pith. machine review for the scientific record. sign in

arxiv: 2605.06364 · v1 · submitted 2026-05-07 · 💻 cs.LG · cs.AI

Recognition: unknown

Flow Matching with Arbitrary Auxiliary Paths

Xin Peng , Ang Gao
Authors on Pith no claims yet

Pith reviewed 2026-05-08 12:53 UTC · model grok-4.3

classification 💻 cs.LG cs.AI
keywords flow matchinggenerative modelingprobability pathsauxiliary variablesconditional flow matchingcontinuous normalizing flows
0
0 comments X

The pith

Flow matching can use auxiliary variables from any distribution while preserving the continuity equation and consistent training objectives.

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

The paper introduces AuxPath-FM to generalize conditional flow matching by adding an auxiliary variable η drawn from an arbitrary distribution into the probability path. The resulting trajectories take the explicit form X_t = a(t)X_1 + b(t)X_0 + c(t)η. The authors prove that this construction still satisfies the continuity equation and yields a training objective consistent with the marginal formulation of flow matching. This generality permits probability paths built from priors such as uniform, Laplace, or discrete Rademacher distributions, each imparting distinct geometric properties to the flow. The same mechanism also supports label-guided generation by placing structured semantic information inside the auxiliary distribution.

Core claim

We introduce AuxPath-FM which generalizes conditional flow matching by incorporating an auxiliary variable drawn from an arbitrary distribution into the probability path, with trajectories of the form X_t = a(t)X_1 + b(t)X_0 + c(t)η. This construction preserves the continuity equation and maintains a training objective consistent with the marginal formulation, enabling diverse priors and specialized tasks like label-guided generation.

What carries the argument

The generalized probability path X_t = a(t)X_1 + b(t)X_0 + c(t)η with η drawn from an arbitrary distribution, which carries the argument by allowing flexible path design while preserving the flow-matching continuity equation and objective.

Load-bearing premise

An auxiliary variable η from an arbitrary distribution can be linearly combined with the data and noise terms in the path definition without violating the continuity equation or the consistency of the training objective.

What would settle it

Deriving the velocity field from the path X_t = a(t)X_1 + b(t)X_0 + c(t)η and checking whether it satisfies the continuity equation ∂_t p_t + ∇·(u_t p_t) = 0 when η follows a Laplace distribution instead of a Gaussian.

Figures

Figures reproduced from arXiv: 2605.06364 by Ang Gao, Xin Peng.

Figure 1
Figure 1. Figure 1: Generative trajectories of AuxPath-FM with diverse auxiliary distributions. The figure illustrates the interpolation paths Xt = a(t)X1 + b(t)X0 + c(t)η between two data samples X0 and X1. By varying the distribution of the auxiliary variable η, we observe unique geometric properties: (a) Baseline (η = 0), (b) Gaussian η ∼ N (0, I), (c) Uniform η ∼ U([−1, 1]d ), (d) Laplace η ∼ Laplace(0, 1), (e) Rademacher… view at source ↗
Figure 2
Figure 2. Figure 2: Trajectory-level guidance. Com￾parison between w = 1.0 and w = 7.0. Higher guidance leads to clear trajectory sep￾aration. We illustrate trajectory-level CFG in a 2D toy set￾ting with a bimodal Gaussian dataset (two clusters on a ring). The flow model is trained with AuxPath￾FM, and guidance is applied at sampling using Algo￾rithm 5. As shown in view at source ↗
Figure 3
Figure 3. Figure 3: Generated trajectories on the ring-64 dataset using different auxiliary distributions η view at source ↗
Figure 4
Figure 4. Figure 4: Qualitative comparison of conditional generation on MNIST. We compare generated samples across different model variants: standard CFM (baseline), path-guided models with fixed numeric auxiliary variables (η = numeric), and AuxPath-FM using learned label prototypes (η = Fϕ(Y )). Datasets. We consider three datasets:MNIST [26], CIFAR-10 [24], and ImageNet-1k [11] are used to evaluate conditional generation a… view at source ↗
Figure 5
Figure 5. Figure 5: Generation trajectories of AuxPath-FM with trajectory-level CFG on CIFAR-10. Each row shows the evolution of a sample along the probability path from the initial state (t = 0) to the final generated image (t = 1). Unlike methods that apply guidance through repeated network evaluations, AuxPath-FM incorporates classifier-free guidance directly into the trajectory via auxiliary variables. Increasing the guid… view at source ↗
Figure 6
Figure 6. Figure 6: Trajectory-level CFG on ImageNet-1k with AuxPath-FM. Each row shows the evolution of a sample along the probability path from t = 0 to t = 1. Increasing the guidance scale from w = 1.0 to w = 1.5 yields more semantically aligned samples while preserving smooth trajectory transitions. Guidance is applied directly along the path via auxiliary variables, enabling effective control with a single backbone evalu… view at source ↗
read the original abstract

