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

arXiv: https://arxiv.org/abs/2412.03487 · HTML: https://mlsanity.com/paper/2412.03487/

## Key facts

- Cited by 57 papers in the map (data through 14 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.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 29 of 29)

- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Llama 2: Open Foundation and Fine-Tuned Chat Models](https://mlsanity.com/paper/2307.09288/) (2023)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Scalable Diffusion Models with Transformers](https://mlsanity.com/paper/2212.09748/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [On the Opportunities and Risks of Foundation Models](https://mlsanity.com/paper/2108.07258/) (2021)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Pointer Sentinel Mixture Models](https://mlsanity.com/paper/1609.07843/) (2016)
- [XLNet: Generalized Autoregressive Pretraining for Language Understanding](https://mlsanity.com/paper/1906.08237/) (2019)
- [Make-A-Video: Text-to-Video Generation without Text-Video Data](https://mlsanity.com/paper/2209.14792/) (2022)
- [The LAMBADA dataset: Word prediction requiring a broad discourse context](https://mlsanity.com/paper/1606.06031/) (2016)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Chameleon: Mixed-Modal Early-Fusion Foundation Models](https://mlsanity.com/paper/2405.09818/) (2024)
- [Simple and Effective Masked Diffusion Language Models](https://mlsanity.com/paper/2406.07524/) (2024)
- [Autoregressive Model Beats Diffusion: Llama for Scalable Image Generation](https://mlsanity.com/paper/2406.06525/) (2024)
- [Simplified and Generalized Masked Diffusion for Discrete Data](https://mlsanity.com/paper/2406.04329/) (2024)
- [Transfusion: Predict the Next Token and Diffuse Images with One Multi-Modal Model](https://mlsanity.com/paper/2408.11039/) (2024)
- [A Downsampled Variant of ImageNet as an Alternative to the CIFAR datasets](https://mlsanity.com/paper/1707.08819/) (2017)
- [Discrete Flow Matching](https://mlsanity.com/paper/2407.15595/) (2024)
- [One Billion Word Benchmark for Measuring Progress in Statistical Language Modeling](https://mlsanity.com/paper/1312.3005/) (2013)
- [Generative Flows on Discrete State-Spaces: Enabling Multimodal Flows with Applications to Protein Co-Design](https://mlsanity.com/paper/2402.04997/) (2024)
- [Emu: Enhancing Image Generation Models Using Photogenic Needles in a Haystack](https://mlsanity.com/paper/2309.15807/) (2023)
- [Autoregressive Diffusion Models](https://mlsanity.com/paper/2110.02037/) (2021)
- [Masked Diffusion Models are Secretly Time-Agnostic Masked Models and Exploit Inaccurate Categorical Sampling](https://mlsanity.com/paper/2409.02908/) (2024)
- [Crystal Structure Prediction by Joint Equivariant Diffusion](https://mlsanity.com/paper/2309.04475/) (2023)
- [Unlocking Guidance for Discrete State-Space Diffusion and Flow Models](https://mlsanity.com/paper/2406.01572/) (2024)
- [Audiobox: Unified Audio Generation with Natural Language Prompts](https://mlsanity.com/paper/2312.15821/) (2023)
- [Fine-Tuned Language Models Generate Stable Inorganic Materials as Text](https://mlsanity.com/paper/2402.04379/) (2024)
- [Language models can generate molecules, materials, and protein binding sites directly in three dimensions as XYZ, CIF, and PDB files](https://mlsanity.com/paper/2305.05708/) (2023)

