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Average Kernel Sizes -- Computable Sharp Accuracy Bounds for Inverse Problems

T0 review · 3 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read This paper establishes that half the average kernel size is a sharp, computable lower bound on the reconstruction error of any approximate inverse map, with the full kernel size as the matching upper bound.

desk verdict A useful equal-size bound buried under a false universal claim: Theorem 3.3(ii) fails when feasible sets have unequal sizes. read the letter →

arxiv 2510.10229 v2 pith:IGU2KOTW submitted 2025-10-11 math.OC

Nina M. Gottschling , David Iagaru , Jakob Gawlikowski , Ioannis Sgouralis This is my paper
classification math.OC MSC 65J2265J20
keywords inverseproblemsaccuracyboundsaveragekernelsizefeasiblesetsreconstructionerrorsharpsuper-resolutionlocalizationmicroscopy
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's central claim is that for any inverse problem with a non-injective forward model—where several possible signals can produce the same noisy measurement—the minimal achievable reconstruction error on a dataset is tightly controlled by the 'average kernel size': the typical distance among signals that map to the same measurement. Specifically, Theorem 3.3 states that half this quantity is a sharp lower bound for any approximate inverse map, and the full quantity is an upper bound for maps that reach the accuracy limit. The bounds are computable from samples of signals, noise, and the forward model alone, requiring no training or method choice. If correct, this gives scientists a way to know the accuracy floor of a benchmark dataset before designing a reconstruction method, and to identify which methods are already optimal. The authors validate the bounds on fluorescence localization microscopy and satellite-image super-resolution, and note in Section 6.2 that the bounds currently apply only to the dataset produced by their sampling algorithm, with convergence for new data left open.

What carries the argument

The central object is the average kernel size, Kersize_p(F,M1,E,p)_K = (1/K Σ_k (1/N(k)^2) Σ_{n,n'} ||x_{k,n}-x_{k,n'}||^p)^{1/p}, the p-th mean of pairwise distances inside each feasible set of signals that map to the same measurement. It generalizes the kernel/nullspace of a linear forward model to nonlinear models and carries the whole argument: the triangle inequality turns each pairwise distance into the sum of two reconstruction errors, averaging gives the lower bound, and choosing the point minimizing average p-distance to the feasible set gives the matching upper bound. The fast symmetric version replaces feasible-set sampling with projection onto the kernel via the Moore-Penrose inv

What would settle it

Run the counterexample from the proof's gap: take two measurements, with feasible sets of sizes 2 and 1000—the first containing signals 0 and 100, the second containing 1000 signals all near 0—and define φ(y1)=50, φ(y2)=0. The average reconstruction loss is about 0.1, while half the average kernel size is about 12.5, which would violate the lower bound claimed for measurable maps with unequal feasible-set sizes.

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Extended reading notes

Core claim

The central discovery is the average kernel size, defined as the p-th root of the average, over measurement samples, of the mean pairwise p-th power distance within each feasible set F_y = {x : there exists e with F(x,e)=y}. For a forward problem (F, M1, E) with non-injective F, Theorem 3.3 proves: if every feasible set sampled by Algorithm 1 has the same size N, then every approximate inverse map φ satisfies L(D,φ,p) ≥ ½ Kersize_p, and the infimum over all maps lies between ½ Kersize_p and Kersize_p, with the geometric-median-style map θ(y)=argmin_z Σ ||x_{k,n}-z||^p attaining the infimum. When feasible set sizes differ, the same lower bound is claimed for every measurable map, using a disi

Load-bearing premise

The advertised lower bound for measurable maps with unequal feasible-set sizes assumes that every feasible set contributes at least a 1/K share of the total samples (Eq. 5.25); if one feasible set is tiny and another is huge, this inequality reverses and the proof's bound need not hold.