We introduce a new generative modeling framework, \textbf{Flow Matching with Arbitrary Auxiliary Paths (AuxPath-FM)}, which generalizes conditional flow matching by incorporating an auxiliary variable drawn from an arbitrary distribution into the probability path. Unlike prior methods that restrict auxiliary components to Gaussian noise, AuxPath-FM allows the variable $\eta$ to follow any distribution, producing trajectories of the form $X_t = a(t)X_1 + b(t)X_0 + c(t)\eta$. We theoretically demonstrate that this construction preserves the continuity equation and maintains a training objective consistent with the marginal formulation. This flexibility enables the design of diverse probability paths using various priors, including Gaussian, Uniform, Laplace, and discrete Rademacher distributions, each offering unique geometric properties for generative flows. Furthermore, our framework allows for specialized tasks such as label-guided generation by encoding structured semantic information into the auxiliary distribution. Overall, AuxPath-FM provides a principled and general foundation for probability path design, offering both theoretical generality and practical flexibility for diverse generative modeling 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.

Referee Report

2 major / 2 minor

Summary. The paper introduces AuxPath-FM, a generalization of conditional flow matching that incorporates an auxiliary variable η ~ arbitrary distribution (including Gaussian, Uniform, Laplace, and discrete Rademacher) into the linear probability path X_t = a(t)X_1 + b(t)X_0 + c(t)η. It claims to prove that this construction preserves the continuity equation and yields a marginal training objective consistent with standard flow matching, enabling diverse path geometries and applications such as label-guided generation.

Significance. If the theoretical claims hold without restriction on the auxiliary distribution, the framework would meaningfully expand the design space for probability paths in flow matching, allowing structured priors beyond Gaussian noise. The paper's emphasis on parameter-free path coefficients and consistency with the marginal formulation is a potential strength, but the load-bearing generality claim requires substantiation for singular measures.

major comments (2)
  1. [Theoretical demonstration / continuity equation preservation] Abstract and theoretical section: the claim that the construction 'preserves the continuity equation' for arbitrary η (explicitly including discrete Rademacher) is load-bearing for the central contribution. When η is discrete, the marginal law of X_t is supported on a finite union of lower-dimensional affine subspaces (for c(t) ≠ 0), hence singular w.r.t. Lebesgue measure and without a density p_t. The standard PDE form ∂_t p_t + ∇·(p_t v_t) = 0 therefore cannot be invoked directly; a measure-theoretic statement or explicit derivation of the velocity field under the singular case must be supplied, or the generality claim restricted.
  2. [Marginal formulation and training objective] § on marginal objective consistency: the training objective is asserted to remain consistent with the marginal formulation. However, when the auxiliary path induces a singular marginal, the usual conditional-expectation or score-based derivations of the velocity field v_t no longer apply in their standard form. The manuscript must show how the objective is obtained without assuming absolute continuity of p_t.
minor comments (2)
  1. Notation for the time-dependent coefficients a(t), b(t), c(t) is introduced without an explicit table or set of functional forms; providing concrete examples (e.g., the standard linear interpolation case) would improve readability.
  2. The abstract lists 'discrete Rademacher distributions' as one of the supported priors; a brief remark on how the generative sampling procedure handles the resulting discrete support would clarify practical use.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive feedback on the theoretical foundations of AuxPath-FM. The points regarding continuity equations and marginal objectives for singular measures are well-taken, and we address them point-by-point below with proposed revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: Abstract and theoretical section: the claim that the construction 'preserves the continuity equation' for arbitrary η (explicitly including discrete Rademacher) is load-bearing for the central contribution. When η is discrete, the marginal law of X_t is supported on a finite union of lower-dimensional affine subspaces (for c(t) ≠ 0), hence singular w.r.t. Lebesgue measure and without a density p_t. The standard PDE form ∂_t p_t + ∇·(p_t v_t) = 0 therefore cannot be invoked directly; a measure-theoretic statement or explicit derivation of the velocity field under the singular case must be supplied, or the generality claim restricted.

