pith. sign in

arxiv: 2605.21094 · v1 · pith:NF53VDJ4new · submitted 2026-05-20 · 💻 cs.LG

UOTIP: Unbalanced Optimal Transport Map for Unpaired Inverse Problems

Donggyu Lee , Taekyung Lee , Jaewoong Choi This is my paper

Pith reviewed 2026-05-21 05:17 UTC · model grok-4.3

classification 💻 cs.LG
keywords unbalanced optimal transportunpaired inverse problemsimage reconstructionlikelihood costtwist conditiontransport mapill-posed problems
0
0 comments X

The pith

Unbalanced optimal transport learns a unique map from noisy measurements to clean signals without paired examples by adding a quadratic cost that satisfies the twist condition.

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

The paper sets out to show that image inverse problems such as denoising or deblurring can be solved when only separate collections of noisy measurements and clean images are available, rather than matched pairs. It does this by casting the task as learning an unbalanced optimal transport map between the two distributions, guided by a likelihood-based cost. A quadratic term is added to the cost so that the learned map is guaranteed to exist and to be unique, satisfying the twist condition even when the underlying inverse problem is ill-posed. The unbalanced relaxation further lets the method tolerate varying noise levels, mismatched dataset sizes, and different noise types. If the approach holds, practitioners could train reconstruction models on readily available but unpaired data instead of having to collect perfectly aligned examples.

Core claim

We formulate the reconstruction task as learning a UOT map from the noisy measurement distribution to the clean signal distribution using a likelihood-based cost. Incorporating a quadratic cost term ensures the existence and uniqueness of the transport map by satisfying the twist condition, even for ill-posed inverse problems. The unbalanced framework provides robustness to multi-level noise, adaptability to class imbalance, and generalizability across noise types. Experiments show state-of-the-art results on unpaired benchmarks for both linear and nonlinear inverse problems.

What carries the argument

The UOT map from noisy to clean distributions, driven by a likelihood cost plus a quadratic term that enforces the twist condition for uniqueness.

If this is right

  • The same map can reconstruct images at multiple noise levels without retraining.
  • The method remains effective when the number of noisy samples differs from the number of clean samples.
  • It applies equally to linear operators such as blurring and to nonlinear forward models.
  • The theoretical guarantee on map uniqueness holds for a range of ill-posed settings once the quadratic cost is included.

Where Pith is reading between the lines

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

  • The distribution-level matching idea could be tested on non-image modalities such as audio or sensor time series where paired data are also scarce.
  • Replacing the quadratic term with other costs that still meet the twist condition might yield maps with different smoothness properties.
  • If the transport map generalizes across noise types, one could pre-train on synthetic noise and fine-tune on real measurements without new pairings.

Load-bearing premise

The overall distribution of noisy measurements can be mapped to the overall distribution of clean signals in a way that still yields accurate reconstructions for individual ill-posed instances.

What would settle it

Finding a concrete ill-posed inverse problem where the learned map produces visibly incorrect reconstructions or where multiple distinct maps satisfy the same objective would falsify the uniqueness claim.

Figures

Figures reproduced from arXiv: 2605.21094 by Donggyu Lee, Jaewoong Choi, Taekyung Lee.

