pith. sign in

arxiv: 2512.10857 · v2 · pith:I33NTXR4new · submitted 2025-12-11 · 💻 cs.LG · cs.AI· stat.ML

Generative Modeling from Black-box Corruptions via Self-Consistent Stochastic Interpolants

Chirag Modi , Jiequn Han , Eric Vanden-Eijnden , Joan Bruna This is my paper

Pith reviewed 2026-05-16 22:59 UTC · model grok-4.3

classification 💻 cs.LG cs.AIstat.ML
keywords generative modelingstochastic interpolantsinverse problemsblack-box corruptionstransport mapsself-consistent iterationimage reconstructiondistribution learning
0
0 comments X

The pith

An iterative procedure with stochastic interpolants learns a transport map that inverts black-box corruptions to generate clean data from corrupted observations alone.

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

The paper introduces a method that builds generative models when only corrupted measurements are available, rather than clean data. It does so by repeatedly refining a transport map that connects corrupted samples to synthetic clean ones, using the corrupted dataset and black-box calls to the corruption process. Under suitable conditions the iteration settles on a self-consistent map that effectively undoes the corruption, producing a generative model for the original data. The approach is shown to be more efficient than variational baselines, to accommodate arbitrary nonlinear corruptions, and to carry convergence guarantees, with demonstrations on image and scientific reconstruction tasks.

Core claim

Under appropriate conditions, iteratively updating a transport map between corrupted observations and clean samples via stochastic interpolants, using only the corrupted dataset and black-box access to the corruption channel, converges to a self-consistent transport map that inverts the corruption channel and thereby enables generative sampling of the clean data distribution.

What carries the argument

Self-consistent stochastic interpolant (SCSI), an iterative update rule that refines a transport map until it is consistent with the observed corrupted distribution and the black-box corruption operator.

If this is right

  • Generative models can be trained directly on corrupted scientific or imaging datasets without requiring paired clean examples.
  • The method applies to any corruption that can be simulated as a black-box forward operator, including nonlinear ones.
  • Training cost is lower than variational alternatives because it avoids explicit density estimation or variational optimization.
  • Convergence of the iteration is guaranteed when the corruption channel and data distributions satisfy the stated conditions.
  • The learned map can be used for both generation and for solving inverse problems by pushing corrupted samples toward the clean distribution.

Where Pith is reading between the lines

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

  • The same iterative scheme could be applied to real-world sensor data where the corruption is only approximately known, provided a simulator can still be queried.
  • Combining the SCSI map with existing diffusion or flow models might allow hybrid pipelines that first correct corruption and then refine samples.
  • The convergence guarantees suggest that the method could serve as a building block for distribution-level inverse problems in other domains such as tomography or cryo-EM.
  • If the corruption is time-varying, an online version of the iteration might track drifting distributions without retraining from scratch.

Load-bearing premise

The iteration converges to a self-consistent transport map under suitable but unspecified conditions on the corruption channel and the underlying data distributions.

What would settle it

A concrete test would be to run the procedure on a known synthetic corruption (for example, a strongly nonlinear blur plus heavy noise) and check whether the generated clean samples match the true clean distribution in total variation or Wasserstein distance; failure to match would falsify the convergence claim.

Figures

Figures reproduced from arXiv: 2512.10857 by Chirag Modi, Eric Vanden-Eijnden, Jiequn Han, Joan Bruna.

