Expanding Flow Maps

2026-07-23

AI Summary

Existing flow-based generative models struggle with tasks involving variable output dimensions. This work proposes Extended Flows (EFlows) and Extended Flow Mappings (EFMs), establishing the first continuous flow framework that supports dynamically expanding state spaces. By treating output dimensionality as a learnable and controllable degree of freedom, the approach unifies conventional fixed-dimensional flow models as a special case. The method jointly optimizes augmentation operators and transport mappings along interpolation paths, enabling efficient generation of variable-size graphs and sequences over both continuous spaces and discrete simplices. Experiments demonstrate that the proposed framework consistently produces high-quality, controllable data of varying sizes across both continuous and discrete modalities.

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. 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.

Problem

• flow-based generative models
• variable-size generation
• expanding dimensionality
• sequence length variability
• discrete simplex

Innovation

• Expanding Flow Maps
• generative flows
• variable-size generation
• expand operator
• discrete simplex