Figure 1
Figure 1. Figure 1: OT Maps under different cost functions (A: projection onto x-axis along d = (1, −1)) where M+(Y × X ) denotes the set of positive Radon measures on Y × X . The terms DΨ1 and DΨ2 are f￾divergences induced by convex functions Ψi , penalizing deviations of the marginals of π from µ and ν, respectively , i.e., DΨi (πj∥η) = R Ψi  dπj (x) dη(x)  dη(x). Note that the standard OT enforces exact matching of the m… view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of inverse problem solvers on FFHQ for four tasks: Gaussian deblurring (GB), Super-resolution (SR), High dynamic range reconstruction (HDR), and Nonlinear deblurring (NB). Our model produces higher fidelity images with well-preserved textures [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Quantitative results under class imbalance with imbalance ratio k between AFHQ-cat and AFHQ-dog for the Gaussian deblurring (cat:dog 1 : 1 → k : 1). Our method achieves superior performance and demonstrates greater robustness across various ratios. 10 15 20 25 30 P S N R ( " ) Gaussian Deblurring Super-Resolution Laplace noise Poisson noise Gaussian noise Laplace noise Poisson noise Gaussian noise NOT OTUR… view at source ↗
Figure 4
Figure 4. Figure 4: PSNR (↑) results for diverse noise types and cost ablation. Complete results on all three metrics are provided in Appendix E.2. In Fig.4a, our model generalizes well across different noise types. In Fig.4b, the full model achieves the best overall performance, and even the blind-corruption variant (using only cq) outperforms the previous state-of-the-art OTUR model. lowest results. Diverse Noise Types We i… view at source ↗
Figure 5
Figure 5. Figure 5: Conceptual illustration of the multi-level observation noise problem. A single target signal x may be observed under multiple Gaussian noise levels. Unlike the OT map, the UOT map can rescale each sample, enabling the alignment of multiple noisy observations with a single target signal. This flexibility makes our UOT-based model more robust to multi-level observation noise. Algorithm 1 Training algorithm o… view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of inverse problem solvers on AFHQ for four tasks: Gaussian deblurring (GB), Super-resolution (SR), High dynamic range reconstruction (HDR), and Nonlinear deblurring (NB). Our model reconstructs images of superior fidelity with well￾preserved structural details. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Additional qualitative results of our model for the Gaussian deblurring task on FFHQ (degraded (Left) → clean (Right)) [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Additional qualitative results of our model for the Gaussian deblurring task on AFHQ (degraded (Left) → clean (Right)) [PITH_FULL_IMAGE:figures/full_fig_p017_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Additional qualitative results of our model for the super-resolution task on FFHQ (degraded (Left) → clean (Right)). 17 [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Additional qualitative results of our model for the super-resolution task on AFHQ (degraded (Left) → clean (Right)) [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Additional qualitative results of our model for the HDR reconstruction task on FFHQ (degraded (Left) → clean (Right)) [PITH_FULL_IMAGE:figures/full_fig_p018_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Additional qualitative results of our model for the HDR reconstruction task on AFHQ (degraded (Left) → clean (Right)) [PITH_FULL_IMAGE:figures/full_fig_p018_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Additional qualitative results of our model for the nonlinear deblurring task on FFHQ (degraded (Left) → clean (Right)). 18 [PITH_FULL_IMAGE:figures/full_fig_p018_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Additional qualitative results of our model for the nonlinear deblurring task on AFHQ (degraded (Left) → clean (Right)) [PITH_FULL_IMAGE:figures/full_fig_p019_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Qualitative results of our quadratic-cost-only variant for the Gaussian deblurring task on FFHQ (degraded (Left) → clean (Right)) [PITH_FULL_IMAGE:figures/full_fig_p019_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Qualitative results of our quadratic-cost-only variant for the super resolution task on FFHQ (degraded (Left) → clean (Right)). 19 [PITH_FULL_IMAGE:figures/full_fig_p019_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Comparison of inverse problem solvers under multi-level observation noise on FFHQ for the Gaussian deblurring. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Comparison of inverse problem solvers under multi-level observation noise on FFHQ for the super-resolution. y = 0.025 source NOT OTUR Ours-OT Ours target y = 0.05 y = 0.1 y = 0.2 [PITH_FULL_IMAGE:figures/full_fig_p021_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Comparison of inverse problem solvers under multi-level observation noise on FFHQ for the HDR reconstruction. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Comparison of inverse problem solvers under multi-level observation noise on FFHQ for the nonlinear deblurring. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Ablation study on the cost intensity parameter τ . For clarity, only PSNR (↑) and SSIM (↑) are visualized [PITH_FULL_IMAGE:figures/full_fig_p023_21.png] view at source ↗
read the original abstract

We investigate unpaired image inverse problems, a challenging setting where only independent, non-paired sets of noisy measurements and clean target signals are available for training. We propose a novel inverse problem solver based on Unbalanced Optimal Transport, called Unbalanced Optimal Transport Map for Inverse Problems (UOTIP). Our method formulates the reconstruction task, predicting clean target signals from noisy measurements, as learning a UOT Map from noisy measurement distribution to clean signal distribution by incorporating a likelihood-based cost function. By relaxing the exact marginal constraint, the UOT framework provides key advantages to our model: robustness to multi-level observation noise, adaptability to class imbalance between noisy and clean datasets, and generalizability to diverse noise-type scenarios. Furthermore, we theoretically demonstrate that incorporating a quadratic cost term ensures the existence and uniqueness of the transport map by satisfying the twist condition, even for ill-posed inverse problems. Our experiments demonstrate that UOTIP achieves state-of-the-art performance on unpaired image inverse problem benchmarks, across linear and nonlinear inverse problems.