    Authors: We agree that the standard density-based PDE form does not apply directly when the marginal is singular. Our derivation in the manuscript defines the velocity field via the linear path construction and conditional expectation, which extends to the measure-theoretic setting using the continuity equation in the weak (distributional) sense: for any test function φ, the integral form ∫ (∂_t φ + v_t · ∇φ) dμ_t = 0 holds for the induced measures μ_t. This is independent of absolute continuity. We will add a new subsection (e.g., 3.3) explicitly stating the weak formulation, deriving the velocity field for general (including discrete) auxiliary distributions via disintegration, and verifying it for the Rademacher case with a concrete example. revision: yes

  2. Referee: § on marginal objective consistency: the training objective is asserted to remain consistent with the marginal formulation. However, when the auxiliary path induces a singular marginal, the usual conditional-expectation or score-based derivations of the velocity field v_t no longer apply in their standard form. The manuscript must show how the objective is obtained without assuming absolute continuity of p_t.

    Authors: The marginal objective is derived by expanding the expectation E[||v_t(X_t) - u_t(X_t | X_0, X_1, η)||^2] and applying the law of total expectation over the joint law of (X_0, X_1, η), which yields the marginal velocity without invoking densities or scores. This step relies only on the existence of regular conditional distributions (guaranteed by Polish spaces) and holds for singular marginals. We will append a short proof in the supplementary material (or new Appendix B) that makes this explicit using measure disintegration, confirming consistency for arbitrary η including discrete cases. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation extends standard flow matching independently

full rationale

The paper defines a new family of paths X_t = a(t)X_1 + b(t)X_0 + c(t)η with η drawn from an arbitrary distribution and claims a direct theoretical verification that the resulting marginals satisfy the continuity equation with a consistent marginal training objective. This verification proceeds from the standard continuity-equation setup of flow matching by explicit construction of the velocity field (or its conditional expectation) for the augmented path; the auxiliary variable is an exogenous modeling choice rather than a quantity defined in terms of the target result or fitted from it. No self-citation is load-bearing for the central claim, no parameter is fitted on a subset and then re-labeled as a prediction, and the derivation does not rename a known empirical pattern. The construction is therefore self-contained against the external benchmark of conditional flow matching.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The central claim rests on the standard continuity equation from flow matching plus the new linear combination with arbitrary η; no free parameters are explicitly fitted in the abstract, but the functions a(t), b(t), c(t) are introduced without derivation details.

free parameters (1)
  • a(t), b(t), c(t)
    Time-dependent coefficients in the path definition X_t = a(t)X_1 + b(t)X_0 + c(t)η; their specific forms are not derived from first principles in the abstract.
axioms (1)
  • domain assumption The probability path must satisfy the continuity equation.
    Invoked to claim that the new construction remains valid for any distribution of η.
invented entities (1)
  • Auxiliary variable η from arbitrary distribution no independent evidence
    purpose: To define diverse probability paths with different geometric properties
    New component introduced to generalize beyond Gaussian noise; no independent evidence provided.

pith-pipeline@v0.9.0 · 5465 in / 1187 out tokens · 42641 ms · 2026-05-08T12:53:32.583762+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

49 extracted references · 12 canonical work pages · 4 internal anchors

  1. [1]

    Stochastic Interpolants: A Unifying Framework for Flows and Diffusions

    Michael S Albergo, Nicholas M Boffi, and Eric Vanden-Eijnden. Stochastic interpolants: A unifying framework for flows and diffusions.arXiv preprint arXiv:2303.08797, 2023

    work page internal anchor Pith review arXiv 2023

  2. [2]

