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arxiv: 2512.17051 · v2 · submitted 2025-12-18 · 💻 cs.LG

SFBD-OMNI: Bridge models for lossy measurement restoration with limited clean samples

Haoye Lu , Yaoliang Yu , Darren Lo This is my paper

Pith reviewed 2026-05-16 21:14 UTC · model grok-4.3

classification 💻 cs.LG
keywords distribution restorationentropic optimal transportbridge modelsEM algorithmnoisy samplesrecoverability testmeasurement modelsdeconvolution
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The pith

Distribution restoration from noisy samples reduces to a one-sided entropic optimal transport problem solved by an EM-like algorithm.

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

The paper shows that restoring an unknown data distribution from many noisy observations is feasible when the corruption process can be sampled arbitrarily as a black box. Casting the task as a one-sided entropic optimal transport problem yields an EM-like iterative solver that maps corrupted samples to the ground-truth distribution. A recoverability test identifies cases where per-sample information loss prevents full recovery, yet the addition of only a small number of clean samples renders the distribution largely recoverable in those cases. These elements are realized in the SFBD-OMNI framework, which extends bridge models to arbitrary measurement corruptions beyond Gaussian noise and reports improved performance on benchmark datasets.

Core claim

We show that this task can be framed as a one-sided entropic optimal transport problem and solved via an EM-like algorithm. We further provide a test criterion to determine whether the true underlying distribution is recoverable under per-sample information loss, and show that in otherwise unrecoverable cases, a small number of clean samples can render the distribution largely recoverable. Building on these insights, we introduce SFBD-OMNI, a bridge model-based framework that maps corrupted sample distributions to the ground-truth distribution and generalizes Stochastic Forward-Backward Deconvolution to handle arbitrary measurement models.

What carries the argument

one-sided entropic optimal transport problem solved via an EM-like algorithm

If this is right

  • The restoration task admits an efficient EM-like iterative solver.
  • A recoverability test identifies when the true distribution can be recovered from noisy samples alone.
  • A small number of clean samples suffices to recover distributions that are otherwise unrecoverable.
  • Bridge models trained under this formulation generalize to arbitrary measurement models beyond Gaussian corruption.
  • The method produces measurable gains in both qualitative and quantitative metrics across benchmark datasets.

Where Pith is reading between the lines

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

  • The black-box corruption assumption enables direct use of physics simulators or sensor models without deriving explicit likelihoods.
  • The recoverability test could guide experimental design by indicating when additional clean samples are worth collecting.
  • Similar transport-based reasoning may apply to other inverse problems such as image deblurring or sensor fusion with partial observations.
  • Hybrid datasets containing mostly noisy samples plus a few clean ones may become standard for practical distribution learning tasks.

Load-bearing premise

The corruption process must be exactly known and freely sampleable as a black-box generator, with the one-sided entropic optimal transport model faithfully representing the true data-generating process.

What would settle it

Generate synthetic data from a known ground-truth distribution using a known corruption black-box, run the EM-like algorithm, and check whether the restored distribution matches the ground truth to within small error when the recoverability test predicts success.

Figures

Figures reproduced from arXiv: 2512.17051 by Darren Lo, Haoye Lu, Yaoliang Yu.

Figure 1
Figure 1. Figure 1: Effect of λ on p ∗ λ . As λ → 0, the first term in Eq (11) ensures that p remains within S(q), while the second term selects the element h † ∈ S(q) clos￾est to h. Consequently, p ∗ λ converges to h † , which represents the projection of h onto the feasible set S(q). All proofs are deferred to the appendix. We highlight sev￾eral common corruption operators Tr together with their injectivity properties: Addi… view at source ↗
Figure 2
Figure 2. Figure 2: FID scores of Online SFBD-OMNI under different clean sample weights [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FID scores of SFBD-OMNI under different settings. (a) Online SFBD-OMNI FIDs [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Pixel Masking (a) SFBD-OMNI (FID: 10.81) (b) Online SFBD-OMNI (FID: 11.06) [PITH_FULL_IMAGE:figures/full_fig_p023_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Addictive Gauss. 23 [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Grayscale H.2 CELEBA (a) SFBD-OMNI (FID: 11.60) (b) Online SFBD-OMNI (FID: 10.28) [PITH_FULL_IMAGE:figures/full_fig_p024_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Gauss. Blur 24 [PITH_FULL_IMAGE:figures/full_fig_p024_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Grayscale I DISCUSSION ON AMBIENT DIFFUSION OMNI Ambient diffusion-Omni (Ambient-o) incorporates corrupted samples by injecting additional Gaus￾sian noise. The key idea is that once sufficient Gaussian noise is added, the corrupted-noisy distribu￾tion and the clean-noisy distribution become harder to distinguish. This observation suggests that a corrupted sample, after being further noised, can effectively… view at source ↗
Figure 9
Figure 9. Figure 9: Reconstructed Satellite Images – Poisson Noise (photo budget [PITH_FULL_IMAGE:figures/full_fig_p027_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Reconstructed MRI – Compressive Sensing and after online iterative refinement. Across all evaluated settings, the online phase yields a clear improvement over the pretrained model, demonstrating the effectiveness of SFBD-OMNI as a general reconstruction framework for real-world corruption processes. We provide qualitative reconstructions in Figures 9 and 10. On satellite images, SFBD-OMNI visibly recovers… view at source ↗
read the original abstract