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

1 major / 2 minor

Summary. The paper proposes UOTIP, a method for unpaired image inverse problems that formulates reconstruction as learning an unbalanced optimal transport map from the distribution of noisy measurements to clean signals. It incorporates a likelihood-based cost augmented by a quadratic term, claims that this guarantees existence and uniqueness of the map by satisfying the twist condition even for ill-posed problems, and reports state-of-the-art empirical performance on benchmarks for linear and nonlinear inverse problems.

Significance. If the central theoretical claim holds, the work offers a principled way to handle unpaired data in inverse problems with built-in robustness to noise levels and class imbalance via the unbalanced relaxation. This could be impactful for applications where paired training data is unavailable, extending OT techniques beyond balanced settings.

major comments (1)
  1. [Theoretical Analysis] Theoretical section (proof of twist condition): The argument that adding the quadratic term to the likelihood cost ensures the twist condition (injectivity of y ↦ ∇_x c(x,y)) does not explicitly treat the case of a forward operator A with nontrivial kernel. When A has a null space, the likelihood gradient is invariant along ker(A) directions; without an explicit domination argument or calculation of the mixed Hessian showing the quadratic term renders it nondegenerate for all x in the support, the uniqueness guarantee for ill-posed inverses is not yet established.
minor comments (2)
  1. [Abstract and Experiments] The abstract and experimental section should specify the exact forward operators, noise models, and datasets used for the linear and nonlinear benchmarks to facilitate direct comparison.
  2. [Method] Notation for the cost function c(x,y) and the weighting between likelihood and quadratic terms should be introduced earlier and used consistently throughout the theoretical development.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thoughtful and constructive review. The feedback on the theoretical analysis is particularly helpful, and we address the major comment in detail below. We believe the suggested clarification will strengthen the manuscript.

read point-by-point responses

  1. Referee: [Theoretical Analysis] Theoretical section (proof of twist condition): The argument that adding the quadratic term to the likelihood cost ensures the twist condition (injectivity of y ↦ ∇_x c(x,y)) does not explicitly treat the case of a forward operator A with nontrivial kernel. When A has a null space, the likelihood gradient is invariant along ker(A) directions; without an explicit domination argument or calculation of the mixed Hessian showing the quadratic term renders it nondegenerate for all x in the support, the uniqueness guarantee for ill-posed inverses is not yet established.

    Authors: We agree with the referee that the current write-up of the twist-condition argument would benefit from an explicit treatment of the nontrivial-kernel case. While the quadratic term is designed to ensure non-degeneracy, the manuscript does not presently supply a detailed domination argument or mixed-Hessian calculation that covers directions in ker(A). In the revised version we will add a short lemma that computes the relevant second derivatives of the composite cost and shows that the quadratic component strictly dominates the (possibly degenerate) likelihood term along the kernel, thereby establishing injectivity of y ↦ ∇_x c(x,y) for all x in the support. This addition will make the uniqueness claim fully rigorous for ill-posed operators. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central theory is an independent proof claim

full rationale

The derivation introduces UOTIP as a UOT map from noisy measurements to clean signals using a composite cost (likelihood-based plus quadratic term). The key theoretical step asserts that the quadratic term guarantees the twist condition for existence/uniqueness even under ill-posed operators. This is presented as a demonstration/proof rather than a definitional reduction, fitted-parameter renaming, or load-bearing self-citation. No equations in the provided text equate the claimed result to its own inputs by construction, and the method's advantages (robustness to noise imbalance) are argued separately from the twist-condition claim. The result is therefore self-contained against external OT theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on the abstract, the central claim rests on standard optimal transport theory assumptions and the specific choice of cost function; no free parameters or invented entities are explicitly introduced in the summary.