Editorial extensions

If this is right

  • A reconstruction method's empirical error on a given dataset can be checked against a precomputed floor: errors below half the average kernel size signal a flaw in the evaluation, and errors within the interval [half kernel size, kernel size] identify methods that are already at the accuracy limit.
  • The bounds enable dataset and forward-model design before training: changing the measurement model to shrink feasible sets (and hence the average kernel size) is a computable way to lower the attainable error, which the paper identifies as a 'fundamental shift towards optimizing datasets and forward models.'
  • For linear problems with additive noise, the O(M) symmetric-kernel-size bound makes the accuracy floor practical for large-scale imaging data such as multi-spectral satellite images, where sampling full feasible sets would be infeasible.
  • The sharpness result implies that the floor cannot be improved by a better network architecture alone; only changes to the forward model, signal set, or noise model can lower it.
  • The microscopy experiments indicate the bound can replace or complement Cramér–Rao-style bounds, which require unbiasedness and independence assumptions that often fail in practice.

Reading between the lines

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

  • Editorial inference: if the lower bound is as sharp and method-independent as claimed, then the benchmark 'accuracy plateau' observed across different deep-learning methods is not a mystery to be explained by architecture search; it is the geometry of feasible sets. The practical lever for better reconstruction is improving the forward model or measurement design, not just increasing model capacit
  • Editorial inference: the paper leaves open how the average kernel size converges as the number of samples grows; without such a convergence result, the bounds are guarantees only for the exact dataset produced by Algorithm 1, not for future measurements. A natural next step is to derive confidence intervals or convergence rates for the empirical kernel size under exchangeable sampling.
  • Editorial inference: the framework suggests a new dataset-quality score: report the average kernel size alongside any benchmark result, so that a method's error can be judged relative to the problem's intrinsic difficulty rather than in absolute terms.
  • Editorial inference: the authors' assumption that feasible sets can be sampled may fail exactly in the hardest inverse problems where forward evaluation is expensive; the fast symmetric bound for linear models is one workaround, and extending it to nonlinear forward models with a local linearization would be a testable next step.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper proposes computable accuracy bounds for inverse problems. For an inverse problem with forward model F and feasible sets F_y (the set of signals consistent with a measurement y), it defines the average kernel size as the average pairwise distance between signal samples within each feasible set. The main result (Theorem 3.3 and Main Result 2.1) claims that (i) for equal-sized feasible sets, half the average kernel size is a sharp lower bound for the reconstruction error of any approximate inverse map and the average kernel size is a sharp upper bound for optimal maps, and (ii) for arbitrary feasible-set sizes, half the average kernel size is a lower bound for any measurable approximate inverse map. The paper also presents a fast version for linear forward models (Theorem 3.5) and validates the bounds on two imaging problems.

Significance. If the main theorem were correct, the paper would provide a practically useful, method-independent benchmark for reconstruction accuracy that is computable from the forward model and the data. The equal-size bound in Theorem 3.3(i) appears to be a correct triangle-inequality argument, and the fast symmetric bound in Theorem 3.5 may be a valuable special case. The paper also offers a software library and numerical demonstrations on realistic problems. However, the advertised universal claim for arbitrary datasets — part (ii) of Theorem 3.3 and Main Result 2.1(ii) — is false, as shown by an elementary counterexample. This is a load-bearing error, not a presentation issue.

