mlsanityData to 7 Oct 2026

Flow Matching with General Discrete Paths: A Kinetic-Optimal Perspective

arXiv 2412.03487 ↗First posted 4 Dec 2024cs.LG cs.AIcited by 54

On the map: Diffusion models → Flow matching, time series and science

Abstract (arXiv)

The design space of discrete-space diffusion or flow generative models are significantly less well-understood than their continuous-space counterparts, with many works focusing only on a simple masked construction. In this work, we aim to take a holistic approach to the construction of discrete generative models based on continuous-time Markov chains, and for the first time, allow the use of arbitrary discrete probability paths, or colloquially, corruption processes. Through the lens of optimizing the symmetric kinetic energy, we propose velocity formulas that can be applied to any given probability path, completely decoupling the probability and velocity, and giving the user the freedom to specify any desirable probability path based on expert knowledge specific to the data domain. Furthermore, we find that a special construction of mixture probability paths optimizes the symmetric kinetic energy for the discrete case. We empirically validate the usefulness of this new design space across multiple modalities: text generation, inorganic material generation, and image generation. We find that we can outperform the mask construction even in text with kinetic-optimal mixture paths, while we can make use of domain-specific constructions of the probability path over the visual domain.

Key facts. Cited by 54 papers in the map (data through 7 Oct 2026). Placed in Flow matching, time series and science, one of 14 directions in Diffusion models. Builds on 29 of its 68 references that are in the map. First posted on arXiv 4 Dec 2024.

Builds on

29 of 68 references.

Closest ancestors · 5

Same direction, newest first

  1. 2024Discrete Flow Matching22 Jul
  2. 2024Generative Flows on Discrete State-Spaces: Enabling Multimodal Flows with Applications to Protein Co-Design7 Feb
  3. 2023Crystal Structure Prediction by Joint Equivariant Diffusion30 Jul
  4. 2022Flow Matching for Generative Modeling6 Oct
  5. 2022Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow7 Sep

Other references · 24

  1. 2018BERT: Pre-training of Deep Bidirectional Transformers for Language Understandingcited by 40,204
  2. 2023Llama 2: Open Foundation and Fine-Tuned Chat Modelscited by 12,865
  3. 2021Diffusion Models Beat GANs on Image Synthesiscited by 8,185
  4. 2022Scalable Diffusion Models with Transformerscited by 5,931
  5. 2021On the Opportunities and Risks of Foundation Modelscited by 3,961
  6. 2016Pointer Sentinel Mixture Modelscited by 2,992
  7. 2019XLNet: Generalized Autoregressive Pretraining for Language Understandingcited by 2,801
  8. 2022Make-A-Video: Text-to-Video Generation without Text-Video Datacited by 1,725
  9. 2016The LAMBADA dataset: Word prediction requiring a broad discourse contextcited by 960
  10. 2024Chameleon: Mixed-Modal Early-Fusion Foundation Modelscited by 904
  11. 2023Stochastic Interpolants: A Unifying Framework for Flows and Diffusionscited by 897
  12. 2024Simple and Effective Masked Diffusion Language Modelscited by 800
12 more
  1. 2024Autoregressive Model Beats Diffusion: Llama for Scalable Image Generationcited by 760
  2. 2024Simplified and Generalized Masked Diffusion for Discrete Datacited by 606
  3. 2024Transfusion: Predict the Next Token and Diffuse Images with One Multi-Modal Modelcited by 468
  4. 2017A Downsampled Variant of ImageNet as an Alternative to the CIFAR datasetscited by 374
  5. 2013One Billion Word Benchmark for Measuring Progress in Statistical Language Modelingcited by 302
  6. 2023Emu: Enhancing Image Generation Models Using Photogenic Needles in a Haystackcited by 278
  7. 2021Autoregressive Diffusion Modelscited by 208
  8. 2024Masked Diffusion Models are Secretly Time-Agnostic Masked Models and Exploit Inaccurate Categorical Samplingcited by 204
  9. 2024Unlocking Guidance for Discrete State-Space Diffusion and Flow Modelscited by 124
  10. 2023Audiobox: Unified Audio Generation with Natural Language Promptscited by 120
  11. 2024Fine-Tuned Language Models Generate Stable Inorganic Materials as Textcited by 92
  12. 2023Language models can generate molecules, materials, and protein binding sites directly in three dimensions as XYZ, CIF, and PDB filescited by 59

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2025: 31; 2026: 23.

54 papers.

