differ across frequency bands (so the tensor cannot be written as $\sum_r \mathcal{S}_r \circ \mathbf{c}_r$ with one set of spatial fields), sample it at increasing rates with noiseless measurements, and run LaPnP with the true $R$: if the relative error does not monotonically decrease toward zero as the sampling rate grows, the exact-factorization assumption is indispensable and the central claim fails for such maps.

Watch

Extended reading notes

Core claim

This paper establishes that plug-and-play denoising can be carried out in the latent factor space of a radio map rather than in the data domain, and that this change is not merely a computational trick but preserves, and in some cases strengthens, theoretical guarantees. The authors model the radio map as $\mathcal{X} = \sum_{r=1}^{R} \mathcal{S}_r \circ \mathbf{c}_r$, where $\mathcal{S}_r$ is the spatial loss field (an $M \times N$ image) and $\mathbf{c}_r$ is the frequency spectrum of emitter $r$. Their LaPnP algorithm solves the resulting nonconvex problem by ADMM whose $\mathcal{Z}_r$ update is exactly a denoising step applied to $\mathcal{S}_r + \Psi_r$, allowing any grayscale image denoiser to serve as the prior. Under a spectral condition on the denoising matrix, the implicit denoiser-regularization is identified with an explicit quadratic regularizer, which lets them prove that every optimal solution of the latent formulation lies within a bound of the ground truth that decreases with the number of sensors, and that the ADMM iterates converge to a fixed point, or a KKT point for linear denoisers. The reported experiments with statistical simulations, ray-traced maps, and real indoor measurements show the method matching deep-learning baselines that were trained on hundreds of thousands of radio-map examples.

Load-bearing premise

The load-bearing premise is that the true radio map is exactly a sum of $R$ outer products of a nonnegative spatial field and a nonnegative frequency profile, with $R$ known; if real shadowing is frequency-selective or $R$ is wrong, the denoiser regularizes a latent structure that does not match the data, and the recoverability and convergence guarantees no longer apply.

Editorial extensions

If this is right

  • If the central claim is correct, spectrum cartography can deploy sophisticated image denoisers with zero radio-map training, allowing instant adaptation when a sensing scenario changes.
  • The recoverability bound (Theorem 1) gives an RIP-free sample-complexity statement for PnP-based radio map estimation: for linear symmetric denoisers, the estimation error is controlled by the number of samples $|\Omega|$, the noise, and a gap term that shrinks with more measurements.
  • The convergence results mean practitioners can trust the ADMM output to be a fixed point, and with linear denoisers actually a KKT point of the latent-regularized problem.
  • Because only $R$ denoising calls are needed per iteration instead of the number of frequency bands $K$, the method becomes substantially cheaper than data-domain PnP, especially when the number of frequency bands is large.
  • Overestimating the number of emitters $R$ degrades performance only mildly, while underestimating $R$ causes a clear drop, so calibration can err on the side of larger $R$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the exact-factorization premise holds only approximately, the same latent-domain scheme might still work with an overestimated $R$; the paper's own sensitivity experiment hints at this, but it is not proven in the paper.
  • The latent-domain idea should transfer to other tensor-completion problems whose physics provides a natural outer-product structure, such as hyperspectral imaging with endmember and abundance factors, or multi-channel biomedical arrays.
  • A testable extension would be to replace the fixed $R$ with an automatic rank-selection rule, since the recoverability analysis assumes $R$ is known.
  • The theory is established for linear symmetric denoisers; the positive results with BM3D and DRUnet are not yet supported by the same recoverability guarantee, so extending the explicit-regularizer argument to nonlinear denoisers is a natural next step.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper proposes LaPnP, a plug-and-play ADMM algorithm for radio map estimation. It models the spatio-spectral radio map as a sum of R outer products of spatial loss fields (SLFs) and emitter PSDs, applies a grayscale image denoiser to the SLFs in the latent domain, and uses a fixed number J of HALS inner iterations for the factor updates. The authors provide a recoverability analysis for linear denoisers (Theorem 1), a fixed-point convergence result (Theorem 2), and a KKT convergence result for linear denoisers (Theorem 3). Experiments on statistical-model data, ray-tracing data, and the Mannheim real-world dataset compare LaPnP with interpolation, tensor-decomposition, learned, and data-domain PnP baselines, and show competitive accuracy with lower runtime and no radio-map training data.