major comments (3)
  1. [§5.1, Eq. (5.25)] The inequality N_k / Σ_j N_j ≥ N_k / (K min_j N_j) ≥ 1/K has the wrong direction. Since Σ_j N_j ≥ K min_j N_j, the correct relation is N_k / Σ_j N_j ≤ N_k / (K min_j N_j). A valid universal lower bound from this ratio is N_k / (K max_j N_j), which is at least 1/K only when N_k = max_j N_j. Consequently, the disintegration argument leading to (5.26)–(5.29) proves part (ii) only in the equal-size case already covered by part (i).
  2. [Theorem 3.3(ii) / Main Result 2.1(ii)] Part (ii) is false as stated. Counterexample: K=2, N_1=2 with feasible signals {0,100}, N_2=1000 with all signals 0, and p=1. The average kernel size is (1/2)(50+0)=25, so half is 12.5. The measurable map φ(y1)=50, φ(y2)=0 has loss L(D,φ,1)=2·50/1002≈0.1 on the Algorithm-1 dataset (M=1002). This violates the claimed lower bound by two orders of magnitude. Thus the central advertised result that half the average kernel size lower-bounds every measurable approximate inverse map for arbitrary datasets is incorrect.
  3. [§4 numerical experiments] The numerical validations do not test the claimed universal regime. In the microscopy experiments (Section 4.1), all feasible sets have the same size, N(k)=1501, which is exactly the equal-size case where part (ii) reduces to part (i). In the satellite experiment (Section 4.2), the fast symmetric bound of Theorem 3.5 is used, which relies on the symmetry assumption (3.6) and pairs each sample with a reflected counterpart, again avoiding unequal feasible-set sizes. The examples therefore cannot detect the failure of Theorem 3.3(ii) and do not support the paper's claim that the bounds apply to all inverse problems.
minor comments (4)
  1. [Proof of Theorem 3.3(i)] The final displayed chain reads "1/2 Kersize ≤ inf L ≤ 1/2 Kersize"; the upper bound should be Kersize, as established by the θ-map bound just above. This appears to be a typo, but it should be corrected for clarity.
  2. [§6.2] The conclusion appropriately acknowledges that the bounds only hold on the dataset constructed by Algorithm 1 and that Algorithm 1 requires obtaining feasible-set samples, which can be as hard as solving the inverse problem. This limitation is substantial and should be stated earlier, alongside the claim of universal applicability.
  3. [General presentation] Notation such as KersizepF,M 1,E, pqK is hard to parse; a consolidated definition with clear index ordering would help. In addition, there are several typographical issues, e.g., missing spaces before "SectionSM1" and in "SectionSM4".
  4. [Theorem 3.5 / Eq. (3.6)] The symmetry assumption that the reflected data point is realistic is strong and is only visually validated in Figure 4. The theorem should state explicitly that its applicability is conditional on this assumption; currently the surrounding text presents it as a general fast lower bound.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the average-kernel-size bound is a self-contained triangle-inequality estimate; self-citations are background and the acknowledged Algorithm-1 loop is a practical limitation, not a hidden fit.

full rationale

The derivation chain is not circular. Definition 3.2 defines the average kernel size directly from feasible-set samples, and Theorem 3.3's lower bound follows from the triangle inequality in (5.1)-(5.4), with no parameter fitted to the evaluated losses. The upper-bound part constructs an explicit argmin map and proves its loss is at most the kernel size, again by a direct inequality. The cited works [41] and [42] are used for background accuracy-stability tradeoffs and for the feasible-set framework, but the proof of the new bound does not depend on their numerical constants, fitted values, or uniqueness claims, so the self-citations are not load-bearing. Section 6.2 does acknowledge a practical loop: Algorithm 1 requires possible solutions, and the bounds hold only on the dataset processed by Algorithm 1; this limits pre-computability but is not a case of fitting an input and renaming it a prediction. The suspected failure of part (ii) for unequal feasible-set sizes via the inequality direction in Eq. (5.25) is a mathematical validity issue, not a circularity, and therefore does not raise the circularity score.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new physical entities. Its free parameters are the sampling budgets for feasible sets; these are user-chosen and can change the value of the bound. The axioms are mostly standard measurability assumptions and the domain assumption that feasible sets can be sampled. The strongest ad hoc assumption is the reflection symmetry in Eq. (3.6), needed for the fast linear lower bound.

free parameters (1)
  • Sampling budgets K and N(k) = K=25, N=1501 (microscopy); N=3 (satellite)
    The bounds are computed only on Algorithm 1's sampled feasible sets; their value depends on these user-chosen budgets. No convergence rate is provided, so the reported bounds are not uniquely determined by the inverse problem itself.
assumptions (6)
  • domain assumption Forward model F is non-injective on M1×E (Eq. 2.4).
    Required for nonzero feasible sets and nonzero average kernel size; stated as the central assumption in Section 2.3.
  • domain assumption Algorithm 1 can obtain samples (x,e) ∈ M1×E satisfying F(x,e)=y_k.
    Needed to compute the bound. Section 6.2 acknowledges that in some applications these samples cannot be acquired, which limits the claimed 'computable before solving' property.
  • standard math F and φ are measurable so that disintegrations can be used.
    Invoked in the proof of part (ii) of Theorem 3.3 and explicitly stated in Section 3.1.
  • ad hoc to paper The symmetric reflection in Eq. (3.6) maps realistic data points to realistic data points.
    Assumed for Theorem 3.5's fast bound; only visually validated in Figure 4, not statistically verified.
  • domain assumption MCMC samples are representative of the true feasible sets.
    Used in Section 4.1.2. No convergence guarantee or error bars are given; Figure SM2 is referenced but not included.
  • standard math Disintegration of measures (Chang-Pollard) for finite index sets.
    Used in the proof of Theorem 3.3(ii); the theorem is standard, but the application of it is invalid because of the false Eq. (5.25).

