Expanding Flow Maps

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.