Significance. If the theoretical results are made sound, the contribution is significant: LaPnP offers a training-free RME method that leverages powerful natural-image denoisers in a latent domain, reducing the number of denoising operations per ADMM iteration and improving noise robustness. The empirical study is fairly extensive, covering multiple datasets, sampling rates, shadowing levels, SNRs, and a sensitivity analysis for the number of emitters, and the code is publicly released. The recoverability and convergence results would address an important gap in PnP theory for tensor-structured inverse problems. However, the two theoretical gaps identified below affect the central formal claims and need to be resolved before the paper can be accepted.

major comments (3)
  1. [Sec. IV-A, Lemma 2, Eq. (34), Theorem 1] Lemma 2 and Theorem 1 are not supported as stated because v♮_obj in Eq. (34) is infinite whenever a ground-truth SLF S♮_r is not in the range of Q~_r. Lemma 1 (Eq. (25)) defines r(·) with the indicator 1[z∈R(Q~)], and the explicit problem (26) therefore includes the constraint sr ∈ R(Q~_r). However, Eq. (34) evaluates vobj at {S♮_r,c♮_r} using only the quadratic term plus ζ∥c♮∥² and the noise term, omitting this constraint. Since Assumption 1 permits zero eigenvalues of W, R(Q~_r) can be a proper subspace; for example, the rank-one projection W=(1/MN)11^T satisfies Assumption 1 but maps every image to its constant mean, so any non-constant S♮_r gives v♮_obj=+∞ and the bounds (27a), (27b), and (30) are vacuous. The theorem needs an explicit assumption that S♮_r∈R(Q~_r) for all r (or that W is full rank), and the same issue propagates into Lemma 2 and Theorem 1.
  2. [Algorithm 1, Sec. IV-B, Appendix C (Eqs. (57)–(59))] The proof of Theorem 2 does not cover Algorithm 1 with J>1 HALS inner iterations. In Appendix C, the update of sr is analyzed through the single stationary equation (57), which corresponds to one exact block-coordinate minimization of the subproblem (17), and Eq. (59) then bounds s_r^{(t+1)}−s_r^{(t)} using that equation. But Algorithm 1 runs J=20 inner HALS sweeps before the outer iteration advances; the final s_r^{(t+1)} is the output of J block-coordinate steps, and its relation to s_r^{(t)} involves gradients evaluated at intermediate inner iterates, not at the fixed point of a single stationary equation. Thus the displayed bound (59) is not established for the actual algorithm. The authors should either state and prove Theorem 2 for one HALS step and then handle J>1 by a separate sub-iteration argument, or modify the proof to account for the cumulative effect of the J inner steps.
  3. [Sec. IV-B, Assumption 3] Assumption 3 is stated as a boundedness condition on partial gradients of f, but the only justification given ('a large error term will be penalized by f... while large values of cr will be prevented by ζc^T c') is not a proof and does not follow for the iterates of Algorithm 1, which are not shown to remain in a bounded set. Since the fixed-point convergence of Theorem 2 directly relies on the uniform bound L in Eq. (32), the theorem is at present conditional on an unverified property of the algorithm's trajectory. The authors should either prove that the iterates satisfy Assumption 3, or replace it by a verifiable condition such as boundedness of {c_r^{(t)}} and of the measured residuals.
minor comments (5)
  1. [Sec. IV-A, Theorem 1] The probability space for the statement 'with probability 1−δ' is not specified; please state that Ω is drawn uniformly at random without replacement with fixed cardinality |Ω| and that the probability is over this choice.
  2. [Sec. III-B, Algorithm 1] The order of the HALS updates (18)–(20) is not specified; state whether sr(Ωvec), cr, and sr(Ω∁) are updated cyclically within each of the J inner iterations.
  3. [Sec. V-D] The Mannheim experiment sets R=7 for all methods following [10,11]; since the true number of emitters is unknown, a sensitivity analysis for R on the real data would help interpret the comparison.
  4. [Lemma 1, Eq. (24)] The variables E and Z in Lemma 1 are used before their relation to the denoising argument is made explicit; please define E and Z clearly.
  5. [Appendix B, Eq. (51)] The use of Serfling's inequality for sampling without replacement should include the exact statement or a citation with conditions, since the proof relies on it for the high-probability bound.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; the central derivation is self-contained, with only minor non-load-bearing self-citations.