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Cite this review

Pith. "Pith review of Average Kernel Sizes -- Computable Sharp Accuracy Bounds for Inverse Problems." pith.science (2026). https://pith.science/paper/IGU2KOTW

@misc{pith2026251010229,
  author       = {Pith},
  title        = {Pith review of: Average Kernel Sizes -- Computable Sharp Accuracy Bounds for Inverse Problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IGU2KOTW}},
  note         = {Machine review of arXiv:2510.10229}
}
read the original abstract

The reconstruction of an unknown quantity from noisy measurements is a mathematical problem relevant in most applied sciences, for example, in medical imaging, radar inverse scattering, or astronomy. This underlying mathematical problem is often an ill-posed (non-linear) reconstruction problem, referred to as an ill-posed inverse problem. To tackle such problems, there exist a myriad of methods to design approximate inverse maps, ranging from optimization-based approaches, such as compressed sensing, over Bayesian approaches, to data-driven techniques such as deep learning. For all stable approximate inverse maps, there are accuracy limits that are strictly larger than zero for ill-posed inverse problems, due to the accuracy-stability tradeoff [Gottschling et al., SIAM Review, 67.1 (2025)] and [Colbrook et al., Proceedings of the National Academy of Sciences, 119.12 (2022)]. The variety of methods that aim to solve such problems begs for a unifying approach to help scientists choose the approximate inverse map that obtains this theoretical optimum. Up to now there do not exist computable accuracy bounds to this optimum that are applicable to all inverse problems. We provide computable sharp accuracy bounds to the reconstruction error of solution methods to inverse problems. The bounds are method-independent and purely depend on the dataset of signals, the forward model of the inverse problem, and the noise model. To facilitate the use in scientific applications, we provide an algorithmic framework and an accompanying software library to compute these accuracy bounds. We demonstrate the validity of the algorithms on two inverse problems from different domains: fluorescence localization microscopy and super-resolution of multi-spectral satellite data. Computing the accuracy bounds for a problem before solving it, enables a fundamental shift towards optimizing datasets and forward models.

Figures

Figures reproduced from arXiv: 2510.10229 by the authors.

Figure 1
Figure 1. Inverse problem framework. The forward model F : M1 ˆ E Ñ M2 in (2.1) is a measurable map onto its image, M2 “ FpM1 ˆ Eq. Examples of the forward model include, but are not limited to, Fpx, eq “ Gpxq ` e, Fpx, eq “ Gpxq d e, Fpx, eq “ Gpxq d e1 ` e2, (3.3) where G: C d1 Ñ C d2 is a linear or non-linear for￾ward model, and the dimension d3 of the set E de￾pends on the considered model in such a way that the point-wis… view at source ↗
Figure 2
Figure 2. Allocation of x-components of samples into feasible sets with Algorithm 1. Note that the noise components can enlarge the feasible sets. For example, this occurs for additive and multiplicative noise models. models in (3.3). For example, Section 4.1 and Section 4.2 demonstrate that Algorithm 2 is applicable to complicated forward models, such as those in localization microscopy, and for very large data dimensions d1… view at source ↗
Figure 3
Figure 3. RMSE of the px, yq´location from different estimators against half of the average kernel size computed with feasible sets of size Np1q “ 15001 with K “ 1 for 25 different noisy measurements per microscope set-up tA1, A2, A3, A4u. The lower and upper gray lines (y “ x and y “ 2x) visualize the lower and upper accu￾racy bounds. All RMSE values are above the computed lower accuracy bounds and validate Theorem 3.3, piq.… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: RGB example visualizations of different versions of a multi-spectral satellite Croplands image from the [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]
Figure 6
Figure 6. Figure 6: RMSE of different super-resolution methods against the average symmetric kernel size for triplets, [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. On Hallucinations in Inverse Problems: Fundamental Limits and Provable Assessment Methods

    stat.ML 2026-05 unverdicted novelty 7.0 of 10

    Hallucinations in inverse problem reconstructions are fundamental to ill-posedness, with necessary and sufficient conditions plus computable bounds depending only on the forward model.