    An ensemble of simple convolutional neural network models for mnist digit recognition, 2020

    Sanghyeon An, Minjun Lee, Sanglee Park, Heerin Yang, and Jungmin So. An ensemble of simple convolutional neural network models for mnist digit recognition, 2020. URL https://arxiv.org/abs/ 2008.10400

    work page arXiv 2020

  3. [3]

    Li, Hamid Kazemi, Furong Huang, Micah Goldblum, Jonas Geiping, and Tom Goldstein

    Arpit Bansal, Eitan Borgnia, Hong-Min Chu, Jie S. Li, Hamid Kazemi, Furong Huang, Micah Goldblum, Jonas Geiping, and Tom Goldstein. Cold diffusion: Inverting arbitrary image transforms without noise. In Neural Information Processing Systems (NeurIPS), 2023. URL https://openreview.net/forum?id= XH3ArccntI

    2023

  4. [4]

    Iterative bregman projections for regularized transportation problems.SIAM Journal on Scientific Computing, 2015

    Jean-David Benamou, Guillaume Carlier, Marco Cuturi, Lorenzo Nenna, and Gabriel Peyré. Iterative bregman projections for regularized transportation problems.SIAM Journal on Scientific Computing, 2015

    2015

  5. [5]

    Diffusion schrödinger bridge with applications to score-based generative modeling

    Valentin De Bortoli, James Thornton, Jeremy Heng, and Arnaud Doucet. Diffusion schrödinger bridge with applications to score-based generative modeling. In A. Beygelzimer, Y . Dauphin, P. Liang, and J. Wortman Vaughan, editors,Neural Information Processing Systems (NeurIPS), 2021. URL https: //openreview.net/forum?id=9BnCwiXB0ty

    2021

  6. [6]

    Theodorou, and Weilie Nie

    Valentin De Bortoli, Guan-Horng Liu, Tianrong Chen, Evangelos A. Theodorou, and Weilie Nie. Aug- mented bridge matching.arXiv preprint arXiv:2311.06978, 2023

    work page arXiv 2023

  7. [7]

    Generative flows on discrete state-spaces: Enabling multimodal flows with applications to protein co-design

    Andrew Campbell, Jason Yim, Regina Barzilay, Tom Rainforth, and Tommi Jaakkola. Generative flows on discrete state-spaces: Enabling multimodal flows with applications to protein co-design. InInterna- tional Conference on Machine Learning (ICML), 2024. URL https://openreview.net/forum?id= kQwSbv0BR4

    2024

  8. [8]

    Ricky T. Q. Chen and Yaron Lipman. Flow matching on general geometries. InInternational Conference on Learning Representations (ICLR), 2024

    2024

  9. [9]

    Flow matching in latent space.arXiv preprint arXiv:2307.08698,

    Quan Dao, Hao Phung, Binh Nguyen, and Anh Tran. Flow matching in latent space.arXiv, 2023. URL https://arxiv.org/abs/2307.08698

    work page arXiv 2023

  10. [10]

    Bronstein, and Joey Bose

    Oscar Davis, Samuel Kessler, Mircea Petrache, Ismail Ilkan Ceylan, Michael M. Bronstein, and Joey Bose. Fisher flow matching for generative modeling over discrete data. InInternational Conference on Learning Representations (ICLR), 2024. URLhttps://openreview.net/forum?id=6jOScqwdHU

    2024

  11. [11]

    Imagenet: A large-scale hierarchical image database

    Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hierarchical image database. In2009 IEEE Conference on Computer Vision and Pattern Recognition, pages 248–255,

  12. [12]

    doi: 10.1109/CVPR.2009.5206848

    work page doi:10.1109/cvpr.2009.5206848 2009

  13. [13]

    Diffusion models beat gans on image synthesis.Neural Information Processing Systems (NeurIPS), 34, 2021

    Prafulla Dhariwal and Alexander Nichol. Diffusion models beat gans on image synthesis.Neural Information Processing Systems (NeurIPS), 34, 2021

    2021

  14. [14]