full rationale

The proposed LaPnP estimator, its recoverability bound, and its convergence results are derived from the algorithm's own updates and from standard external characterizations of linear denoisers, rather than from fitted parameters or from the authors' prior results. The spatio-spectral model X = sum_r S_r o c_r is presented as a physical model and is supported by a mix of external and prior references; the load-bearing regularization identity in Lemma 1 comes from the external line of work [27,28,32-34]. Theorem 1 is an oracle-style bound: v♮_obj is the objective evaluated at the ground truth, not a fitted or predicted quantity, and the gap/covering-number argument is a standard uniform-convergence calculation. No parameter is fit to a subset and then renamed a prediction. Self-citations such as [10,11,36] supply initialization, experimental context, and lineage, but none is invoked to forbid alternatives or to carry a central premise. One non-circular correctness concern is that Eq. (34) drops the indicator in Eq. (25), so Lemma 2 and Theorem 1 implicitly require S♮_r in R(Q~_r); if that fails, v♮_obj is infinite and the bound is unsupported. This is a feasibility gap, not circularity. Accordingly the circularity score is 1.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The method introduces no new physical entities. It relies on a standard low-rank factorization model and the PnP prior-transfer assumption. Several hyperparameters are hand-chosen or adapted during optimization, and the theory uses stated assumptions that are not always verified for the nonlinear denoisers used in practice.

free parameters (5)
  • R (number of emitters) = Set to ground truth in synthetic experiments; R=7 for Mannheim data
    Model order for the spatio-spectral factorization; method is sensitive to underestimation of R.
  • lambda (regularization on SLFs) = Not reported
    Regularization weight in the objective (10); hand-chosen, affects denoiser noise level via sqrt(lambda/rho).
  • zeta (regularization on PSDs) = Not reported
    Regularization weight on c_r in (10); appears in the recoverability bounds.
  • rho (ADMM penalty) = Adaptively updated with gamma=1.1; initial value not reported
    Augmented Lagrangian parameter; influences step sizes and theoretical bounds.
  • J (HALS inner iterations) = 20
    Fixed number of HALS updates per ADMM iteration; the convergence proof does not fully account for the resulting inexactness.
assumptions (6)
  • domain assumption The radio map satisfies the rank-R spatio-spectral decomposition X = sum S_r ◦ c_r with nonnegative factors (Eq. 9).
    Central to the latent-domain design and all theoretical analysis; stated in Sec. III-A.
  • domain assumption Natural-image denoisers are effective regularizers for radio-map SLFs.
    The core PnP postulate; empirically validated but not theoretically justified for nonlinear denoisers.
  • ad hoc to paper Assumption 2: the denoiser is bounded, ||D_sigma(X)-X||^2/(MN) <= sigma^2 C.
    Used to prove Theorem 2; may not hold for deep denoisers used in the experiments.
  • ad hoc to paper Assumption 3: partial gradients of f are bounded.
    Used to prove Theorem 2 and Theorem 3; the paper argues it is generally valid but does not verify it for specific data.
  • domain assumption Assumption 1: linear denoisers have nonnegative, symmetric, irreducible matrices with eigenvalue constraints.
    Used in Lemma 2 and Theorem 1; satisfied by several classical denoisers but not by BM3D or deep denoisers.
  • standard math Standard concentration inequalities and covering-number bounds.
    Used in Appendix B for the sample-complexity result; standard results from [58],[59].

how reviews work

0 comments
Cite this review

Pith. "Pith review of Radio Map Estimation via Latent Domain Plug-and-Play Denoising." pith.science (2026). https://pith.science/paper/J3VI7JDP

@misc{pith2026250113472,
  author       = {Pith},
  title        = {Pith review of: Radio Map Estimation via Latent Domain Plug-and-Play Denoising},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/J3VI7JDP}},
  note         = {Machine review of arXiv:2501.13472}
}
read the original abstract

