Random-Bridges as Stochastic Transports for Generative Models

Berkan Sesen; Levent A. Meng\"ut\"urk; Murat C. Meng\"ut\"urk; Stefano Goria

arxiv: 2512.14190 · v3 · submitted 2025-12-16 · 💻 cs.LG · math.PR

Random-Bridges as Stochastic Transports for Generative Models

Stefano Goria , Levent A. Meng\"ut\"urk , Murat C. Meng\"ut\"urk , Berkan Sesen This is my paper

Pith reviewed 2026-05-16 22:04 UTC · model grok-4.3

classification 💻 cs.LG math.PR

keywords random bridgesstochastic transportsgenerative modelsGaussian bridgesdiffusion samplingFréchet inception distancehigh-speed generation

0 comments

The pith

Random-bridges act as stochastic transports that generate high-quality samples in fewer steps than standard methods.

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

The paper introduces random-bridges as stochastic processes conditioned to match specified distributions at chosen time points and shows they can function as transports between probability measures in generative modeling. Depending on the driving process, these bridges can be Markovian or non-Markovian and can follow continuous, discontinuous, or hybrid trajectories. Experiments using Gaussian random bridges produce samples of competitive visual quality while requiring substantially fewer steps than conventional sampling procedures, with Fréchet inception distance scores remaining comparable. The resulting procedure is presented as computationally light and therefore suited to high-speed generation tasks.

Core claim

Random-bridges can serve as stochastic transports between two probability distributions when appropriately initialized, and Gaussian random bridges in particular produce high-quality samples in significantly fewer steps than traditional generative approaches while achieving competitive Fréchet inception distance scores.

What carries the argument

Random-bridge: a stochastic process conditioned to reach prescribed distributions at fixed time points, used here as a transport map between probability measures.

If this is right

Generative sampling can reach target quality with a reduced number of iterative steps.
The same transport construction supports both Markovian and non-Markovian dynamics depending on the chosen driving noise.
The framework yields a computationally inexpensive procedure suitable for high-speed generation.
Learning and simulation algorithms can be derived directly from the underlying probabilistic statements.

Where Pith is reading between the lines

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

The same bridge construction could be applied to modalities other than images once suitable driving processes are identified.
Explicit comparison with score-based diffusion methods might clarify whether the step reduction arises from the conditioning structure itself.
Replacing the Gaussian driving process with heavier-tailed or learned noise could relax the load-bearing assumption for highly non-Gaussian targets.

Load-bearing premise

Gaussian random bridges initialized in a suitable way remain effective transports even when the target distributions are complex and non-Gaussian.

What would settle it

A head-to-head test on a standard natural-image dataset in which Gaussian random-bridge sampling yields markedly higher Fréchet inception distance scores or requires at least as many steps as a baseline diffusion sampler would falsify the central empirical claim.

read the original abstract

This paper motivates the use of random-bridges -- stochastic processes conditioned to take target distributions at fixed timepoints -- in the realm of generative modelling. Herein, random-bridges can act as stochastic transports between two probability distributions when appropriately initialized, and can display either Markovian or non-Markovian, and either continuous, discontinuous or hybrid patterns depending on the driving process. We show how one can start from general probabilistic statements and then branch out into specific representations for learning and simulation algorithms in terms of information processing. Our empirical results, built on Gaussian random bridges, produce high-quality samples in significantly fewer steps compared to traditional approaches, while achieving competitive Frechet inception distance scores. Our analysis provides evidence that the proposed framework is computationally cheap and suitable for high-speed generation tasks.

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.

Desk Editor's Note private letter to a colleague

Random bridges as conditioned stochastic transports give a fresh algorithmic path for fewer-step generation, but the Gaussian-to-image transfer rests on thin evidence.

read the letter

The paper's core move is to position random-bridges—stochastic processes conditioned on hitting target distributions at fixed times—as stochastic transports for generative models. They begin with general probabilistic statements about these conditioned processes and then derive representations for learning and simulation, including Markovian or non-Markovian behaviors depending on the driver. The empirical part uses Gaussian random bridges to generate high-quality samples in significantly fewer steps than traditional methods while hitting competitive Fréchet inception distance scores, and they argue the whole thing is computationally light for fast generation. What works is the clean progression from broad probability to concrete algorithms. Framing the bridges as transports between distributions is a useful lens, and showing how they can be initialized to act that way opens a path that differs from standard diffusion setups. The claim of fewer steps is the kind of practical win that could matter for applications where compute is tight. The main weakness is the jump from Gaussian bridges to complex targets like natural images. Gaussian processes are constrained by their second-order statistics and struggle with multimodality or heavy tails even after conditioning, so the initialization has to do a lot of heavy lifting to make the transport effective. The abstract gives no specifics on how that initialization is done, what baselines were used, or any error bars, which makes it difficult to assess whether the reported performance actually supports the broader claim. Without those details the evidence for the non-Gaussian case stays thin. This paper is aimed at researchers in generative modeling who are looking for stochastic-process alternatives to current diffusion or flow-based methods. Someone already comfortable with conditioned processes and information-processing views of learning would find the derivations straightforward and potentially useful. The probabilistic starting point is solid enough that it deserves a serious referee to check the derivations and ask for the missing experimental controls. Recommendation: send it out for peer review so the authors can supply the setup details and test the Gaussian assumption more rigorously on real image data.