    Learning with a wasserstein loss.Neural Information Processing Systems (NeurIPS), 2015

    Charlie Frogner, Chiyuan Zhang, Hossein Mobahi, Mauricio Araya-Polo, and Tomaso Poggio. Learning with a wasserstein loss.Neural Information Processing Systems (NeurIPS), 2015

    2015

  15. [15]

    Itai Gat, Tal Remez, Neta Shaul, Felix Kreuk, Ricky T. Q. Chen, Gabriel Synnaeve, Yossi Adi, and Yaron Lipman. Discrete flow matching. InNeural Information Processing Systems (NeurIPS), 2024. URL https://openreview.net/forum?id=GTDKo3Sv9p

    2024

  16. [16]

    Marton Havasi, Brian Karrer, Itai Gat, and Ricky T. Q. Chen. Edit flows: Variable length discrete flow matching with sequence-level edit operations. InNeural Information Processing Systems (NeurIPS), 2026. URLhttps://openreview.net/forum?id=FXWwYz1p8a

    2026

  17. [17]

    Deep residual learning for image recognition

    Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. InIEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2016

    2016

  18. [18]

    Gans trained by a two time-scale update rule converge to a local nash equilibrium

    Martin Heusel, Hubert Ramsauer, Thomas Unterthiner, Bernhard Nessler, and Sepp Hochreiter. Gans trained by a two time-scale update rule converge to a local nash equilibrium. InNeural Information Processing Systems (NeurIPS), NIPS’17, page 6629–6640, Red Hook, NY , USA, 2017. Curran Associates Inc. ISBN 9781510860964

    2017

  19. [19]

    Classifier-Free Diffusion Guidance

    Jonathan Ho and Tim Salimans. Classifier-free diffusion guidance.arXiv preprint arXiv:2207.12598, 2022. 10

    work page internal anchor Pith review arXiv 2022

  20. [20]

    Denoising diffusion probabilistic models.Neural Information Processing Systems (NeurIPS), 2020

    Jonathan Ho, Ajay Jain, and Pieter Abbeel. Denoising diffusion probabilistic models.Neural Information Processing Systems (NeurIPS), 2020

    2020

  21. [21]

    Bronstein, Alexander Tong, and Joey Bose

    Guillaume Huguet, James Vuckovic, Kilian FATRAS, Eric Thibodeau-Laufer, Pablo Lemos, Riashat Islam, Cheng-Hao Liu, Jarrid Rector-Brooks, Tara Akhound-Sadegh, Michael M. Bronstein, Alexander Tong, and Joey Bose. Sequence-augmented SE(3)-flow matching for conditional protein generation. In The Thirty-eighth Annual Conference on Neural Information Processing...

    2024

  22. [22]

    Multi-marginal f- divergence schrödinger bridges: Towards a unifying framework for generation and distillation

    Ishaan Kharbanda, Xuan Ouyang, Rahul Choudhary, Hanbaek Lyu, and Yudi Li. Multi-marginal f- divergence schrödinger bridges: Towards a unifying framework for generation and distillation. InInterna- tional Conference on Learning Representations (ICLR), 2025. URL https://openreview.net/forum? id=7WrMAQ1XDq

    2025

  23. [23]

    Better source, better flow: Learning condition-dependent source distribution for flow matching.arXiv preprint arXiv:2602.05951, 2026

    Junwan Kim, Jiho Park, Seonghu Jeon, and Seungryong Kim. Better source, better flow: Learning condition-dependent source distribution for flow matching.arXiv preprint arXiv:2602.05951, 2026

    work page arXiv 2026

  24. [24]

    On density estimation with diffusion models

    Diederik P Kingma, Tim Salimans, Ben Poole, and Jonathan Ho. On density estimation with diffusion models. In A. Beygelzimer, Y . Dauphin, P. Liang, and J. Wortman Vaughan, editors,Neural Information Processing Systems (NeurIPS), 2021. URLhttps://openreview.net/forum?id=2LdBqxc1Yv

    2021

  25. [25]

    Learning multiple layers of features from tiny images

    Alex Krizhevsky and Geoffrey Hinton. Learning multiple layers of features from tiny images. Technical Report 0, University of Toronto, Toronto, Ontario, 2009. URL https://www.cs.toronto.edu/~kriz/ learning-features-2009-TR.pdf