Radio map estimation (RME), also known as spectrum cartography, aims to reconstruct the strength of radio interference across different domains (e.g., space and frequency) from sparsely sampled measurements. To tackle this typical inverse problem, state-of-the-art RME methods rely on handcrafted or data-driven structural information of radio maps. However, the former often struggles to model complex radio frequency (RF) environments and the latter requires excessive training -- making it hard to quickly adapt to in situ sensing tasks. This work presents a spatio-spectral RME approach based on plug-and-play (PnP) denoising, a technique from computational imaging. The idea is to leverage the observation that the denoising operations of signals like natural images and radio maps are similar -- despite the nontrivial differences of the signals themselves. Hence, sophisticated denoisers designed for or learned from natural images can be directly employed to assist RME, avoiding using radio map data for training. Unlike conventional PnP methods that operate directly in the data domain, the proposed method exploits the underlying physical structure of radio maps and proposes an ADMM algorithm that denoises in a latent domain. This design significantly improves computational efficiency and enhances noise robustness. Theoretical aspects, e.g., recoverability of the complete radio map and convergence of the ADMM algorithm are analyzed. Synthetic and real data experiments are conducted to demonstrate the effectiveness of our approach.

Figures

Figures reproduced from arXiv: 2501.13472 by the authors.

Figure 1
Figure 1. Illustration of the RME task. Upper: the spatio-spectral [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Illustration of the decomposition model of radio maps. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Illustration of matricization of the measurements. Left: [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Recovered radio maps under the 10-th frequency bin; dc = 50, τ = 10% transformations on linear denoisers. For the baselines Nasdac and Dowjons, training is per￾formed using 500,000 SLFs generated by (33). Each SLF is generated by uniformly sampling the parameters γ ∈ […
Figure 5
Figure 5. Figure 5: Recovered radio maps in the 10-th frequency bin; τ = 5%, R = 3. TABLE IV: Performance under RTM and different τ ; R = 3. RSE τ LL1 TPS NN LaPnP-NLM LaPnP-DRU LaPnP-BM3D 5% 0.911 0.907 0.979 0.907 0.873 0.871 10% 0.782 0.777 0.807 0.689 0.680 0.634 15% 0.725 0.726 0.771…
Figure 6
Figure 6. Figure 6: Performance under RTM and different estimated [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

59 extracted references · 39 canonical work pages

  1. [1]

    A tutorial on environment-aware communications via channel knowledge map for 6G,

    Y . Zeng, J. Chen, J. Xu, D. Wu, X. Xu, S. Jin, X. Gao, D. Gesbert, S. Cui, and R. Zhang, “A tutorial on environment-aware communications via channel knowledge map for 6G,” IEEE Commun. Surveys Tuts. , 2024

    work page 2024

  2. [2]

    Engineering radio maps for wireless resource management,

    S. Bi, J. Lyu, Z. Ding, and R. Zhang, “Engineering radio maps for wireless resource management,” IEEE Wireless Commun., vol. 26, no. 2, pp. 133–141, 2019

    work page 2019

  3. [3]

    Radio map estimation: A data-driven approach to spectrum cartography,

    D. Romero and S.-J. Kim, “Radio map estimation: A data-driven approach to spectrum cartography,” IEEE Signal Process. Mag., vol. 39, no. 6, pp. 53–72, 2022

    work page 2022

  4. [4]

    Group-lasso on splines for spectrum cartography,

    J. A. Bazerque, G. Mateos, and G. B. Giannakis, “Group-lasso on splines for spectrum cartography,” IEEE Trans. Signal Process., vol. 59, no. 10, pp. 4648–4663, 2011

    2011

  5. [5]

    Cooperative spectrum sensing for cognitive radios using kriged kalman filtering,

    S.-J. Kim, E. Dall’Anese, and G. B. Giannakis, “Cooperative spectrum sensing for cognitive radios using kriged kalman filtering,” IEEE J. Sel. Topics Signal Process., vol. 5, no. 1, pp. 24–36, 2010

    2010

  6. [6]

    Wire- less sensor network for spectrum cartography based on kriging inter- polation,

    G. Boccolini, G. Hern ´andez-Pe˜naloza, and B. Beferull-Lozano, “Wire- less sensor network for spectrum cartography based on kriging inter- polation,” in Proc. IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC) , 2012

    work page 2012

  7. [7]