In many real-world scenarios, obtaining fully observed samples is prohibitively expensive or even infeasible, while partial and noisy observations are comparatively easy to collect. In this work, we study distribution restoration with abundant noisy samples, assuming the corruption process is available as a black-box generator. We show that this task can be framed as a one-sided entropic optimal transport problem and solved via an EM-like algorithm. We further provide a test criterion to determine whether the true underlying distribution is recoverable under per-sample information loss, and show that in otherwise unrecoverable cases, a small number of clean samples can render the distribution largely recoverable. Building on these insights, we introduce SFBD-OMNI, a bridge model-based framework that maps corrupted sample distributions to the ground-truth distribution. Our method generalizes Stochastic Forward-Backward Deconvolution (SFBD; Lu et al., 2025) to handle arbitrary measurement models beyond Gaussian corruption. Experiments across benchmark datasets and diverse measurement settings demonstrate significant improvements in both qualitative and quantitative performance.

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

3 major / 1 minor

Summary. The manuscript frames distribution restoration from abundant lossy measurements (with corruption available as a black-box generator) as a one-sided entropic optimal transport problem solved by an EM-like algorithm. It introduces a recoverability test for the true distribution under per-sample information loss and shows that a small number of clean samples can render otherwise unrecoverable cases largely recoverable. SFBD-OMNI is proposed as a bridge-model generalization of prior SFBD work to arbitrary (non-Gaussian) measurement models, with claimed qualitative and quantitative improvements on benchmarks.

Significance. If the OT framing, EM procedure, and recoverability criterion hold with the stated assumptions, the work would offer a principled extension of deconvolution methods to general corruption settings with limited clean data, potentially useful in imaging or sensor applications. The generalization of SFBD and the explicit recoverability test are the primary potential contributions, though the absence of derivations or quantitative results in the visible text constrains the assessed impact.

major comments (3)
  1. [Abstract and Methods] Abstract and Methods: the central claim that the task can be framed as a one-sided entropic optimal transport problem and solved via an EM-like algorithm is stated without any derivation details, error bounds, or proof sketches, which is load-bearing for validating the theoretical contribution and the subsequent recoverability test.
  2. [Abstract and Experiments] Abstract and Experiments: the recoverability test and the claim that 'a small number of clean samples can render the distribution largely recoverable' rest on the black-box corruption generator exactly reproducing the true per-sample loss and on the OT formulation being faithful; neither is independently verified or derived in the provided text.
  3. [Experiments] Experiments: significant improvements are asserted across benchmark datasets and diverse measurement settings, yet the visible text contains no quantitative tables, specific metrics, or baseline comparisons, preventing assessment of the empirical support.
minor comments (1)
  1. [Abstract] Abstract: the statement that the method 'generalizes Stochastic Forward-Backward Deconvolution (SFBD; Lu et al., 2025) to handle arbitrary measurement models' would benefit from an explicit list of the new assumptions introduced by the one-sided entropic OT formulation.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive feedback, which helps clarify the presentation of our theoretical and empirical contributions. We address each major comment point by point below and will revise the manuscript to incorporate additional details where needed.

read point-by-point responses

  1. Referee: [Abstract and Methods] Abstract and Methods: the central claim that the task can be framed as a one-sided entropic optimal transport problem and solved via an EM-like algorithm is stated without any derivation details, error bounds, or proof sketches, which is load-bearing for validating the theoretical contribution and the subsequent recoverability test.

    Authors: We agree that the abstract and main Methods section present the one-sided entropic OT framing and EM-like algorithm at a high level. The full manuscript derives the formulation from the per-sample corruption model in Section 3, but to strengthen the theoretical contribution we will add an expanded subsection with explicit derivation steps, a proof sketch for the EM procedure, and error bounds under the stated assumptions in the revised version. revision: yes

  2. Referee: [Abstract and Experiments] Abstract and Experiments: the recoverability test and the claim that 'a small number of clean samples can render the distribution largely recoverable' rest on the black-box corruption generator exactly reproducing the true per-sample loss and on the OT formulation being faithful; neither is independently verified or derived in the provided text.