## Built on by (most cited first, 30 of 57)

- [Discrete Diffusion in Large Language and Multimodal Models: A Survey](https://mlsanity.com/paper/2506.13759/) (2025)
- [Edit Flows: Flow Matching with Edit Operations](https://mlsanity.com/paper/2506.09018/) (2025)
- [Any-Order Flexible Length Masked Diffusion](https://mlsanity.com/paper/2509.01025/) (2025)
- [FUDOKI: Discrete Flow-based Unified Understanding and Generation via Kinetic-Optimal Velocities](https://mlsanity.com/paper/2505.20147/) (2025)
- [Coevolutionary Continuous Discrete Diffusion: Make Your Diffusion Language Model a Latent Reasoner](https://mlsanity.com/paper/2510.03206/) (2025)
- [LEAPS: A discrete neural sampler via locally equivariant networks](https://mlsanity.com/paper/2502.10843/) (2025)
- [Continuous Diffusion Model for Language Modeling](https://mlsanity.com/paper/2502.11564/) (2025)
- [Discrete Flow Maps](https://mlsanity.com/paper/2604.09784/) (2026)
- [Target Concrete Score Matching: A Holistic Framework for Discrete Diffusion](https://mlsanity.com/paper/2504.16431/) (2025)
- [MDNS: Masked Diffusion Neural Sampler via Stochastic Optimal Control](https://mlsanity.com/paper/2508.10684/) (2025)
- [Structuring The Future: Diffusion LLM Speculative Decoding via Calibrated Draft Graphs](https://mlsanity.com/paper/2509.18085/) (2025)
- [NExT-OMNI: Towards Any-to-Any Omnimodal Foundation Models with Discrete Flow Matching](https://mlsanity.com/paper/2510.13721/) (2025)
- [Beyond Masked and Unmasked: Discrete Diffusion Models via Partial Masking](https://mlsanity.com/paper/2505.18495/) (2025)
- [DiFFPO: Training Diffusion LLMs to Reason Fast and Furious via Reinforcement Learning](https://mlsanity.com/paper/2510.02212/) (2025)
- [α-Flow: A Unified Framework for Continuous-State Discrete Flow Matching Models](https://mlsanity.com/paper/2504.10283/) (2025)
- [FLOWR: Flow Matching for Structure-Aware De Novo, Interaction- and Fragment-Based Ligand Generation](https://mlsanity.com/paper/2504.10564/) (2025)
- [OneFlow: Concurrent Mixed-Modal and Interleaved Generation with Edit Flows](https://mlsanity.com/paper/2510.03506/) (2025)
- [WAM-Flow: Parallel Coarse-to-Fine Motion Planning via Discrete Flow Matching for Autonomous Driving](https://mlsanity.com/paper/2512.06112/) (2025)
- [Flow Matching Meets Biology and Life Science: A Survey](https://mlsanity.com/paper/2507.17731/) (2025)
- [Uniform Discrete Diffusion with Metric Path for Video Generation](https://mlsanity.com/paper/2510.24717/) (2025)
- [The Diffusion Duality, Chapter II: Psi-Samplers](https://mlsanity.com/paper/2602.21185/) (2026)
- [Discrete Adjoint Matching](https://mlsanity.com/paper/2602.07132/) (2026)
- [Error Analysis of Discrete Flow with Generator Matching](https://mlsanity.com/paper/2509.21906/) (2025)
- [OAT-FM: Optimal Acceleration Transport for Improved Flow Matching](https://mlsanity.com/paper/2509.24936/) (2025)
- [Discrete Guidance Matching: Exact Guidance for Discrete Flow Matching](https://mlsanity.com/paper/2509.21912/) (2025)
- [Soft-Di[M]O: Improving One-Step Discrete Image Generation with Soft Embeddings](https://mlsanity.com/paper/2509.22925/) (2025)
- [Topology Sculptor, Shape Refiner: Discrete Diffusion Model for High-Fidelity 3D Meshes Generation](https://mlsanity.com/paper/2510.21264/) (2025)
- [DFM-VLA: Iterative Action Refinement for Robot Manipulation via Discrete Flow Matching](https://mlsanity.com/paper/2603.26320/) (2026)
- [Accelerating Discrete Diffusion Models with Parallel-In-Time Sampling](https://mlsanity.com/paper/2607.00773/) (2026)
- [Flow to Learn: Flow Matching on Neural Network Parameters](https://mlsanity.com/paper/2503.19371/) (2025)

## 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.