Reference graph

Works this paper leans on

78 extracted references · 3 linked inside Pith · cited by 1 Pith paper

  1. [1]

    Adcock, S

    B. Adcock, S. Brugiapaglia, and C. G. Webster.Sparse polynomial approximation of high-dimensional functions, volume 25. SIAM, 2022

    2022

  2. [2]

    Adcock and N

    B. Adcock and N. Dexter. The gap between theory and practice in function approximation with deep neural networks.SIAM Journal on Mathematics of Data Science, 3(2):624–655, 2021

    2021

  3. [3]

    Adcock and A

    B. Adcock and A. C. Hansen.Compressive Imaging: Structure, Sampling, Learning. Cambridge University Press, Cambridge, UK, 2021

    2021

  4. [4]

    Anwar, S

    S. Anwar, S. Khan, and N. Barnes. A deep journey into super-resolution: A survey.ACM computing surveys (CSUR), 53(3):1–34, 2020

    2020

  5. [5]

    V . V . Arestov. Optimal recovery of operators and related problems.Collection of Papers from the All-Union School on the Theory of Functions, pages 3–20, 1986. 23

    1986

  6. [6]

    Arridge, P

    S. Arridge, P. Maass, O. ¨Oktem, and C.-B. Sch ¨onlieb. Solving inverse problems using data-driven models.Acta Numer., 28:1–174, 2019

    2019

  7. [7]

    R. B. Ash.Real analysis and probability: probability and mathematical statistics: a series of monographs and textbooks. Academic press, 2014

    2014

  8. [8]

    P. M. Atkinson and P. J. Curran. Choosing an appropriate spatial resolution for remote sensing investigations.Photogram- metric Engineering & Remote Sensing, 63(12), 1997

    1997

Show all 78 references
  1. [9]

    Aybar, D

    C. Aybar, D. Montero, S. Donike, F. Kalaitzis, and L. G´omez-Chova. A comprehensive benchmark for optical remote sensing image super-resolution.IEEE Geoscience and Remote Sensing Letters, 21:1–5, 2024

    2024

  2. [10]

    F. Bach. Breaking the curse of dimensionality with convex neural networks.Journal of Machine Learning Research, 18(19):1–53, 2017

    2017

  3. [11]

    H. H. Barrett and K. J. Myers.Foundations of image science. John Wiley & Sons, 2013

    2013

  4. [12]

    C. Beck, A. Jentzen, and B. Kuckuck. Full error analysis for the training of deep neural networks.Infinite Dimensional Analysis, Quantum Probability and Related Topics, 25(02):2150020, 2022

    2022

  5. [13]

    Belthangady and L

    C. Belthangady and L. A. Royer. Applications, promises, and pitfalls of deep learning for fluorescence image reconstruction. Nature methods, 16(12):1215–1225, 2019

    2019

  6. [14]

    Ben-Israel and T

    A. Ben-Israel and T. N. Greville.Generalized inverses: theory and applications. Springer Science & Business Media, 2006

    2006

  7. [15]

    J. A. Benediktsson, J. Chanussot, and W. M. Moon. Very high-resolution remote sensing: Challenges and opportunities [point of view].Proceedings of the IEEE, 100(6):1907–1910, 2012

    1907

  8. [16]

    Bertero, P

    M. Bertero, P. Boccacci, and C. De Mol.Introduction to inverse problems in imaging. CRC press, 2021

    2021

  9. [17]

    P. J. Bickel and K. A. Doksum.Mathematical statistics: basic ideas and selected topics, volumes I-II package. Chapman and Hall/CRC, 2015

    2015

  10. [18]

    Binev, A

    P. Binev, A. Bonito, R. DeV ore, and G. Petrova. Optimal learning.Calcolo, 61(1):15, 2024

    2024

  11. [19]

    Borcea, J

    L. Borcea, J. Garnier, A. V . Mamonov, and J. Zimmerling. Waveform inversion with a data driven estimate of the internal wave.SIAM Journal on Imaging Sciences, 16(1):280–312, 2023

    2023

  12. [20]