Figure 1
Figure 1. Figure 1: Schematic of the SCSI: the fixed point Θ ∗ satisfies K𝜋Θ∗ = 𝜇, which in turns implies 𝜋Θ∗ = 𝜋. No samples from 𝜋 are required—the approach only uses corrupted samples from 𝜇 and the map F. Algorithm 1: Training of Self-Consistent SI 1 Θ ← Θ (0) // Initialize 2 for 𝑘 in 1 . . . 𝐾 do 3 for 𝑖 in 1 . . . 𝑇tr do 4 Sample 𝐼𝑡 in (8) with Θ (𝑘) 5 SGD update of Θ via losses (3)(4) 6 Θ (𝑘) ← Θ // Update transport ma… view at source ↗
Figure 2
Figure 2. Figure 2: Convergence of ∥Σ − Σ𝑘 ∥ 2 using either ODE (𝜖 = 0, in red) or the EM algorithm (𝜖 = 1, green curve), for Σ drawn from the Wishart distribution. In dashed we plot the rates 1/𝑘 2 and 1/𝑘 respectively. where 𝛾𝑏,Σ denotes the Gaussian density with mean 𝑏 and covariance Σ. This posterior is then used to update the prior via the ‘M’ step, defining the mixture d𝜋𝑘+1 (𝑥) = 𝔼𝑦∼𝜇 [𝑝𝑘 (d𝑥|𝑦)] = 𝛾0,Σ˜ 𝑘 (𝑥)𝔼𝑦∼𝛾0,Σ+𝐼… view at source ↗
Figure 3
Figure 3. Figure 3: Scatterplot of 𝑊2 2 (𝐴, 𝐵) (y-axis) versus T (𝐴; 𝐵) (x-axis), for pairs of independent Wishart matrices 𝐴 and 𝐵. Red line corresponds to 𝑅 = 1. Left: Low SNR, Center: Moderate SNR, Right: High SNR. We observe that 𝑅 < 1 with high probability. from the Wishart ensemble and verify numerically that T (𝐴; 𝐵) < 𝑊2 2 (𝐴, 𝐵) with overwhelming probability, even for moderately small SNRs; see [PITH_FULL_IMAGE:figu… view at source ↗
Figure 4
Figure 4. Figure 4: AWGN channel: Comparing ODE and SDE restoration for different noise levels (𝜎𝑛). While the SDE formalism can be more ro￾bust for highly corrupting forward models in synthetic examples, we find that for high dimen￾sional experiments, it is also more sensitive to hyperparameters like noise schedule and num￾ber of transport steps. On the other hand, ODE approach works well for moderate corruptions, is largely… view at source ↗
Figure 5
Figure 5. Figure 5: Restored samples for different forward maps from our interpolants and DPS. [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: JPEG + noise: results for different compression level (Top: Corrupted; Bottom: Restored). [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Restored quasar spectra for different scenarios. Legends show MSE over 1000 samples. [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Randomly drawn images from the large diffusion model trained on cleaned images. [PITH_FULL_IMAGE:figures/full_fig_p029_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Randomly drawn images from the smaller diffusion model trained on cleaned images. [PITH_FULL_IMAGE:figures/full_fig_p030_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Training loss curves. D.2 SDE Results We present results for restoration using SDE in this section. We experiment with two different strategies: i) learning a combined drift for the velocity drift and score terms using the objective (5), and ii) learning different networks for drift and score separately using losses (3)(4). While the latter offers more flexibility in varying the noise schedule during infe… view at source ↗
Figure 11
Figure 11. Figure 11: Restored samples for different forward maps with SDE based SI. [PITH_FULL_IMAGE:figures/full_fig_p031_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Restored quasar spectra using SDE. The numbers in the legend represent the mean squared error [PITH_FULL_IMAGE:figures/full_fig_p031_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Restoring images with SI for varying fractions of masked pixels (levels of corruptions). [PITH_FULL_IMAGE:figures/full_fig_p033_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Samples from the diffusion model trained on the restored samples of random masking experiment [PITH_FULL_IMAGE:figures/full_fig_p034_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Restoring images with SI for varying size of motion blur kernel (levels of corruptions). [PITH_FULL_IMAGE:figures/full_fig_p034_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Additional images for JPEG restoration for the model trained on samples with [PITH_FULL_IMAGE:figures/full_fig_p035_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: F: JPEG compression + noise (𝜎𝑛 = 0.01): results for different compression levels (Top: Corrupted; Bottom: Restored). Model is trained only on samples with 𝑞 ∼ U [0.1, 0.5]. Results for higher qualities are in extrapolation regime. Original Image JPEG Compress Restored [PITH_FULL_IMAGE:figures/full_fig_p035_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Additional images for JPEG restoration for the model trained on samples with [PITH_FULL_IMAGE:figures/full_fig_p035_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Restoring images with SI for Gaussian blurring with Poisson noise for different noise levels. [PITH_FULL_IMAGE:figures/full_fig_p036_19.png] view at source ↗
read the original abstract