    Spectrum cartography via coupled block-term tensor decomposition,

    G. Zhang, X. Fu, J. Wang, X.-L. Zhao, and M. Hong, “Spectrum cartography via coupled block-term tensor decomposition,” IEEE Trans. Signal Process., vol. 68, pp. 3660–3675, 2020

    2020

  8. [8]

    Integrated interpolation and block-term tensor decomposition for spectrum map construction,

    H. Sun and J. Chen, “Integrated interpolation and block-term tensor decomposition for spectrum map construction,” IEEE Trans. Signal Process., vol. 72, pp. 3896–3911, 2024

    2024

Show all 59 references
  1. [9]

    Propagation map reconstruction via interpolation assisted matrix completion,

    ——, “Propagation map reconstruction via interpolation assisted matrix completion,” IEEE Trans. Signal Process., vol. 70, pp. 6154–6169, 2022

    2022

  2. [10]

    Deep spectrum cartography: com- pleting radio map tensors using learned neural models,

    S. Shrestha, X. Fu, and M. Hong, “Deep spectrum cartography: com- pleting radio map tensors using learned neural models,” IEEE Trans. Signal Process., vol. 70, pp. 1170–1184, 2022

    2022

  3. [11]

    Quantized radio map estimation using tensor and deep generative models,

    S. Timilsina, S. Shrestha, and X. Fu, “Quantized radio map estimation using tensor and deep generative models,” IEEE Trans. Signal Process., vol. 72, pp. 173–189, 2024

    2024

  4. [12]

    RadioUnet: Fast radio map estimation with convolutional neural networks,

    R. Levie, C ¸ . Yapar, G. Kutyniok, and G. Caire, “RadioUnet: Fast radio map estimation with convolutional neural networks,” IEEE Trans. Wireless Commun., vol. 20, no. 6, pp. 4001–4015, 2021

    2021

  5. [13]

    Deep completion autoencoders for radio map estimation,

    Y . Teganya and D. Romero, “Deep completion autoencoders for radio map estimation,” IEEE Trans. Wireless Commun. , vol. 21, no. 3, pp. 1710–1724, 2021

    2021

  6. [14]

    Deep-learning-based radio map reconstruction for V2X communications,

    S. Roger, M. Brambilla, B. C. Tedeschini, C. Botella-Mascarell, M. Co- bos, and M. Nicoli, “Deep-learning-based radio map reconstruction for V2X communications,” IEEE Trans. Veh. Technol. , vol. 73, no. 3, pp. 3863–3871, 2024

    2024

  7. [15]

    Plug-and-play methods for integrating physical and learned models in computational imaging: Theory, algorithms, and applications,

    U. S. Kamilov, C. A. Bouman, G. T. Buzzard, and B. Wohlberg, “Plug-and-play methods for integrating physical and learned models in computational imaging: Theory, algorithms, and applications,” IEEE Signal Process. Mag. , vol. 40, no. 1, pp. 85–97, 2023

    2023

  8. [16]

    Plug-and-play priors for model based reconstruction,

    S. V . Venkatakrishnan, C. A. Bouman, and B. Wohlberg, “Plug-and-play priors for model based reconstruction,” inProc. IEEE Global Conference on Signal and Information Processing (GlobalSIP) , 2013

    2013

  9. [17]

    Zero-shot reconstruction of ocean sound speed field tensors: A deep plug-and-play approach,

    S. Li, L. Cheng, X. Fu, and J. Li, “Zero-shot reconstruction of ocean sound speed field tensors: A deep plug-and-play approach,” The Journal of the Acoustical Society of America , vol. 155, no. 5, pp. 3475–3489, 2024

    2024

  10. [18]

    Hy- perspectral image restoration by tensor fibered rank constrained opti- mization and plug-and-play regularization,

    Y .-Y . Liu, X.-L. Zhao, Y .-B. Zheng, T.-H. Ma, and H. Zhang, “Hy- perspectral image restoration by tensor fibered rank constrained opti- mization and plug-and-play regularization,” IEEE Trans. Geosci. Remote Sens., vol. 60, pp. 1–17, 2022

    2022

  11. [19]