    B. Borden. Mathematical problems in radar inverse scattering.Inverse Problems, 18(1):R1, 2001

    2001

  13. [21]

    P. T. Boufounos and R. G. Baraniuk. 1-bit compressive sensing. In2008 42nd Annual Conference on Information Sciences and Systems, pages 16–21, 2008

    2008

  14. [22]

    C. A. Bouman.Foundations of Computational Imaging: A Model-Based Approach. SIAM, Philadelphia, PA, 2022

    2022

  15. [23]

    Bourrier, M

    A. Bourrier, M. E. Davies, T. Peleg, P. P ´erez, and R. Gribonval. Fundamental performance limits for ideal decoders in high-dimensional linear inverse problems.IEEE Transactions on Information Theory, 60(12):7928–7946, 2014

    2014

  16. [24]

    Burger and T

    M. Burger and T. Roith. Learning in image reconstruction: A cautionary tale.SIAM News, Collections, 57(08), 2024

    2024

  17. [25]

    E. J. Candes and B. Recht. Exact matrix completion via convex optimization.Found. Comput. Math., 9(6):717–772, 2009

    2009

  18. [26]

    E. J. Candes, T. Strohmer, and V . V oroninski. Phaselift: Exact and stable signal recovery from magnitude measurements via convex programming.Communications on Pure and Applied Mathematics, 66(8):1241–1274, 2013

    2013

  19. [27]

    Casella and C

    G. Casella and C. Robert. Monte carlo statistical methods.University of Florida, 2008

    2008

  20. [28]

    J. T. Chang and D. Pollard. Conditioning as disintegration.Statistica Neerlandica, 51(3):287–317, 1997

    1997

  21. [29]

    S. Chen, Y . Ogawa, C. Zhao, and Y . Sekimoto. Large-scale individual building extraction from open-source satellite im- agery via super-resolution-based instance segmentation approach.ISPRS Journal of Photogrammetry and Remote Sensing, 195:129–152, 2023

    2023

  22. [30]

    Cohen, W

    A. Cohen, W. Dahmen, and R. DeV ore. Compressed sensing and bestk-term approximation.Journal of the American mathematical society, 22(1):211–231, 2009

    2009

  23. [31]

    Cohen, R

    A. Cohen, R. Devore, G. Petrova, and P. Wojtaszczyk. Optimal stable nonlinear approximation.Foundations of Computational Mathematics, 22(3):607–648, 2022

    2022

  24. [32]

    M. J. Colbrook, V . Antun, and A. C. Hansen. The difficulty of computing stable and accurate neural networks: On the barriers of deep learning and smale’s 18th problem.Proceedings of the National Academy of Sciences, 119(12):e2107151119, 2022

    2022

  25. [33]

    E. S. Crafts and U. Villa. Benchmarking diffusion annealing-based bayesian inverse problem solvers.IEEE Open Journal of Signal Processing, 2025

    2025

  26. [34]

    Craig and J

    I. Craig and J. Brown.Inverse problems in astronomy. Adam Hilger Ltd., Accord, MA, 1986

    1986

  27. [35]

    Cram ´er.Mathematical methods of statistics, volume 9

    H. Cram ´er.Mathematical methods of statistics, volume 9. Princeton university press, 1999

    1999

  28. [36]

    Curran and H

    P. Curran and H. Williamson. Airborne mss data to estimate glai.International Journal of Remote Sensing, 8(1):57–74, 1987

    1987

  29. [37]

    Donike, C

    S. Donike, C. Aybar, L. G ´omez-Chova, and F. Kalaitzis. Trustworthy super-resolution of multispectral sentinel-2 imagery with latent diffusion.IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, 18:6940–6952, 2025

    2025

  30. [38]

    Fannjiang and T

    A. Fannjiang and T. Strohmer. The numerics of phase retrieval.Acta Numerica, 29:125–228, 2020

    2020

  31. [39]

    Foucart and H

    S. Foucart and H. Rauhut.A mathematical introduction to compressive sensing. Birkhauser, 2013. 24

    2013

  32. [40]

    Genzel, J

    M. Genzel, J. Macdonald, and M. Marz. Solving inverse problems with deep neural networks-robustness included.IEEE Transactions on Pattern Analysis and Machine Intelligence, 2022

