# Expanding Flow Maps

arXiv: https://arxiv.org/abs/2607.21585 · HTML: https://mlsanity.com/paper/2607.21585/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 51 of its 66 references that are in the map.
- First posted on arXiv 23 Jul 2026.

## Builds on (most cited first, 30 of 51)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [MoleculeNet: A Benchmark for Molecular Machine Learning](https://mlsanity.com/paper/1703.00564/) (2017)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Simplified and Generalized Masked Diffusion for Discrete Data](https://mlsanity.com/paper/2406.04329/) (2024)
- [Mean Flows for One-step Generative Modeling](https://mlsanity.com/paper/2505.13447/) (2025)
- [GeoDiff: a Geometric Diffusion Model for Molecular Conformation Generation](https://mlsanity.com/paper/2203.02923/) (2022)
- [Consistency Trajectory Models: Learning Probability Flow ODE Trajectory of Diffusion](https://mlsanity.com/paper/2310.02279/) (2023)
- [Improved Techniques for Training Consistency Models](https://mlsanity.com/paper/2310.14189/) (2023)
- [One Step Diffusion via Shortcut Models](https://mlsanity.com/paper/2410.12557/) (2024)
- [Your Absorbing Discrete Diffusion Secretly Models the Conditional Distributions of Clean Data](https://mlsanity.com/paper/2406.03736/) (2024)
- [Better & Faster Large Language Models via Multi-token Prediction](https://mlsanity.com/paper/2404.19737/) (2024)
- [One Billion Word Benchmark for Measuring Progress in Statistical Language Modeling](https://mlsanity.com/paper/1312.3005/) (2013)
- [GEOM: Energy-annotated molecular conformations for property prediction and molecular generation](https://mlsanity.com/paper/2006.05531/) (2020)
- [Continuous diffusion for categorical data](https://mlsanity.com/paper/2211.15089/) (2022)
- [Masked Diffusion Models are Secretly Time-Agnostic Masked Models and Exploit Inaccurate Categorical Sampling](https://mlsanity.com/paper/2409.02908/) (2024)
- [Fr'echet ChemNet Distance: A metric for generative models for molecules in drug discovery](https://mlsanity.com/paper/1803.09518/) (2018)
- [Learning Gradient Fields for Molecular Conformation Generation](https://mlsanity.com/paper/2105.03902/) (2021)
- [The Diffusion Duality](https://mlsanity.com/paper/2506.10892/) (2025)
- [Dirichlet Flow Matching with Applications to DNA Sequence Design](https://mlsanity.com/paper/2402.05841/) (2024)
- [Flow map matching with stochastic interpolants: A mathematical framework for consistency models](https://mlsanity.com/paper/2406.07507/) (2024)
- [GeoMol: Torsional Geometric Generation of Molecular 3D Conformer Ensembles](https://mlsanity.com/paper/2106.07802/) (2021)
- [How to build a consistency model: Learning flow maps via self-distillation](https://mlsanity.com/paper/2505.18825/) (2025)
- [Learning Neural Generative Dynamics for Molecular Conformation Generation](https://mlsanity.com/paper/2102.10240/) (2021)
- [Align Your Flow: Scaling Continuous-Time Flow Map Distillation](https://mlsanity.com/paper/2506.14603/) (2025)
- [Multistep Distillation of Diffusion Models via Moment Matching](https://mlsanity.com/paper/2406.04103/) (2024)
- [DeFoG: Discrete Flow Matching for Graph Generation](https://mlsanity.com/paper/2410.04263/) (2024)
- [Beyond Autoregression: Fast LLMs via Self-Distillation Through Time](https://mlsanity.com/paper/2410.21035/) (2024)

## Built on by (most cited first, 1 of 1)

- [Discrete Beckmann Transport Models for One-Step Language Modeling and Reasoning](https://mlsanity.com/paper/2609.15903/) (2026)

## Abstract (arXiv)

Flow-based generative models have enabled remarkable progress in fast and controllable generation across continuous and discrete state spaces, yet existing parameterizations are constrained to fixed dimensions or fixed sequence lengths. Here, we introduce Expanding Generative Flows (EFlows), which define flows between distributions of increasing dimensionality along an expanding interpolant that grows the state by augmenting it with conditional noise. Building on this construction, we propose Expanding Flow Maps (EFMs), a new class of flow maps that distill the expanding interpolant into efficient few-step generative models. Each EFM factors the map between any two timesteps into two learnable operations: an expand operator, which augments the state space with new coordinates or tokens conditioned on the current state, and a transport map, which pushes the expanded state forward along the interpolant. Composing these operators yields a single map that jointly expands and denoises the state, recovering existing fixed-canvas flows and flow maps as the special case in which the expand operator is the identity. We further extend the framework to the discrete simplex, enabling variable-size graph generation and variable-length sequence generation. Across both continuous and discrete modalities, we establish EFlows and EFMs as a principled framework for settings in which output size is itself a learned, controllable degree of freedom.