    Tuning-free plug-and-play proximal algorithm for inverse imaging problems,

    K. Wei, A. Aviles-Rivero, J. Liang, Y . Fu, C.-B. Sch ¨onlieb, and H. Huang, “Tuning-free plug-and-play proximal algorithm for inverse imaging problems,” in Proc. International Conference on Machine Learning (ICML). PMLR, 2020

    2020

  12. [20]

    Rare: Image reconstruction using deep priors learned without groundtruth,

    J. Liu, Y . Sun, C. Eldeniz, W. Gan, H. An, and U. S. Kamilov, “Rare: Image reconstruction using deep priors learned without groundtruth,” IEEE J. Sel. Topics Signal Process. , vol. 14, no. 6, pp. 1088–1099, 2020

    2020

  13. [21]

    A non-local algorithm for image denoising,

    A. Buades, B. Coll, and J.-M. Morel, “A non-local algorithm for image denoising,” in Proc. IEEE Conference on Computer Vision and Pattern Recognition (CVPR). IEEE, 2005

    2005

  14. [22]

    Image denoising by sparse 3-D transform-domain collaborative filtering,

    K. Dabov, A. Foi, V . Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-D transform-domain collaborative filtering,” IEEE Trans. Image Process., vol. 16, no. 8, pp. 2080–2095, 2007

    2007

  15. [23]

    FFDNet: Toward a fast and flexible solution for CNN-based image denoising,

    K. Zhang, W. Zuo, and L. Zhang, “FFDNet: Toward a fast and flexible solution for CNN-based image denoising,” IEEE Trans. Image Process., vol. 27, no. 9, pp. 4608–4622, 2018

    2018

  16. [24]

    Plug-and-play image restoration with deep denoiser prior,

    K. Zhang, Y . Li, W. Zuo, L. Zhang, L. Van Gool, and R. Timofte, “Plug-and-play image restoration with deep denoiser prior,” IEEE Trans. Pattern Anal. Mach. Intell. , vol. 44, no. 10, pp. 6360–6376, 2021

    2021

  17. [25]

    Plug-and-play ADMM for image restoration: Fixed-point convergence and applications,

    S. H. Chan, X. Wang, and O. A. Elgendy, “Plug-and-play ADMM for image restoration: Fixed-point convergence and applications,” IEEE Trans. Comput. Imag., vol. 3, no. 1, pp. 84–98, 2016

    2016

  18. [26]

    Plug-and- play methods provably converge with properly trained denoisers,

    E. Ryu, J. Liu, S. Wang, X. Chen, Z. Wang, and W. Yin, “Plug-and- play methods provably converge with properly trained denoisers,” in Proc. International Conference on Machine Learning (ICML) . PMLR, 2019

    2019

  19. [27]

    Plug-and-play priors for bright field electron tomography and sparse interpolation,

    S. Sreehari, S. V . Venkatakrishnan, B. Wohlberg, G. T. Buzzard, L. F. Drummy, J. P. Simmons, and C. A. Bouman, “Plug-and-play priors for bright field electron tomography and sparse interpolation,” IEEE Trans. Comput. Imag., vol. 2, no. 4, pp. 408–423, 2016

    2016

  20. [28]

    A con- vergent image fusion algorithm using scene-adapted Gaussian-mixture- based denoising,

    A. M. Teodoro, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “A con- vergent image fusion algorithm using scene-adapted Gaussian-mixture- based denoising,” IEEE Trans. Image Process., vol. 28, no. 1, pp. 451– 463, 2019

    2019

  21. [29]

    Distributed spectrum sensing for cognitive radio networks by exploiting sparsity,

    J. A. Bazerque and G. B. Giannakis, “Distributed spectrum sensing for cognitive radio networks by exploiting sparsity,” IEEE Trans. Signal Process., vol. 58, no. 3, pp. 1847–1862, 2009

    2009

  22. [30]

    Learn- ing power spectrum maps from quantized power measurements,

    D. Romero, S.-J. Kim, G. B. Giannakis, and R. L ´opez-Valcarce, “Learn- ing power spectrum maps from quantized power measurements,” IEEE Trans. Signal Process., vol. 65, no. 10, pp. 2547–2560, 2017

    2017

  23. [31]