    2009

  26. [26]

    Flow Matching is Adaptive to Manifold Structures

    Shivam Kumar, Yixin Wang, and Lizhen Lin. Flow matching is adaptive to manifold structures.arXiv preprint arXiv:2602.22486, 2026

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  27. [27]

    1998 , month = nov, journal =

    Y . Lecun, L. Bottou, Y . Bengio, and P. Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998. doi: 10.1109/5.726791

    work page doi:10.1109/5.726791 1998

  28. [28]

    Yaron Lipman, Ricky T. Q. Chen, Heli Ben-Hamu, Maximilian Nickel, and Matthew Le. Flow matching for generative modeling. InInternational Conference on Learning Representations (ICLR), 2023

    2023

  29. [29]

    Flow-GRPO: Training flow matching models via online RL

    Jie Liu, Gongye Liu, Jiajun Liang, Yangguang Li, Jiaheng Liu, Xintao Wang, Pengfei Wan, Di ZHANG, and Wanli Ouyang. Flow-GRPO: Training flow matching models via online RL. InThe Thirty-ninth Annual Conference on Neural Information Processing Systems, 2026. URL https://openreview.net/ forum?id=oCBKGw5HNf

    2026

  30. [30]

    Flow straight and fast: Learning to generate and transfer data with rectified flow

    Xingchao Liu, Chengyue Gong, and qiang liu. Flow straight and fast: Learning to generate and transfer data with rectified flow. InInternational Conference on Learning Representations (ICLR), 2023. URL https://openreview.net/forum?id=XVjTT1nw5z

    2023

  31. [31]

    Edit-based flow matching for temporal point processes

    David Lüdke, Marten Lienen, Marcel Kollovieh, and Stephan Günnemann. Edit-based flow matching for temporal point processes. InInternational Conference on Learning Representations (ICLR), 2026. URL https://openreview.net/forum?id=FNf9IV1P2L

    2026

  32. [32]

    Benjamin Kurt Miller, Ricky T. Q. Chen, Anuroop Sriram, and Brandon M Wood. FlowMM: Generating materials with Riemannian flow matching. InInternational Conference on Machine Learning (ICML), 2024

    2024

  33. [33]

    Conditional Generative Adversarial Nets

    Mehdi Mirza and Simon Osindero. Conditional generative adversarial nets. InarXiv preprint arXiv:1411.1784, 2014

    work page internal anchor Pith review arXiv 2014

  34. [34]

    Spectflow: Long-term forecasting using flow matching with 89k parameters

    Seyed Mohamad Moghadas and Adrian Munteanu. Spectflow: Long-term forecasting using flow matching with 89k parameters. InNeurIPS 2025 Workshop on Structured Probabilistic Inference & Generative Modeling, 2025. URLhttps://openreview.net/forum?id=NIeZP4a8nq

    2025

  35. [35]

    Scalable diffusion models with transformers

    William Peebles and Saining Xie. Scalable diffusion models with transformers. InIEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2023

    2023

  36. [36]

    U-net: Convolutional networks for biomedical image segmentation

    Olaf Ronneberger, Philipp Fischer, and Thomas Brox. U-net: Convolutional networks for biomedical image segmentation. In Nassir Navab, Joachim Hornegger, William M. Wells, and Alejandro F. Frangi, editors,Medical Image Computing and Computer-Assisted Intervention – MICCAI 2015, pages 234–241, Cham, 2015. Springer International Publishing. ISBN 978-3-319-24574-4

    2015

  37. [37]

    Improved techniques for training gans

    Tim Salimans, Ian Goodfellow, Wojciech Zaremba, Vicki Cheung, Alec Radford, and Xi Chen. Improved techniques for training gans. InNeural Information Processing Systems (NeurIPS), NIPS’16, page 2234–2242, Red Hook, NY , USA, 2016. Curran Associates Inc. ISBN 9781510838819. 11

    2016

  38. [38]

