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Y-shaped Generative Flows

Arip Asadulaev, Semyon Semenov, Abduragim Shtanchaev, Eric Moulines, Fakhri Karray, Martin Takac

TL;DR

Y-shaped generative flows replace traditional V-shaped transport with shared trunk pathways that branch toward targets by optimizing a velocity-powered action with a concave exponent $\alpha\in(0,1)$. The authors show that this velocity-based formulation is equivalent (up to constants) to branched transport costs under bounded density and enable a scalable neural-ODE training objective augmented by a Sinkhorn boundary loss to enforce endpoint matching. They provide theoretical results (equivalence, time-compression, well-posedness) and demonstrate through synthetic, image, and biology experiments that Y-flows learn hierarchical, branching trajectories, improve distributional metrics over strong baselines, and achieve target distributions with fewer integration steps. The approach highlights a path toward structure-aware generative modeling where computation adapts to data topology, potentially informing future comparisons with diffusion models and broader applications in high-dimensional transport.

Abstract

Modern continuous-time generative models often induce V-shaped transport: each sample travels independently along nearly straight trajectories from prior to data, overlooking shared structure. We introduce Y-shaped generative flows, which move probability mass together along shared pathways before branching to target-specific endpoints. Our formulation is based on novel velocity-powered objective with a sublinear exponent (between zero and one). this concave dependence rewards joint and fast mass movement. Practically, we instantiate the idea in a scalable neural ODE training objective. On synthetic, image, and biology datasets, Y-flows recover hierarchy-aware structure, improve distributional metrics over strong flow-based baselines, and reach targets with fewer integration steps.

Y-shaped Generative Flows

TL;DR

Y-shaped generative flows replace traditional V-shaped transport with shared trunk pathways that branch toward targets by optimizing a velocity-powered action with a concave exponent . The authors show that this velocity-based formulation is equivalent (up to constants) to branched transport costs under bounded density and enable a scalable neural-ODE training objective augmented by a Sinkhorn boundary loss to enforce endpoint matching. They provide theoretical results (equivalence, time-compression, well-posedness) and demonstrate through synthetic, image, and biology experiments that Y-flows learn hierarchical, branching trajectories, improve distributional metrics over strong baselines, and achieve target distributions with fewer integration steps. The approach highlights a path toward structure-aware generative modeling where computation adapts to data topology, potentially informing future comparisons with diffusion models and broader applications in high-dimensional transport.

Abstract

Modern continuous-time generative models often induce V-shaped transport: each sample travels independently along nearly straight trajectories from prior to data, overlooking shared structure. We introduce Y-shaped generative flows, which move probability mass together along shared pathways before branching to target-specific endpoints. Our formulation is based on novel velocity-powered objective with a sublinear exponent (between zero and one). this concave dependence rewards joint and fast mass movement. Practically, we instantiate the idea in a scalable neural ODE training objective. On synthetic, image, and biology datasets, Y-flows recover hierarchy-aware structure, improve distributional metrics over strong flow-based baselines, and reach targets with fewer integration steps.

Paper Structure

This paper contains 20 sections, 3 theorems, 41 equations, 8 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Flow Shape Control
  4. Method
  5. Practical Implementation
  6. Time discretization and MC estimator.
  7. Lemma 3 (Well posedness).
  8. Related Work
  9. Experiments
  10. Branching Illustrations
  11. Biology Data
  12. Image data
  13. Conclusion
  14. Background (Extended)
  15. Modica-Mortola Branched Flows
  16. ...and 5 more sections

Key Result

Lemma 1

Let the exponent $\alpha \in (0,1)$. Suppose the density $\rho(x,t)$ is bounded such that for constants $m, M$: Then, the following inequality holds:

Figures (8)

  • Figure 1: Blueprint of flow shapes. Conceptually in a V-flow, the mass separates and moves from $x$ along straight lines. In Y and T flows, the mass moves together and then splits into targets.
  • Figure 2: Comparison of Gaussian mixture toy tasks. Each column is a task (T-shape, 4,6,18 Branches). Each row is a method: top = Flow Matching (FM), bottom = Y-Flows (ours). The color gradient represents the flow steps. A monotone color indicates that the number of steps on this region was equal to 1. As shown in the 2, 4, and 18-branch cases, our model initially made a significant jump toward the target before splitting the mass. Subsequent movement involved reaching the targets via small steps
  • Figure 3: Result of Y-Flows on LiDAR dataset.
  • Figure 4: PCA projected results of Y-Flows on Tedsim dataset (50D).
  • Figure 5: Female-to-male domain translation in the ALAE latent space. Two-step ODE results. Our method produces compatible results in a higher space, as do other generative models.
  • ...and 3 more figures

Theorems & Definitions (6)

  • Lemma 1: Equivalence under bounded density
  • proof
  • Lemma 2: Time-compression
  • proof
  • Lemma 3: Well-posedness in the neural-ODE setting
  • proof