    Channel gain cartography for cognitive radios leveraging low rank and sparsity,

    D. Lee, S.-J. Kim, and G. B. Giannakis, “Channel gain cartography for cognitive radios leveraging low rank and sparsity,” IEEE Trans. Wireless Commun., vol. 16, no. 9, pp. 5953–5966, 2017

    2017

  24. [32]

    On exact and robust recovery for plug-and-play compressed sensing,

    R. G. Gavaskar, C. D. Athalye, and K. N. Chaudhury, “On exact and robust recovery for plug-and-play compressed sensing,” Signal Processing, vol. 211, p. 109100, 2023

    2023

  25. [33]

    On plug-and-play regularization using linear denoisers,

    ——, “On plug-and-play regularization using linear denoisers,” IEEE Trans. Image Process., vol. 30, pp. 4802–4813, 2021

    2021

  26. [34]

    Performance analysis of plug-and-play ADMM: A graph signal processing perspective,

    S. H. Chan, “Performance analysis of plug-and-play ADMM: A graph signal processing perspective,” IEEE Trans. Comput. Imag., vol. 5, no. 2, pp. 274–286, 2019

    2019

  27. [35]

    Recovery analysis for plug-and-play priors using the restricted eigenvalue condition,

    J. Liu, S. Asif, B. Wohlberg, and U. Kamilov, “Recovery analysis for plug-and-play priors using the restricted eigenvalue condition,” in Proc. Advances in Neural Information Processing Systems (NeuIPS) , 2021. 12

    2021

  28. [36]

    Radio map estimation via latent-domain plug-and-play denoisers,

    L. Xu, L. Cheng, J. Chen, W. Pu, and X. Fu, “Radio map estimation via latent-domain plug-and-play denoisers,” in Proc. IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2025

    2025

  29. [37]

    Goldsmith, Wireless Communications

    A. Goldsmith, Wireless Communications. Cambridge University Press, 2005

    2005

  30. [38]

    Sionna: An open-source library for next-generation physical layer research,

    J. Hoydis, S. Cammerer, F. Ait Aoudia, A. Vem, N. Binder, G. Marcus, and A. Keller, “Sionna: An open-source library for next-generation physical layer research,” arXiv preprint, Mar. 2022

    2022

  31. [39]

    A method to reconstruct coverage loss maps based on matrix completion and adaptive sampling,

    S. Chouvardas, S. Valentin, M. Draief, and M. Leconte, “A method to reconstruct coverage loss maps based on matrix completion and adaptive sampling,” in Proc. IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) , 2016

    2016

  32. [40]

    Distributed optimization and statistical learning via the alternating direction method of multipliers,

    S. Boyd, N. Parikh, E. Chu, B. Peleato, J. Eckstein et al., “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Foundations and Trends® in Machine Learning , vol. 3, no. 1, pp. 1–122, 2011

    2011

  33. [41]

    Multi-slice fusion for sparse-view and limited-angle 4d ct reconstruc- tion,

    S. Majee, T. Balke, C. A. J. Kemp, G. T. Buzzard, and C. A. Bouman, “Multi-slice fusion for sparse-view and limited-angle 4d ct reconstruc- tion,” IEEE Tran. Comput. Imag. , vol. 7, pp. 448–462, 2021

    2021

  34. [42]

    A tour of modern image filtering: New insights and methods, both practical and theoretical,

    P. Milanfar, “A tour of modern image filtering: New insights and methods, both practical and theoretical,” IEEE Signal Process. Mag. , vol. 30, no. 1, pp. 106–128, 2013

    2013

  35. [43]

    Szeliski, Computer vision: algorithms and applications

    R. Szeliski, Computer vision: algorithms and applications . Springer Nature, 2022

    2022

  36. [44]

    Solving inverse problems with piecewise linear estimators: From gaussian mixture models to structured sparsity,

    G. Yu, G. Sapiro, and S. Mallat, “Solving inverse problems with piecewise linear estimators: From gaussian mixture models to structured sparsity,” IEEE Trans. Image Process. , vol. 21, no. 5, pp. 2481–2499, 2011

    2011

  37. [45]

    Denoising: A powerful building-block for imaging, inverse problems, and machine learning,

    P. Milanfar and M. Delbracio, “Denoising: A powerful building-block for imaging, inverse problems, and machine learning,” arXiv preprint arXiv:2409.06219, 2024

