Learning Straight Flows: Variational Flow Matching for Efficient Generation

TL;DR

This work tackles the curvature of flows learned by Flow Matching caused by independent sampling couplings. It introduces Straight Variational Flow Matching (S-VFM), which conditions a velocity field on a global generation overview via a variational latent code and enforces straight trajectories by minimizing the time derivative $D_t v$ along generation paths. The authors provide a theoretical foundation for straight interpolants, proving marginal preservation and equivalence to vanishing $D_t v$, and demonstrate strong empirical gains in one-step and few-step generation on CIFAR-10 and ImageNet, along with synthetic visualizations. Collectively, S-VFM improves generation fidelity, sampling efficiency, and training stability by resolving the intrinsic conflict between straight trajectory learning and independent couplings, with practical impact on fast, high-quality generative modeling.

Abstract

Flow Matching has limited ability in achieving one-step generation due to its reliance on learned curved trajectories. Previous studies have attempted to address this limitation by either modifying the coupling distribution to prevent interpolant intersections or introducing consistency and mean-velocity modeling to promote straight trajectory learning. However, these approaches often suffer from discrete approximation errors, training instability, and convergence difficulties. To tackle these issues, in the present work, we propose \textbf{S}traight \textbf{V}ariational \textbf{F}low \textbf{M}atching (\textbf{S-VFM}), which integrates a variational latent code representing the ``generation overview'' into the Flow Matching framework. \textbf{S-VFM} explicitly enforces trajectory straightness, ideally producing linear generation paths. The proposed method achieves competitive performance across three challenge benchmarks and demonstrates advantages in both training and inference efficiency compared with existing methods.

Figures (6)

• Figure 1: Generation Trajectory Visualization in the 2D Synthesized Hexagonal Dataset.
• Figure 2: Generation Trajectory Visualization in the 2D Synthesized Eight-Gaussians-to-Moon Dataset.
• Figure 3: Randomly Selected Generation Results under Different NFE. Each row corresponds to a distinct initial noise set ($X^1_0$ or $X^2_0$) and its associated latent code set ($z^1$ or $z^2$). Within each row, images at the same grid position across panels are generated from the same initial noise and latent code, while panels from left to right correspond to increasing NFE values of $[1, 2, 5, 10]$. Both the noise samples and latent codes are independently drawn from their prior distributions.
• Figure 4: Generation results under the same initial noise but different latent codes. Panels (a) and (b) are generated from the same initial noise set $X^1_0$ with different latent code sets $z^1$ and $z^2$, respectively, while panels (c) and (d) use a different initial noise set $X^2_0$ with corresponding latent code sets $z^3$ and $z^4$. All noise samples and latent codes are independently drawn from their prior distributions, and the number of function evaluations is fixed at $\text{NFE}=1$.
• Figure 5: Comparison of FID-50K Score over Training Iterations on ImageNet $256 \times 256$ Dataset.

Theorems & Definitions (27)

• Definition 1: Rectifiability of $X$
• Definition 2: Non-intersection functional
• Lemma 1: Non-intersection $\equiv$ zero conditional variance
• Definition 3: Straight interpolation compatible with $X$
• Theorem 2: Marginal preservation
• Theorem 3: Convex transport-cost reduction
• Theorem 4: Equivalent characterizations of straight interpolants
• Definition 4: Time derivative
• Theorem 5: Straightness $\equiv$ vanishing time derivative along $X$
• Definition 5: Coupling, linear interpolation, and conditional velocity