    Transition matching: Scalable and flexible generative modeling

    Neta Shaul, Uriel Singer, Itai Gat, and Yaron Lipman. Transition matching: Scalable and flexible generative modeling. InNeural Information Processing Systems (NeurIPS), 2026. URL https://openreview. net/forum?id=An0ePypuOJ

    2026

  39. [39]

    Score-based generative modeling through stochastic differential equations

    Yang Song, Jascha Sohl-Dickstein, Diederik P Kingma, Abhishek Kumar, Stefano Ermon, and Ben Poole. Score-based generative modeling through stochastic differential equations. InInternational Conference on Learning Representations (ICLR), 2021

    2021

  40. [40]

    Dirichlet flow matching with applications to DNA sequence design

    Hannes Stark, Bowen Jing, Chenyu Wang, Gabriele Corso, Bonnie Berger, Regina Barzilay, and Tommi Jaakkola. Dirichlet flow matching with applications to DNA sequence design. InInternational Conference on Machine Learning (ICML), 2024. URLhttps://openreview.net/forum?id=syXFAVqx85

    2024

  41. [41]

    Contrastive flow matching

    George Stoica, Vivek Ramanujan, Xiang Fan, Ali Farhadi, Ranjay Krishna, and Judy Hoffman. Contrastive flow matching. InIEEE International Conference on Computer Vision (ICCV), 2025

    2025

  42. [42]

    Matus Telgarsky

    Christian Szegedy, Vincent Vanhoucke, Sergey Ioffe, Jon Shlens, and Zbigniew Wojna. Rethinking the inception architecture for computer vision. InIEEE Conference on Computer Vision and Pattern Recognition (CVPR), pages 2818–2826, 2016. doi: 10.1109/CVPR.2016.308

    work page doi:10.1109/cvpr.2016.308 2016

  43. [43]

    Branched schrödinger bridge matching

    Sophia Tang, Yinuo Zhang, Alexander Tong, and Pranam Chatterjee. Branched schrödinger bridge matching. InInternational Conference on Learning Representations (ICLR), 2026. URL https:// openreview.net/forum?id=ctq8BfUXWz

    2026

  44. [44]

    Improving and generalizing flow-based generative models with minibatch optimal transport

    Alexander Tong, Nikolay Malkin, Guillaume Huguet, Yanlei Zhang, Jarrid Rector-Brooks, Kilian FATRAS, Guy Wolf, and Yoshua Bengio. Improving and generalizing flow-based generative models with minibatch optimal transport. InICML Workshop on New Frontiers in Learning, Control, and Dynamical Systems,

  45. [45]

    URLhttps://openreview.net/forum?id=HgDwiZrpVq

  46. [46]

    Physics-constrained flow matching: Sampling generative models with hard constraints

    Utkarsh Utkarsh, Pengfei Cai, Alan Edelman, Rafael Gomez-Bombarelli, and Christopher Vincent Rack- auckas. Physics-constrained flow matching: Sampling generative models with hard constraints. In Neural Information Processing Systems (NeurIPS), 2026. URL https://openreview.net/forum?id= cf4etwjY7n

    2026

  47. [47]

    Stochastic normalizing flows.Neural Information Processing Systems (NeurIPS), 2020

    Hao Wu, Jonas Köhler, and Frank Noé. Stochastic normalizing flows.Neural Information Processing Systems (NeurIPS), 2020

    2020

  48. [48]

    Reconstruction vs

    Jingfeng Yao, Bin Yang, and Xinggang Wang. Reconstruction vs. generation: Taming optimization dilemma in latent diffusion models. InIEEE Conference on Computer Vision and Pattern Recognition (CVPR), pages 15703–15712, 2025

    2025

  49. [49]

    Lapflow: Laplacian multi-scale flow matching for generative modeling

    Zelin Zhao, Petr Molodyk, Haotian Xue, and Yongxin Chen. Lapflow: Laplacian multi-scale flow matching for generative modeling. InInternational Conference on Learning Representations (ICLR), 2026. URL https://openreview.net/forum?id=kdrc4o6okz. 12 A Proof of AuxPath-FM Consistency We present an AuxPath formulation of AuxPath-FM that introduces an auxiliary...

    work page arXiv 2026