    2024 arXiv

  38. [46]

    Nonnegative matrix and tensor factorizations: An algorithmic perspective,

    G. Zhou, A. Cichocki, Q. Zhao, and S. Xie, “Nonnegative matrix and tensor factorizations: An algorithmic perspective,” IEEE Signal Process. Mag., vol. 31, no. 3, pp. 54–65, 2014

    2014

  39. [47]

    Fast and robust recursive algorithmsfor separable nonnegative matrix factorization,

    N. Gillis and S. A. Vavasis, “Fast and robust recursive algorithmsfor separable nonnegative matrix factorization,” IEEE Trans. Pattern Anal. Mach. Intell., vol. 36, no. 4, pp. 698–714, 2014

    2014

  40. [48]

    Self-dictionary sparse regression for hyperspectral unmixing: Greedy pursuit and pure pixel search are related,

    X. Fu, W.-K. Ma, T.-H. Chan, and J. M. Bioucas-Dias, “Self-dictionary sparse regression for hyperspectral unmixing: Greedy pursuit and pure pixel search are related,” IEEE J. Sel. Topics Signal Process. , vol. 9, no. 6, pp. 1128–1141, 2015

    2015

  41. [49]

    Fast alternat- ing direction optimization methods,

    T. Goldstein, B. O’Donoghue, S. Setzer, and R. Baraniuk, “Fast alternat- ing direction optimization methods,” SIAM Journal on Imaging Sciences, vol. 7, no. 3, pp. 1588–1623, 2014

    2014

  42. [50]

    R. A. Horn and C. R. Johnson, Matrix analysis. Cambridge university press, 2012

    2012

  43. [51]

    Berman and R

    A. Berman and R. J. Plemmons, Nonnegative matrices in the mathemat- ical sciences. SIAM, 1994

    1994

  44. [52]

    Stability of matrix factorization for collabora- tive filtering,

    Y .-X. Wang and H. Xu, “Stability of matrix factorization for collabora- tive filtering,” in Proc. International Conference on Machine Learning (ICML). PMLR, 2012

    2012

  45. [53]

    Image quality assessment: from error visibility to structural similarity,

    Z. Wang, A. Bovik, H. Sheikh, and E. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process., vol. 13, no. 4, pp. 600–612, 2004

    2004

  46. [54]

    Ray tracing for radio propagation modeling: Principles and applications,

    Z. Yun and M. F. Iskander, “Ray tracing for radio propagation modeling: Principles and applications,” IEEE Access, vol. 3, pp. 1089–1100, 2015

    2015

  47. [55]

    Dataset of pathloss and ToA radio maps with localization application,

    C ¸ . Yapar, R. Levie, G. Kutyniok, and G. Caire, “Dataset of pathloss and ToA radio maps with localization application,” arXiv preprint:2212.11777, 2022. [Online]. Available: https://arxiv.org/abs/ 2212.11777

    2022 arXiv

  48. [56]

    U-net: Convolutional networks for biomedical image segmentation,

    O. Ronneberger, P. Fischer, and T. Brox, “U-net: Convolutional networks for biomedical image segmentation,” in Proc. Medical Image Computing and Computer-Assisted Intervention (MICCAI) , 2015

    2015

  49. [57]

    CRAWDAD dataset mannheim/compass (v. 2008-04- 11),

    T. King, S. Kopf, T. Haenselmann, C. Lubberger, and W. Ef- felsberg, “CRAWDAD dataset mannheim/compass (v. 2008-04- 11),” 2008, [Online]. Available: https://ieee-dataport.org/open-access/ crawdad-mannheimcompass-v-2008-04-11

    2008

  50. [58]

    The covering number in learning theory,

    D.-X. Zhou, “The covering number in learning theory,” Journal of Complexity, vol. 18, no. 3, pp. 739–767, 2002

    2002

  51. [59]

    Shalev-Shwartz and S

    S. Shalev-Shwartz and S. Ben-David, Understanding machine learning: From theory to algorithms . Cambridge university press, 2014. 13 Supplementary Material APPENDIX A PROOF OF LEMMA 2 As discussed in Section IV-A, the problem to solve becomes (26) under Assumption 1, in which ...

    2014

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.