Expanding Flow Maps

Sophia Tang, Pranam Chatterjee

Keywords: flow maps, generative modeling, generative flows, flow matching

TL;DR: Expanding Flow Maps generalize flow maps to distributions of growing dimensionality by composing an expand operator with a transport operator which jointly expands and denoises the state toward the target distribution.

Abstract: Flow maps have enabled remarkable progress in few-step generative modeling across both continuous and discrete state spaces. Despite their promise, existing parameterizations are restricted to flows over fixed dimensions or fixed sequence lengths. Here, we introduce **Expanding Generative Flows** (EFlows), which define flows between distributions of *increasing dimensionality* through an expanding interpolant built from *augmented distributions of conditional noise*. Building on this framework, we propose **Expanding Flow Maps** (EFMs), a new class of flow maps that distill EFlows into efficient few-step generative models. Each EFM factors the map between any two timesteps into two learned components: an *expand operator*, which augments the state with new coordinates or tokens, and a *transport operator*, 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 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-length sequence generation via token insertions. Across domains, EFlows and EFMs provide a principled approach to generative problems in which output size is itself learned, controllable degree of freedom.

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Submission Number: 175