Transport-based methods have emerged as a leading paradigm for building generative models from large, clean datasets. However, in many scientific and engineering domains, clean data are often unavailable: instead, we only observe measurements corrupted through a noisy, ill-conditioned channel. A generative model for the original data thus requires solving an inverse problem at the level of distributions. In this work, we introduce a novel approach to this task based on Stochastic Interpolants: we iteratively update a transport map between corrupted and clean data samples using only access to the corrupted dataset as well as black box access to the corruption channel. Under appropriate conditions, this iterative procedure converges towards a self-consistent transport map that effectively inverts the corruption channel, thus enabling a generative model for the clean data. We refer to the resulting method as the self-consistent stochastic interpolant (SCSI). It (i) is computationally efficient compared to variational alternatives, (ii) highly flexible, handling arbitrary nonlinear forward models with only black-box access, and (iii) enjoys theoretical guarantees. We demonstrate superior performance on inverse problems in natural image processing and scientific reconstruction, and establish convergence guarantees of the scheme under appropriate assumptions. Our source code is publicly available at https://github.com/modichirag/SCSI

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 1 minor

Summary. The manuscript introduces the Self-Consistent Stochastic Interpolant (SCSI) method for generative modeling from black-box corrupted data. It iteratively updates a transport map between corrupted and clean distributions using only corrupted samples and black-box access to the corruption channel, based on stochastic interpolants. The central claim is that this procedure converges to a self-consistent fixed-point map inverting the corruption channel under appropriate conditions, enabling clean-data generation. The work provides theoretical convergence guarantees, claims computational efficiency and flexibility over variational alternatives, and reports superior empirical performance on natural-image inverse problems and scientific reconstruction tasks.

Significance. If the convergence holds with explicit conditions, SCSI would supply an efficient, black-box-compatible alternative to existing transport and variational methods for distribution-level inverse problems. This is potentially impactful in scientific domains where clean data are unavailable. The public code release aids reproducibility and is a positive feature.

major comments (2)
  1. [Abstract] Abstract: The claim that the iterative procedure 'converges towards a self-consistent transport map' under 'appropriate conditions' is load-bearing for the central contribution, yet those conditions (e.g., contractivity of the update operator or uniqueness of the fixed point in the relevant function space) are left unspecified. This prevents assessment of applicability to general nonlinear black-box operators, where non-monotonicity or non-convex support can produce multiple fixed points or non-convergence.
  2. [Theoretical analysis section] Theoretical analysis section: The convergence proof must explicitly verify that the map-update operator is contractive (or possesses a unique fixed point) for the corruption channels used in the experiments; the current presentation leaves these requirements implicit, undermining the guarantee for arbitrary nonlinear forward models.
minor comments (1)
  1. [Method section] Clarify the precise definition of the stochastic interpolant and the update rule in the method section to improve reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their insightful comments on our manuscript. We have revised the abstract and theoretical analysis to more explicitly state the convergence conditions. Our point-by-point responses are provided below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The claim that the iterative procedure 'converges towards a self-consistent transport map' under 'appropriate conditions' is load-bearing for the central contribution, yet those conditions (e.g., contractivity of the update operator or uniqueness of the fixed point in the relevant function space) are left unspecified. This prevents assessment of applicability to general nonlinear black-box operators, where non-monotonicity or non-convex support can produce multiple fixed points or non-convergence.

    Authors: We appreciate this feedback. The appropriate conditions are the contractivity of the map-update operator, which is proven in the theoretical analysis under the assumption that the corruption channel is a contraction mapping (Lipschitz constant <1) in the Wasserstein metric. This guarantees a unique fixed point by the Banach fixed-point theorem. We have updated the abstract to specify: 'Under the contractivity condition on the corruption channel detailed in Section 3, the iterative procedure converges to the self-consistent transport map.' This clarifies the scope and allows assessment for general nonlinear operators, where the condition may or may not hold. revision: yes

  2. Referee: [Theoretical analysis section] Theoretical analysis section: The convergence proof must explicitly verify that the map-update operator is contractive (or possesses a unique fixed point) for the corruption channels used in the experiments; the current presentation leaves these requirements implicit, undermining the guarantee for arbitrary nonlinear forward models.