    Authors: The recoverability test follows directly from the OT objective and the assumption that the black-box generator matches the true corruption distribution, as defined in the problem setup. We will add a dedicated verification subsection with the formal derivation of the test criterion and new experiments quantifying the effect of limited clean samples on recoverability in the revision. revision: yes

  3. Referee: [Experiments] Experiments: significant improvements are asserted across benchmark datasets and diverse measurement settings, yet the visible text contains no quantitative tables, specific metrics, or baseline comparisons, preventing assessment of the empirical support.

    Authors: The complete manuscript includes quantitative tables (e.g., Wasserstein-2 distances, FID scores) and baseline comparisons in Section 5. To address the concern about visibility, we will expand and prominently feature these tables and metrics in the main text of the revised manuscript, adding further details on the diverse measurement settings. revision: yes

Circularity Check

0 steps flagged

No significant circularity; new OT framing, EM algorithm, and recoverability test are derived independently in this paper.

full rationale

The paper claims to frame distribution restoration as a one-sided entropic OT problem solved via EM-like algorithm and to provide a new recoverability test criterion, both presented as contributions of this work. It generalizes the authors' prior SFBD (Lu et al. 2025) to non-Gaussian cases but does not reduce the new claims to definitions or fits from the prior work or from parameters fitted within this manuscript. The black-box corruption assumption is stated explicitly as given, and the recoverability test plus few-clean-sample result are derived from the OT formulation rather than presupposing the conclusion. No equation or step is shown to be equivalent to its input by construction, and the self-citation is not load-bearing for the central novel results. The derivation chain is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the corruption process being exactly known as a black-box sampler and on the one-sided entropic OT formulation accurately capturing the information loss; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Corruption process is available as an exact black-box generator that can be sampled from at will
    Stated in the problem setup and required for both the OT framing and the EM algorithm.

pith-pipeline@v0.9.0 · 5480 in / 1287 out tokens · 23187 ms · 2026-05-16T21:14:22.849292+00:00 · methodology

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Lean theorems connected to this paper

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

  • Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    We show that this task can be framed as a one-sided entropic optimal transport problem and solved via an EM-like algorithm... SFBD-OMNI... generalizes Stochastic Forward-Backward Deconvolution

What do these tags mean?
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supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
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The paper appears to rely on the theorem as machinery.
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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

4 extracted references · 4 canonical work pages

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    Sparse MRI: The application of compressed sensing for rapid MR imaging

    URLhttps://openreview.net/forum?id=a-xFK8Ymz5J. Alex Krizhevsky and Geoffrey Hinton. Learning multiple layers of features from tiny images. Technical report, University of Toronto, 2009. URL https://www.cs.toronto.edu/ ˜kriz/learning-features-2009-TR.pdf. Christian L´eonard. A survey of the schr ¨odinger problem and some of its connections with optimal tr...

    work page doi:10.1002/mrm.21391 2009

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    Diffsound: Discrete diffusion model for text-to-sound generation

    URLhttps://openreview.net/forum?id=vaRCHVj0uGI. Yang Song, Prafulla Dhariwal, Mark Chen, and Ilya Sutskever. Consistency models. InProceedings of the 40th International Conference on Machine Learning, pp. 32211–32252, 2023. URL https://proceedings.mlr.press/v202/song23a.html. 14 Francisco Vargas, Pierre Thodoroff, Austen Lamacraft, and Neil Lawrence. Solv...

    work page arXiv 2023

  3. [3]

    in the second last equation. Lemma 1.Given the cost function be c(x,y) =−logr(y|x) for some corruption kernel r, consider the problem min π∈Πy(q) ZZ π(x,y)c(x,y)dxdy+D KL π∥p⊗q , where Πy(q) is the set of joint distributions with fixed y-marginal q. If q is realizable under p via r, i.e.p∈ S(q), then the optimizer is π⋆(x,y) =p(x|y)q(y), which has margina...

    work page 1983

  4. [4]

    Figure 9: Reconstructed Satellite Images – Poisson Noise (photo budgetα= 10,50,100)

    and MRI scans (compressive sensing). Figure 9: Reconstructed Satellite Images – Poisson Noise (photo budgetα= 10,50,100). therefore corresponds to sampling the Fourier transform of the image. Because clinical MRI protocols routinely undersample k-space to shorten scan time, compressed-sensing MRI accelerates acquisition by collecting only a subset of freq...

    work page 2007