Expanding Flow Maps: Generative Flows Across Increasing Dimensions

Forum topic · 小凯 · 2026-07-27

Summary

Expanding Flow Maps (EFMs), introduced by Sophia Tang and Pranam Chatterjee (arXiv:2507.21743, July 2025), extend flow-based generative models beyond fixed dimensions and fixed sequence lengths. The paper first defines Expanding Generative Flows (EFlows), which establish flows between distributions of increasing dimensionality via an expanding interpolant that grows the state by augmenting it with conditional noise. Building on this, EFMs distill the expanding interpolant into efficient few-step generative models. Each EFM factors the map between any two time steps into two learnable operations: an expand operator that augments the state space with new coordinates or tokens conditioned on the current state, and a transport map that advances the expanded state along the interpolant. Composing these operations yields a single mapping that jointly expands and denoises the state; existing fixed-canvas flows and flow maps are recovered as special cases where the expand operator is the identity. The framework extends to the discrete simplex, enabling variable-size graph generation and variable-length sequence generation. EFMs provide a principled framework for both continuous and discrete modalities where output size is itself a learnable, controllable degree of freedom.

Overview

Field: Machine Learning
Authors: Sophia Tang, Pranam Chatterjee
Published: 2025-07-27
arXiv: 2507.21743

Key Points

Flow-based generative models have enabled fast and controllable generation in continuous and discrete state spaces, but existing parameterizations are constrained to fixed dimensions or fixed sequence lengths.
The paper introduces Expanding Generative Flows (EFlows), which define flows between distributions of increasing dimensionality via an expanding interpolant that grows the state by augmenting it with conditional noise.
Building on EFlows, the authors 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 time steps into two learnable operations:
An expand operator, which augments the state space with new coordinates or tokens conditioned on the current state.
A transport map, which advances the expanded state forward along the interpolant.
Composing these operations produces a single mapping that jointly expands and denoises the state.
Existing fixed-canvas flows and flow maps are recovered as special cases in which the expand operator is the identity mapping.
The framework extends to the discrete simplex, enabling variable-size graph generation and variable-length sequence generation.
Across continuous and discrete modalities, EFlows and EFMs establish a principled framework for settings where the output size itself is a learnable, controllable degree of freedom.

Original Abstract (excerpt)

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 time steps 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 advances the expanded state along the interpolant.

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