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  1. 2025Discrete Diffusion in Large Language and Multimodal Models: A Surveycited by 61
  2. 2025Edit Flows: Flow Matching with Edit Operationscited by 53
  3. 2025Any-Order Flexible Length Masked Diffusioncited by 48
  4. 2025FUDOKI: Discrete Flow-based Unified Understanding and Generation via Kinetic-Optimal Velocitiescited by 46
  5. 2025LEAPS: A discrete neural sampler via locally equivariant networkscited by 30
  6. 2025Continuous Diffusion Model for Language Modelingcited by 29
  7. 2025Coevolutionary Continuous Discrete Diffusion: Make Your Diffusion Language Model a Latent Reasonercited by 28
  8. 2025Target Concrete Score Matching: A Holistic Framework for Discrete Diffusioncited by 25
  9. 2026Discrete Flow Mapscited by 23
  10. 2025Structuring The Future: Diffusion LLM Speculative Decoding via Calibrated Draft Graphscited by 21
  11. 2025MDNS: Masked Diffusion Neural Sampler via Stochastic Optimal Controlcited by 20
  12. 2025Beyond Masked and Unmasked: Discrete Diffusion Models via Partial Maskingcited by 19
38 more
  1. 2025NExT-OMNI: Towards Any-to-Any Omnimodal Foundation Models with Discrete Flow Matchingcited by 19
  2. 2025FLOWR: Flow Matching for Structure-Aware De Novo, Interaction- and Fragment-Based Ligand Generationcited by 18
  3. 2025α-Flow: A Unified Framework for Continuous-State Discrete Flow Matching Modelscited by 17
  4. 2025DiFFPO: Training Diffusion LLMs to Reason Fast and Furious via Reinforcement Learningcited by 17
  5. 2025OneFlow: Concurrent Mixed-Modal and Interleaved Generation with Edit Flowscited by 17
  6. 2025WAM-Flow: Parallel Coarse-to-Fine Motion Planning via Discrete Flow Matching for Autonomous Drivingcited by 15
  7. 2025Flow Matching Meets Biology and Life Science: A Surveycited by 14
  8. 2026The Diffusion Duality, Chapter II: Psi-Samplerscited by 12
  9. 2025Uniform Discrete Diffusion with Metric Path for Video Generationcited by 11
  10. 2026Discrete Adjoint Matchingcited by 10
  11. 2025Error Analysis of Discrete Flow with Generator Matchingcited by 9
  12. 2025OAT-FM: Optimal Acceleration Transport for Improved Flow Matchingcited by 7
  13. 2025Discrete Guidance Matching: Exact Guidance for Discrete Flow Matchingcited by 6
  14. 2025Soft-Di[M]O: Improving One-Step Discrete Image Generation with Soft Embeddingscited by 5
  15. 2025Topology Sculptor, Shape Refiner: Discrete Diffusion Model for High-Fidelity 3D Meshes Generationcited by 5
  16. 2026DFM-VLA: Iterative Action Refinement for Robot Manipulation via Discrete Flow Matchingcited by 5
  17. 2025Flow to Learn: Flow Matching on Neural Network Parameterscited by 3
  18. 2025ReDDiT: Rehashing Noise for Discrete Visual Generationcited by 3
  19. 2025Masked Diffusion Models as Energy Minimizationcited by 3
  20. 2025Drax: Speech Recognition with Discrete Flow Matchingcited by 3
  21. 2025Masked Diffusion Models are Secretly Learned-Order Autoregressive Modelscited by 3
  22. 2026Corrected Samplers for Discrete Flow Modelscited by 3
  23. 2026UDM-GRPO: Stable and Efficient Group Relative Policy Optimization for Uniform Discrete Diffusion Modelscited by 3
  24. 2025Corrector Sampling in Language Modelscited by 1
  25. 2025Flows and Diffusions on the Neural Manifoldcited by 1
  26. 2026Towards Latent Diffusion Suitable For Textcited by 1
  27. 2026Insertion Based Sequence Generation with Learnable Order Dynamicscited by 1
  28. 2026Discrete Flow Matching: Convergence Guarantees Under Minimal Assumptionscited by 1
  29. 2026dFlowGRPO: Rate-Aware Policy Optimization for Discrete Flow Modelscited by 1
  30. 2026Kinetic-Optimal Scheduling with Moment Correction for Metric-Induced Discrete Flow Matching in Zero-Shot Text-to-Speechcited by 1
  31. 2026Accelerating Discrete Diffusion Models with Parallel-In-Time Samplingcited by 1
  32. 2026Discrete Flow Matching Policy Optimizationcited by 0
  33. 2026Discrete Tilt Matchingcited by 0
  34. 2026On the Error-Correcting Effects of Stochasticity in Discrete Diffusioncited by 0
  35. 2026Flexible Flows for Biological Sequence Designcited by 0
  36. 2026Mask, Sample, Revise: A Revisable CTMC Inference Stack for Guided Discrete Flow Matching Text-to-Speechcited by 0
  37. 2026A Time-Reparameterized Cumulative Intensity Extrapolation Sampler for Discrete Flow Matchingcited by 0
  38. 2026Context-weighted Discrete Flow Matchingcited by 0

50 most cited of 54 All 54 (JSON)

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  1. 2025An Introduction to Flow Matching and Diffusion Models
  2. 2025A Theoretical Analysis of Discrete Flow Matching Generative Models
  3. 2025EnfoPath: Energy-Informed Analysis of Generative Trajectories in Flow Matching
  4. 2026A Unified Measure-Theoretic View of Diffusion, Score-Based, and Flow Matching Generative Models
  5. 2025Efficiency vs. Fidelity: A Comparative Analysis of Diffusion Probabilistic Models and Flow Matching on Low-Resource Hardware
  6. 2024Exploring Diffusion and Flow Matching Under Generator Matching
  7. 2025Branching Flows: Discrete, Continuous, and Manifold Flow Matching with Splits and Deletions
  8. 2025From Navigation to Refinement: Revealing the Two-Stage Nature of Flow-based Diffusion Models through Oracle Velocity