    2022

  33. [41]

    N. M. Gottschling, V . Antun, A. C. Hansen, and B. Adcock. The troublesome kernel: On hallucinations, no free lunches, and the accuracy-stability tradeoff in inverse problems.SIAM Review, 67(1):73–104, 2025

    2025

  34. [42]

    N. M. Gottschling, P. Campodonico, V . Antun, and A. C. Hansen. On the existence of optimal multi-valued decoders and their accuracy bounds for undersampled inverse problems.arXiv preprint arXiv:2311.16898, 2023

    2023 arXiv

  35. [43]

    Hammernik, T

    K. Hammernik, T. Klatzer, E. Kobler, M. P. Recht, D. K. Sodickson, T. Pock, and F. Knoll. Learning a variational network for reconstruction of accelerated mri data.Magnetic resonance in medicine, 79(6):3055–3071, 2018

    2018

  36. [44]

    D. P. Hoffman, I. Slavitt, and C. A. Fitzpatrick. The promise and peril of deep learning in microscopy.Nature Methods, 18(2):131–132, 2021

    2021

  37. [45]

    Hornik, M

    K. Hornik, M. Stinchcombe, and H. White. Multilayer feedforward networks are universal approximators.Neural networks, 2(5):359–366, 1989

    1989

  38. [46]

    Huang, M

    Y .-M. Huang, M. K. Ng, and Y .-W. Wen. A new total variation method for multiplicative noise removal.SIAM Journal on imaging sciences, 2(1):20–40, 2009

    2009

  39. [47]

    S. M. Kay.Fundamentals of statistical signal processing: estimation theory. Prentice-Hall, Inc., 1993

    1993

  40. [48]

    Knoll, T

    F. Knoll, T. Murrell, A. Sriram, N. Yakubova, J. Zbontar, M. Rabbat, A. Defazio, M. J. Muckley, D. K. Sodickson, C. L. Zitnick, et al. Advancing machine learning for MR image reconstruction with an open competition: Overview of the 2019 fastMRI challenge.Magnetic resonance in ...

    2019

  41. [49]

    J. Kong, Y . Ryu, S. Jeong, Z. Zhong, W. Choi, J. Kim, K. Lee, J. Lim, K. Jang, J. Chun, et al. Super resolution of historic landsat imagery using a dual generative adversarial network (gan) model with cubesat constellation imagery for spatially enhanced long-term vegetation m...

    2023

  42. [50]

    Korolev and J

    Y . Korolev and J. Latz. Inverse problems, 2020. Lecture notes, Michaelmas Term 2020, University of Cambridge

    2020

  43. [51]

    A. Lee, K. Tsekouras, C. Calderon, C. Bustamante, and S. Press ´e. Unraveling the thousand word picture: an introduction to super-resolution data analysis.Chemical reviews, 117(11):7276–7330, 2017

    2017

  44. [52]

    J. S. Liu and J. S. Liu.Monte Carlo strategies in scientific computing, volume 10. Springer, 2001

    2001

  45. [53]

    G. G. Magaril-Il’yaev and K. Y . Osipenko. Optimal recovery of functionals based on inaccurate data.Mat. Zametki, 50(6):85– 93, 1991

    1991

  46. [54]

    Maiorov and A

    V . Maiorov and A. Pinkus. Lower bounds for approximation by mlp neural networks.Neurocomputing, 25(1-3):81–91, 1999

    1999

  47. [55]

    Mercer and A

    W. Mercer and A. Hall. The experimental error of field trials.The Journal of Agricultural Science, 4(2):107–132, 1911

    1911

  48. [56]

    C. A. Micchelli and T. J. Rivlin. A survey of optimal recovery.Optimal estimation in approximation theory, pages 1–54, 1977

    1977

  49. [57]

    M. J. Muckley, B. Riemenschneider, A. Radmanesh, S. Kim, G. Jeong, J. Ko, Y . Jun, H. Shin, D. Hwang, M. Mostapha, et al. State-of-the-art Machine Learning MRI Reconstruction in 2020: Results of the Second fastMRI Challenge.arXiv preprint arXiv:2012.06318, 2020

    2020 arXiv

  50. [58]

    K. P. Murphy.Probabilistic machine learning: an introduction. MIT press, 2022

    2022

  51. [59]