    Authors: We agree that explicit verification for the experimental settings enhances the presentation. In the revised version, we have added explicit checks in the theoretical analysis section for the corruption channels used (e.g., Gaussian noise with variance sigma^2 and blurring kernels), showing that the induced map-update operator has Lipschitz constant <1 under the parameter ranges in the experiments. For arbitrary black-box models, the guarantee is conditional on this property, which we now emphasize more clearly. This does not change the general proof but makes the connection to experiments explicit. revision: partial

Circularity Check

0 steps flagged

No significant circularity in the SCSI derivation chain

full rationale

The paper defines an iterative fixed-point procedure that updates a transport map using stochastic interpolants, corrupted samples, and black-box corruption access. The central claim is convergence of this iteration to a self-consistent map that inverts the channel under stated assumptions on the distributions and operator. This does not reduce by construction to the inputs: the self-consistency property is a consequence of the fixed-point equation rather than a definitional tautology, and the inversion claim follows from the transport map satisfying the pushforward relation at equilibrium. No load-bearing self-citations, fitted parameters renamed as predictions, or smuggled ansatzes appear in the provided derivation outline. The guarantees are presented as independent theorems relying on contractivity or uniqueness conditions external to the iteration definition itself, making the overall chain self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of a self-consistent transport map and convergence of the iteration; these are stated as holding under appropriate conditions but are not further detailed in the abstract.

axioms (1)
  • domain assumption The iterative procedure converges to a self-consistent transport map under appropriate conditions on the corruption channel and data.
    Stated in the abstract as the basis for the method's validity.

pith-pipeline@v0.9.0 · 5533 in / 1152 out tokens · 28390 ms · 2026-05-16T22:59:02.873512+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

18 extracted references · 18 canonical work pages · 2 internal anchors

  1. [1]

    Stochastic Interpolants: A Unifying Framework for Flows and Diffusions

    [ABVE23] Michael S Albergo, Nicholas M Boffi, and Eric Vanden-Eijnden. Stochastic interpolants: A unifying framework for flows and diffusions.arXiv preprint arXiv:2303.08797,

    work page internal anchor Pith review Pith/arXiv arXiv

  2. [2]

    Stochastic interpolants with data-dependent couplings

    [AGB+23] Michael S Albergo, Mark Goldstein, Nicholas M Boffi, Rajesh Ranganath, and Eric Vanden-Eijnden. Stochastic interpolants with data-dependent couplings.arXiv preprint arXiv:2310.03725,

    work page arXiv

  3. [3]

    Building normalizing flows with stochastic interpolants

    [AVE23] Michael Albergo and Eric Vanden-Eijnden. Building normalizing flows with stochastic interpolants. InICLR 2023 Conference,

    work page 2023

  4. [4]

    Nearly d-linear convergence bounds for diffu- sion models via stochastic localization.arXiv preprint arXiv:2308.03686,

    [BDBDD23] Joe Benton, Valentin De Bortoli, Arnaud Doucet, and George Deligiannidis. Nearly𝑑- linear convergence bounds for diffusion models via stochastic localization.arXiv preprint arXiv:2308.03686,

    work page arXiv

  5. [5]

    Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions.arXiv preprint arXiv:2209.11215,

    17 [CCL+22] Sitan Chen, Sinho Chewi, Jerry Li, Yuanzhi Li, Adil Salim, and Anru R Zhang. Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions. arXiv preprint arXiv:2209.11215,

    work page arXiv

  6. [6]

    S., Boffi, N

    [CGH+24] Yifan Chen, Mark Goldstein, Mengjian Hua, Michael S Albergo, Nicholas M Boffi, and Eric Vanden-Eijnden. Probabilistic forecasting with stochastic interpolants and Föllmer processes. arXiv preprint arXiv:2403.13724,

    work page arXiv

  7. [7]

    Diffusion Posterior Sampling for General Noisy Inverse Problems

    [CKM+22] HyungjinChung, JeongsolKim, MichaelTMccann, MarcLKlasky, andJongChulYe. Diffu- sion posterior sampling for general noisy inverse problems.arXiv preprint arXiv:2209.14687,

    work page internal anchor Pith review Pith/arXiv arXiv

  8. [8]

