Expanding Flow Maps: Teaching Generative Models to Grow Like the Universe

Overview

This post introduces Expanding Flow Maps (EFMs), a paper by Sophia Tang and Pranam Chatterjee (arXiv, 2026) that proposes generative models whose state space can grow during generation, removing the fixed-dimension constraint shared by most modern generative models.

The Problem: The Curse of the Fixed Canvas

Nearly all current generative models — diffusion models like Stable Diffusion, video models like Sora, and LLMs — are constrained to fixed sizes:

• Image models generate fixed resolutions; higher resolutions require upsampling networks
• Video models produce fixed-length clips; longer videos require autoregressive stitching
• Text models have fixed context windows requiring sliding-window workarounds

The paper's diagnosis: "...existing parameterizations are constrained to fixed dimensions or fixed sequence lengths."

Background: Flow Models

Flow-based generative models learn a continuous transformation from a simple distribution (Gaussian noise) to a data distribution, governed by an ODE dz/dt = v(z, t). Flow Matching (Lipman et al., 2022) learns the vector field directly, simplifying training. But all such methods assume the "hillside" — the state space — has a fixed size.

The Core Idea: A Growing Canvas

EFMs make the state space itself a learnable, expandable variable:

• Expanding Generative Flows (EFlows) define interpolants between distributions of *different* dimensions — from a low-dimensional p₀ to a higher-dimensional p₁ (with d₁ > d₀), like cosmic expansion from a singularity
• Expansion uses conditional noise augmentation: new dimensions are generated conditioned on the current state, so added coordinates carry coherent latent information rather than pure noise

Each EFM factors the map between timesteps into two learnable operations:

1. Expand operator — augments the state space with new coordinates or tokens conditioned on the current state 2. Transport map — 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."

This means traditional fixed-size flow models are a special case of EFMs — a hallmark of a good theory.

From Continuous to Discrete

The framework extends to discrete settings:

• Graph generation: starting from an empty or seed graph, the model can decide when to add nodes, assign attributes, and create edges — handling variable size, structure, and permutation invariance
• Variable-length sequences: text length becomes an emergent property of generation rather than a preset, with the model deciding at each step whether to add tokens

> "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."

Why It Matters

• Computational efficiency: no need to allocate resources for the maximum possible size; small outputs use small state spaces
• Creative freedom: size decisions shift from pre-generation planning to emergence during generation
• Multi-scale generation: early stages can plan global structure at lower dimensions, later stages expand to add detail
• Compositional generalization: the model learns *how to build structures at any scale*, not mappings to specific sizes
• Physical resonance: like scale-invariant physical laws, each step depends only on current state, not total output size

Deeper Theory

The post connects EFMs to intrinsic (manifold) dimensionality — generation can proceed at the essential dimension and expand only as needed. Information-theoretically, the expand operator adds dimensions whose conditional entropy given the current state is lower, ensuring meaningful expansion. EFMs generalize diffusion models by allowing dimension changes, learned expansion timing, and smooth dimensional growth.

Outlook

The ultimate vision is an unbounded generative model: images that can zoom indefinitely, text that ends when the story is complete, molecules whose atom count is determined by function. The paper's contribution is reframing output size itself as "a learned, controllable degree of freedom."

References

1. Tang, S., & Chatterjee, P. (2026). *Expanding Flow Maps*. arXiv preprint. 2. Lipman, Y., et al. (2022). Flow Matching for Generative Modeling. arXiv:2210.02747. 3. Ho, J., et al. (2020). Denoising Diffusion Probabilistic Models. NeurIPS. 4. Rezende, D., & Mohamed, S. (2015). Variational Inference with Normalizing Flows. ICML. 5. Liu, X., et al. (2022). Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow. arXiv preprint.