axioms (1)
  • domain assumption Incorporating a quadratic cost term satisfies the twist condition for the transport map even in ill-posed inverse problems.
    Stated in the abstract as the basis for proving existence and uniqueness of the UOT map.

pith-pipeline@v0.9.0 · 5711 in / 1335 out tokens · 37592 ms · 2026-05-21T05:17:20.278292+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

82 extracted references · 82 canonical work pages · 2 internal anchors

  1. [1]

    Proceedings of the IEEE conference on computer vision and pattern recognition , pages=

    Photo-realistic single image super-resolution using a generative adversarial network , author=. Proceedings of the IEEE conference on computer vision and pattern recognition , pages=

    work page

  2. [2]

    European conference on computer vision , pages=

    Colorful image colorization , author=. European conference on computer vision , pages=. 2016 , organization=

    work page 2016

  3. [3]

    Multiscale modeling & simulation , volume=

    A review of image denoising algorithms, with a new one , author=. Multiscale modeling & simulation , volume=. 2005 , publisher=

    work page 2005

  4. [4]

    Image deblurring with blurred/noisy image pairs , author=. Proceedings of the 34th ACM SIGGRAPH Conference on Computer Graphics, 34th Annual Meeting of the Association for Computing Machinery's Special Interest Group on Graphics; San Diego, CA; United States , year=

    work page

  5. [5]

    Proceedings of the 27th annual conference on Computer graphics and interactive techniques , pages=

    Image inpainting , author=. Proceedings of the 27th annual conference on Computer graphics and interactive techniques , pages=

    work page

  6. [6]

    IEEE signal processing magazine , volume=

    An introduction to compressive sampling , author=. IEEE signal processing magazine , volume=. 2008 , publisher=

    work page 2008

  7. [7]

    Statistical science , pages=

    A statistical perspective on ill-posed inverse problems , author=. Statistical science , pages=. 1986 , publisher=

    work page 1986

  8. [8]

    2008 42nd Asilomar Conference on Signals, Systems and Computers , pages=

    Compressed sensing and robust recovery of low rank matrices , author=. 2008 42nd Asilomar Conference on Signals, Systems and Computers , pages=. 2008 , organization=

    work page 2008

  9. [9]

    IEEE Transactions on information theory , volume=

    Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information , author=. IEEE Transactions on information theory , volume=. 2006 , publisher=

    work page 2006

  10. [10]

    Proceedings of the IEEE conference on computer vision and pattern recognition , pages=

    Deep image prior , author=. Proceedings of the IEEE conference on computer vision and pattern recognition , pages=

    work page

  11. [11]

    On the stability of inverse problems , author=. Dokl. Akad. Nauk SSSR , volume=

    work page

  12. [12]

    IEEE Transactions on Signal Processing , volume=

    Sparse reconstruction by separable approximation , author=. IEEE Transactions on Signal Processing , volume=. 2009 , publisher=

    work page 2009

  13. [13]

    Proceedings of the IEEE , volume=

    Computational methods for sparse solution of linear inverse problems , author=. Proceedings of the IEEE , volume=. 2010 , publisher=

    work page 2010

  14. [14]

    Physica D: nonlinear phenomena , volume=

    Nonlinear total variation based noise removal algorithms , author=. Physica D: nonlinear phenomena , volume=. 1992 , publisher=

    work page 1992

  15. [15]

    SIAM Journal on Imaging Sciences , volume=

    A new alternating minimization algorithm for total variation image reconstruction , author=. SIAM Journal on Imaging Sciences , volume=. 2008 , publisher=

    work page 2008

  16. [16]

    Geophysics , volume=

    An overview of full-waveform inversion in exploration geophysics , author=. Geophysics , volume=. 2009 , publisher=

    work page 2009

  17. [17]

    Conference on inverse scattering--Theory and application , volume=

    As a sequence of before stack migrations , author=. Conference on inverse scattering--Theory and application , volume=

    work page

  18. [18]

    Journal of Physics: Conference Series , volume=

    Inverse problems in atmospheric science and their application , author=. Journal of Physics: Conference Series , volume=. 2005 , organization=

    work page 2005

  19. [19]