    Denoising score distillation: From noisy diffusion pretraining to one-step high-quality generation.arXiv preprint arXiv:2503.07578,

    [CZW+25] Tianyu Chen, Yasi Zhang, Zhendong Wang, Ying Nian Wu, Oscar Leong, and Mingyuan Zhou. Denoising score distillation: From noisy diffusion pretraining to one-step high-quality generation.arXiv preprint arXiv:2503.07578,

    work page arXiv

  9. [9]

    Ambient diffusion omni: Training good models with bad data.arXiv preprint arXiv:2506.10038,

    [DRMK+25] Giannis Daras, Adrian Rodriguez-Munoz, Adam Klivans, Antonio Torralba, and Constantinos Daskalakis. Ambient diffusion omni: Training good models with bad data.arXiv preprint arXiv:2506.10038,

    work page arXiv

  10. [10]

    Flow matching for scalable simulation-based inference.arXiv preprint arXiv:2305.17161,

    [DWB+23] Maximilian Dax, Jonas Wildberger, Simon Buchholz, Stephen R Green, Jakob H Macke, and Bernhard Schölkopf. Flow matching for scalable simulation-based inference.arXiv preprint arXiv:2305.17161,

    work page arXiv

  11. [11]

    Diffem: Learning from corrupted data with diffusion models via expectation maximization

    [HCD+25] DanialHosseintabar,FanChen,GiannisDaras,AntonioTorralba,andConstantinosDaskalakis. Diffem: Learning from corrupted data with diffusion models via expectation maximization. arXiv preprint arXiv:2510.12691,

    work page arXiv

  12. [12]

    Any-order flexible length masked diffusion

    [KCKDE+25] Jaeyeon Kim, Lee Cheuk-Kit, Carles Domingo-Enrich, Yilun Du, Sham Kakade, Timothy Ngotiaoco, Sitan Chen, and Michael Albergo. Any-order flexible length masked diffusion. arXiv preprint arXiv:2509.01025,

    work page arXiv

  13. [13]

    Stochastic inverse problem: stability, regularization and wasserstein gradient flow.arXiv preprint arXiv:2410.00229,

    [LOWY24] Qin Li, Maria Oprea, Li Wang, and Yunan Yang. Stochastic inverse problem: stability, regularization and wasserstein gradient flow.arXiv preprint arXiv:2410.00229,

    work page arXiv

  14. [14]

    Inverse problems over probability measure space.arXiv preprint arXiv:2504.18999,

    [LOWY25] Qin Li, Maria Oprea, Li Wang, and Yunan Yang. Inverse problems over probability measure space.arXiv preprint arXiv:2504.18999,

    work page arXiv

  15. [15]

    [MRAM25] GiacomoMeanti,ThomasRyckeboer,MichaelArbel,andJulienMairal.Unsupervisedimaging inverse problems with diffusion distribution matching.arXiv preprint arXiv:2506.14605,

    work page arXiv

  16. [16]

    System-embedded diffusion bridge models.arXiv preprint arXiv:2506.23726,

    [STW+25] Bartlomiej Sobieski, Matthew Tivnan, Yuang Wang, Siyeop Yoon, Pengfei Jin, Dufan Wu, Quanzheng Li, and Przemyslaw Biecek. System-embedded diffusion bridge models.arXiv preprint arXiv:2506.23726,

    work page arXiv

  17. [17]

    Restoration score distillation: From corrupted diffusion pretraining to one-step high-quality generation.arXiv preprint arXiv:2505.13377,

    [ZCW+25] YasiZhang,TianyuChen,ZhendongWang,YingNianWu,MingyuanZhou,andOscarLeong. Restoration score distillation: From corrupted diffusion pretraining to one-step high-quality generation.arXiv preprint arXiv:2505.13377,

    work page arXiv

  18. [18]

    We fix the learning rate to be 0.0005 and use cosine schedule with warmup

    𝑇tr Wasserstein Distance (Mean±Std) 10.0491±0.0038 100.0460±0.0038 1000.0476±0.0047 10000.0593±0.0021 integration, which is the most expensive step in the training process. We fix the learning rate to be 0.0005 and use cosine schedule with warmup. Random masking, motion blur and JPEG experiments are trained for 50,000 iterations while other experiments ar...

    work page arXiv