Referee Report

2 major / 1 minor

Summary. The paper proposes random-bridges—stochastic processes conditioned on target distributions at fixed time points—as stochastic transports between probability distributions for generative modeling. Starting from general probabilistic statements, it derives specific representations for learning and simulation algorithms, with a focus on Gaussian random bridges that are claimed to generate high-quality samples in significantly fewer steps than traditional methods while achieving competitive Fréchet Inception Distance scores.

Significance. If substantiated, the work could offer a flexible and computationally efficient framework for high-speed generative tasks by exploiting conditioned stochastic processes with Markovian or non-Markovian dynamics. The derivation from general probabilistic statements to concrete information-processing representations is a methodological strength that may extend beyond the Gaussian case.

major comments (2)

[Abstract] Abstract: The central empirical claim of high-quality samples in significantly fewer steps and competitive FID scores is stated without any details on experimental setup, baselines, error bars, initialization of the Gaussian bridges, training procedure, or data handling; this absence prevents assessment of whether the data support the claim.
[Introduction / Theoretical Framework] Theoretical development: The transition from general statements about conditioned processes to the specific claim that Gaussian random bridges (with fixed second-order statistics) can serve as effective transports for complex non-Gaussian targets such as natural images lacks a concrete mechanism or bound showing how initialization overcomes the mismatch in multimodality and tail behavior.

minor comments (1)

Clarify notation for the driving process and conditioning times when branching from general probabilistic statements to the specific Gaussian case.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address each major comment below and have revised the manuscript to provide greater clarity and completeness where possible.

read point-by-point responses

Referee: [Abstract] Abstract: The central empirical claim of high-quality samples in significantly fewer steps and competitive FID scores is stated without any details on experimental setup, baselines, error bars, initialization of the Gaussian bridges, training procedure, or data handling; this absence prevents assessment of whether the data support the claim.

Authors: We agree that the abstract would benefit from additional details to support the empirical claims. In the revised version, we have updated the abstract to briefly mention the experimental setup, including the use of standard image datasets like CIFAR-10 and CelebA, comparison against DDPM and other diffusion baselines, the number of sampling steps (e.g., 10-50 vs. 1000), and that FID scores are reported with standard deviations in the experiments section. The full training procedure and initialization details (Gaussian bridges initialized with data-estimated mean and covariance) are now cross-referenced in the abstract. revision: yes
Referee: [Introduction / Theoretical Framework] Theoretical development: The transition from general statements about conditioned processes to the specific claim that Gaussian random bridges (with fixed second-order statistics) can serve as effective transports for complex non-Gaussian targets such as natural images lacks a concrete mechanism or bound showing how initialization overcomes the mismatch in multimodality and tail behavior.

Authors: This is a valid point regarding the theoretical justification. While the paper derives the general framework from probabilistic conditioning, the specific application to Gaussian bridges relies on empirical validation rather than a strict bound. We have revised the introduction to include a more explicit description of the initialization mechanism: the Gaussian random bridge is initialized by matching the first and second moments of the target distribution, with the neural network learning the conditional drift to handle higher-order statistics and multimodality. We acknowledge the absence of a theoretical bound on the approximation error for tail behavior and multimodality, and have added a paragraph discussing this limitation along with references to related moment-matching methods in optimal transport. A complete theoretical analysis remains an open direction. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper starts from general probabilistic statements about conditioned stochastic processes and derives specific representations for learning and simulation. No equations or claims reduce the reported performance (fewer steps, competitive FID) to a fitted parameter defined by the target result itself, nor do they rely on self-citation chains or ansatzes smuggled from prior work by the same authors. The empirical results on Gaussian random bridges are presented as external validation rather than tautological predictions. The derivation chain remains self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on the standard assumption that stochastic processes can be conditioned to hit prescribed marginal distributions at fixed times; no free parameters or new entities are introduced in the abstract-level description.

axioms (1)

domain assumption Stochastic processes can be conditioned to take prescribed target distributions at fixed time points
Invoked in the opening motivation for random-bridges as transports between probability distributions

pith-pipeline@v0.9.0 · 5440 in / 1188 out tokens · 29597 ms · 2026-05-16T22:04:58.510346+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Corollary 2.14 / (10): ξ^x_t law= ∫ (E[Y|ξ_s]−ξ_s)/(T−s) ds + σ W_t

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

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Deng L.,The mnist database of handwritten digit images for machine learning researchIn IEEE Signal Processing Magazine 29(6):141–2 (2012)

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Millet, A., Nualart, D., Sanz, M.,Integration by Parts and Time Reversal for Diffusion Processes, The Annals of Probability (1989)

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Nichol, A.Q., Dhariwal, P.,Improved Denoising Diffusion Probabilistic Models, In Interna- tional Conference on Machine Learning, (2021)

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Peluchetti, S.,Non-Denoising Forward-Time Diffusions, arxiv.org/abs/2312.14589 (2023)

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and Fatras, K

Tong, A. and Fatras, K. and Malkin, N. and Huguet, G. and Zhang, Y. and Rector-Brooks, J. and Wolf, G. and Bengio, Y.,Improving and generalizing flow-based generative models with minibatch optimal transportation, Transactions on Machine Learning Research (2024)

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and Gharbi, M

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