    1996 , publisher=

    The ocean circulation inverse problem , author=. 1996 , publisher=

    work page 1996

  20. [20]

    arXiv preprint arXiv:2402.09821 , year=

    Diffusion models for audio restoration , author=. arXiv preprint arXiv:2402.09821 , year=

    work page arXiv

  21. [21]

    Solving Audio Inverse Problems with a Diffusion Model , year=

    Moliner, Eloi and Lehtinen, Jaakko and Välimäki, Vesa , booktitle=. Solving Audio Inverse Problems with a Diffusion Model , year=

    work page

  22. [22]

    Proceedings of the IEEE/CVF conference on computer vision and pattern recognition , pages=

    Solving 3d inverse problems using pre-trained 2d diffusion models , author=. Proceedings of the IEEE/CVF conference on computer vision and pattern recognition , pages=

    work page

  23. [23]

    Monge, Gaspard , journal=. M

    work page

  24. [24]

    Uspekhi Mat

    On a problem of Monge , author=. Uspekhi Mat. Nauk , pages=

    work page

  25. [25]

    2009 , publisher=

    Optimal transport: old and new , author=. 2009 , publisher=

    work page 2009

  26. [26]

    2015 , publisher=

    Optimal transport for applied mathematicians , author=. 2015 , publisher=

    work page 2015

  27. [27]

    Advances in Neural Information Processing Systems , volume=

    Robust optimal transport with applications in generative modeling and domain adaptation , author=. Advances in Neural Information Processing Systems , volume=

    work page

  28. [28]

    arXiv preprint arXiv:2211.08775 , year=

    Unbalanced Optimal Transport, from theory to numerics , author=. arXiv preprint arXiv:2211.08775 , year=

    work page arXiv

  29. [29]

    The Twelfth International Conference on Learning Representations , year=

    Unbalancedness in Neural Monge Maps Improves Unpaired Domain Translation , author=. The Twelfth International Conference on Learning Representations , year=

    work page

  30. [30]

    Journal of Functional Analysis , volume=

    Unbalanced optimal transport: Dynamic and Kantorovich formulations , author=. Journal of Functional Analysis , volume=. 2018 , publisher=

    work page 2018

  31. [31]

    Inventiones mathematicae , volume=

    Optimal entropy-transport problems and a new Hellinger--Kantorovich distance between positive measures , author=. Inventiones mathematicae , volume=. 2018 , publisher=

    work page 2018

  32. [32]

    , author =

    Semi-Dual Unbalanced Quadratic Optimal Transport: fast statistical rates and convergent algorithm. , author =. Proceedings of the 40th International Conference on Machine Learning , pages =. 2023 , volume =

    work page 2023

  33. [33]

    International Conference on Learning Representations , year=

    Generative Modeling with Optimal Transport Maps , author=. International Conference on Learning Representations , year=

    work page

  34. [34]

    Transactions on Machine Learning Research , issn=

    Neural Monge Map estimation and its applications , author=. Transactions on Machine Learning Research , issn=. 2023 , note=

    work page 2023

  35. [35]

    Thirty-seventh Conference on Neural Information Processing Systems , year=

    Generative Modeling through the Semi-dual Formulation of Unbalanced Optimal Transport , author=. Thirty-seventh Conference on Neural Information Processing Systems , year=

    work page

  36. [36]

    The Eleventh International Conference on Learning Representations , year=

    Neural Optimal Transport , author=. The Eleventh International Conference on Learning Representations , year=

    work page

  37. [37]

    Forty-second International Conference on Machine Learning , year=

    Unpaired Point Cloud Completion via Unbalanced Optimal Transport , author=. Forty-second International Conference on Machine Learning , year=

    work page

  38. [38]

    The Twelfth International Conference on Learning Representations , year=

    Analyzing and Improving Optimal-Transport-based Adversarial Networks , author=. The Twelfth International Conference on Learning Representations , year=

    work page

  39. [39]

    Scalable

    Choi, Jaemoo and Choi, Jaewoong and Kang, Myungjoo , booktitle =. Scalable. 2024 , volume =

    work page 2024

  40. [40]

    International Conference on Machine Learning , pages=

    Optimal transport mapping via input convex neural networks , author=. International Conference on Machine Learning , pages=. 2020 , organization=

    work page 2020

  41. [41]