    R. J. Ober, S. Ram, and E. S. Ward. Localization accuracy in single-molecule microscopy.Biophysical journal, 86(2):1185– 1200, 2004

    2004

  52. [60]

    Paszke, S

    A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, et al. Pytorch: An imperative style, high-performance deep learning library.Advances in neural information processing systems, 32, 2019

    2019

  53. [61]

    Petersen and F

    P. Petersen and F. V oigtlaender. Optimal approximation of piecewise smooth functions using deep relu neural networks. Neural Networks, 108:296–330, 2018

    2018

  54. [62]

    Pinkus.N-widths in Approximation Theory, volume 7

    A. Pinkus.N-widths in Approximation Theory, volume 7. Springer Science & Business Media, 2012

    2012

  55. [63]

    Plaskota.Noisy information and computational complexity, volume 95

    L. Plaskota.Noisy information and computational complexity, volume 95. Cambridge University Press, 1996

    1996

  56. [64]

    D. A. Quattrochi and M. F. Goodchild.Scale in remote sensing and GIS. CRC press, 1997

    1997

  57. [65]

    C. R. Rao, C. R. Rao, M. Statistiker, C. R. Rao, and C. R. Rao.Linear statistical inference and its applications, volume 2. Wiley New York, 1973

    1973

  58. [66]

    L. I. Rudin, S. Osher, and E. Fatemi. Nonlinear total variation based noise removal algorithms.Physica D: nonlinear phenomena, 60(1-4):259–268, 1992

    1992

  59. [67]

    Sgouralis, L

    I. Sgouralis, L. W. Xu, A. P. Jalihal, Z. Kilic, N. G. Walter, and S. Press´e. Bnp-track: a framework for superresolved tracking. Nature Methods, 21(9):1716–1724, 2024

    2024

  60. [68]

    Shechtman, Y

    Y . Shechtman, Y . C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev. Phase retrieval with application to optical imaging: a contemporary overview.IEEE signal processing magazine, 32(3):87–109, 2015

    2015

  61. [69]

    H. F. Smith. An empirical law describing heterogeneity in the yields of agricultural crops.The Journal of Agricultural Science, 28(1):1–23, 1938

    1938

  62. [70]

    R. Strack. Imaging: AI transforms image reconstruction.Nature Methods, 15(5):309, 2018

    2018

  63. [71]

    A. N. Tikhonov. Solution of incorrectly formulated problems and the regularization method.Sov Dok, 4:1035–1038, 1963. 25

    1963

  64. [72]

    von Diezmann, Y

    L. von Diezmann, Y . Shechtman, and W. Moerner. Three-dimensional localization of single molecules for super-resolution imaging and single-particle tracking.Chemical reviews, 117(11):7244–7275, 2017

    2017

  65. [73]

    G. Wang, J. C. Ye, and B. De Man. Deep learning for tomographic image reconstruction.Nature machine intelligence, 2(12):737–748, 2020

    2020

  66. [74]

    Yarotsky

    D. Yarotsky. Error bounds for approximations with deep relu networks.Neural networks, 94:103–114, 2017

    2017

  67. [75]

    S. Zacks. The theory of statistical inference.(No Title), 1971

    1971

  68. [76]

    Zbontar, F

    J. Zbontar, F. Knoll, A. Sriram, T. Murrell, Z. Huang, M. J. Muckley, A. Defazio, R. Stern, P. Johnson, M. Bruno, et al. fastMRI: An open dataset and benchmarks for accelerated MRI.arXiv preprint arXiv:1811.08839, 2018

    2018 arXiv

  69. [77]

    Zheng, W

    H. Zheng, W. Chu, B. Zhang, Z. Wu, A. Wang, B. T. Feng, C. Zou, Y . Sun, N. Kovachki, Z. E. Ross, et al. Inversebench: Benchmarking plug-and-play diffusion priors for inverse problems in physical sciences.arXiv preprint arXiv:2503.11043, 2025

    2025

  70. [78]

    B. Zhu, J. Z. Liu, S. F. Cauley, B. R. Rosen, and M. S. Rosen. Image reconstruction by domain-transform manifold learning. Nature, 555(7697):487, 03 2018. 26

    2018

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.