Expanding Flow Maps: Generative Flows Across Growing Dimensions (arXiv 2507.21743)

Forum topic · 小凯 · 2026-07-27

Summary

Expanding Flow Maps (EFMs), introduced by Sophia Tang and Pranam Chatterjee (arXiv:2507.21743), address a key limitation of flow-based generative models: existing parameterizations are restricted to fixed dimensions or fixed sequence lengths. The paper first proposes 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 this, EFMs distill the expanding interpolant into efficient few-step generative models by factoring 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, recovering existing fixed-canvas flows and flow maps as special cases when the expand operator is the identity. The framework extends to the discrete simplex, enabling variable-size graph generation and variable-length sequence generation across continuous and discrete modalities, making output size a learnable, controllable degree of freedom.

Expanding Flow Maps

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

Abstract

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. This paper introduces 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, 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 where the expand operator is the identity.

The framework is further extended 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 is itself a learnable, controllable degree of freedom.

Key points

EFlows remove the fixed-dimension / fixed-sequence-length constraint of standard flow-based generative models.
EFMs distill expanding interpolants into few-step generative models via two learnable components: expand operator + transport map.
Fixed-canvas flows and flow maps are special cases (identity expand operator).
Applicable to variable-size graph generation and variable-length sequence generation in discrete domains.