    International Conference on Learning Representations , year=

    Wasserstein-2 Generative Networks , author=. International Conference on Learning Representations , year=

    work page

  42. [42]

    The Thirteenth International Conference on Learning Representations , year=

    Improving Neural Optimal Transport via Displacement Interpolation , author=. The Thirteenth International Conference on Learning Representations , year=

    work page

  43. [43]

    The Thirteenth International Conference on Learning Representations , year=

    Robust Barycenter Estimation using Semi-Unbalanced Neural Optimal Transport , author=. The Thirteenth International Conference on Learning Representations , year=

    work page

  44. [44]

    Proceedings of the IEEE/CVF conference on computer vision and pattern recognition , pages=

    A style-based generator architecture for generative adversarial networks , author=. Proceedings of the IEEE/CVF conference on computer vision and pattern recognition , pages=

    work page

  45. [45]

    Proceedings of the IEEE/CVF conference on computer vision and pattern recognition , pages=

    Stargan v2: Diverse image synthesis for multiple domains , author=. Proceedings of the IEEE/CVF conference on computer vision and pattern recognition , pages=

    work page

  46. [46]

    Advances in neural information processing systems , volume=

    Gans trained by a two time-scale update rule converge to a local nash equilibrium , author=. Advances in neural information processing systems , volume=

    work page

  47. [47]

    Proceedings of the IEEE conference on computer vision and pattern recognition , pages=

    The unreasonable effectiveness of deep features as a perceptual metric , author=. Proceedings of the IEEE conference on computer vision and pattern recognition , pages=

    work page

  48. [48]

    Proceedings of the IEEE/CVF conference on computer vision and pattern recognition , pages=

    Explore image deblurring via encoded blur kernel space , author=. Proceedings of the IEEE/CVF conference on computer vision and pattern recognition , pages=

    work page

  49. [49]

    Diffusion Posterior Sampling for General Noisy Inverse Problems

    Diffusion posterior sampling for general noisy inverse problems , author=. arXiv preprint arXiv:2209.14687 , year=

    work page internal anchor Pith review Pith/arXiv arXiv

  50. [50]

    Advances in neural information processing systems , volume=

    Denoising diffusion restoration models , author=. Advances in neural information processing systems , volume=

    work page

  51. [51]

    ICLR , year=

    Pseudoinverse-Guided Diffusion Models for Inverse Problems , author=. ICLR , year=

    work page

  52. [52]

    Israel Journal of Mathematics , volume=

    Optimal transportation on non-compact manifolds , author=. Israel Journal of Mathematics , volume=. 2010 , publisher=

    work page 2010

  53. [53]

    IEEE Transactions on Pattern Analysis and Machine Intelligence , volume=

    Untrained neural network priors for inverse imaging problems: A survey , author=. IEEE Transactions on Pattern Analysis and Machine Intelligence , volume=. 2022 , publisher=

    work page 2022

  54. [54]

    Encyclopedia of applied and computational mathematics , pages=

    Regularization of inverse problems , author=. Encyclopedia of applied and computational mathematics , pages=. 2015 , publisher=

    work page 2015

  55. [55]

    1996 , publisher=

    Regularization of inverse problems , author=. 1996 , publisher=

    work page 1996

  56. [56]

    IEEE transactions on image processing , volume=

    Beyond a gaussian denoiser: Residual learning of deep cnn for image denoising , author=. IEEE transactions on image processing , volume=. 2017 , publisher=

    work page 2017

  57. [57]

    European conference on computer vision , pages=

    Learning a deep convolutional network for image super-resolution , author=. European conference on computer vision , pages=. 2014 , organization=

    work page 2014

  58. [58]

    Proceedings of the IEEE conference on computer vision and pattern recognition , pages=

    Deep multi-scale convolutional neural network for dynamic scene deblurring , author=. Proceedings of the IEEE conference on computer vision and pattern recognition , pages=

    work page

  59. [59]

    International conference on machine learning , pages=

    Do imagenet classifiers generalize to imagenet? , author=. International conference on machine learning , pages=. 2019 , organization=

    work page 2019

  60. [60]

    International conference on machine learning , pages=

    Compressed sensing using generative models , author=. International conference on machine learning , pages=. 2017 , organization=

    work page 2017

  61. [61]

    Solving bayesian inverse 51 problems via variational autoencoders

    Solving Bayesian inverse problems via variational autoencoders , author=. arXiv preprint arXiv:1912.04212 , year=

    work page arXiv 1912

  62. [62]

    Solving inverse problems in med- ical imaging with score-based generative models

    Solving inverse problems in medical imaging with score-based generative models , author=. arXiv preprint arXiv:2111.08005 , year=

    work page arXiv

  63. [63]

    Proceedings of the Computer Vision and Pattern Recognition Conference , pages=

    Improving diffusion inverse problem solving with decoupled noise annealing , author=. Proceedings of the Computer Vision and Pattern Recognition Conference , pages=

    work page

  64. [64]

    A Survey on Diffusion Models for Inverse Problems

    A survey on diffusion models for inverse problems , author=. arXiv preprint arXiv:2410.00083 , year=

    work page internal anchor Pith review Pith/arXiv arXiv

  65. [65]

    IEEE Transactions on Pattern Analysis and Machine Intelligence , volume=

    Optimal transport for unsupervised denoising learning , author=. IEEE Transactions on Pattern Analysis and Machine Intelligence , volume=. 2022 , publisher=

    work page 2022

  66. [66]

    2024 , volume =

    Residual-Conditioned Optimal Transport: Towards Structure-Preserving Unpaired and Paired Image Restoration , author =. 2024 , volume =

    work page 2024

  67. [67]

    Journal of Functional Analysis , pages=

    Regularity theory and geometry of unbalanced optimal transport , author=. Journal of Functional Analysis , pages=. 2025 , publisher=

    work page 2025

  68. [68]

    arXiv preprint arXiv:2502.04583 , year=

    Overcoming Spurious Solutions in Semi-Dual Neural Optimal Transport: A Smoothing Approach for Learning the Optimal Transport Plan , author=. arXiv preprint arXiv:2502.04583 , year=

    work page arXiv

  69. [69]

    arXiv preprint arXiv:2410.02628 , year=

    Inverse Entropic Optimal Transport Solves Semi-supervised Learning via Data Likelihood Maximization , author=. arXiv preprint arXiv:2410.02628 , year=

    work page arXiv

  70. [70]

    Advances in neural information processing systems , volume=

    Adversarial regularizers in inverse problems , author=. Advances in neural information processing systems , volume=

    work page

  71. [71]

    IEEE Transactions on Computational Imaging , volume=

    A neural-network-based convex regularizer for inverse problems , author=. IEEE Transactions on Computational Imaging , volume=. 2023 , publisher=

    work page 2023

  72. [72]

    Proceedings of the 42nd International Conference on Machine Learning , pages =

    Overcoming Spurious Solutions in Semi-Dual Neural Optimal Transport: A Smoothing Approach for Learning the Optimal Transport Plan , author =. Proceedings of the 42nd International Conference on Machine Learning , pages =. 2025 , volume =

    work page 2025

  73. [73]

    arXiv preprint arXiv:2410.02656 , year=

    Scalable simulation-free entropic unbalanced optimal transport , author=. arXiv preprint arXiv:2410.02656 , year=

    work page arXiv

  74. [74]

    Advances in Neural Information Processing Systems , volume=

    Towards prospective medical image reconstruction via knowledge-informed dynamic optimal transport , author=. Advances in Neural Information Processing Systems , volume=

    work page

  75. [75]

    Langley , title =

    P. Langley , title =. Proceedings of the 17th International Conference on Machine Learning (ICML 2000) , address =. 2000 , pages =

    work page 2000

  76. [76]

    T. M. Mitchell. The Need for Biases in Learning Generalizations. 1980

    work page 1980

  77. [77]

    M. J. Kearns , title =

    work page

  78. [78]

    Machine Learning: An Artificial Intelligence Approach, Vol. I. 1983

    work page 1983

  79. [79]

    R. O. Duda and P. E. Hart and D. G. Stork. Pattern Classification. 2000

    work page 2000

  80. [80]

    Suppressed for Anonymity , author=

    work page

Showing first 80 references.