Flow Matching with Arbitrary Auxiliary Paths

2026-05-07

AI Summary

This work addresses the limitations of conventional flow matching methods, which rely on Gaussian noise as auxiliary variables and thus lack flexibility in designing probability paths tailored to diverse generative tasks. The authors propose AuxPath-FM, a novel framework that, for the first time, permits auxiliary variables to follow arbitrary distributions—such as uniform, Laplace, or Rademacher—enabling more general probabilistic paths. Built upon conditional flow matching theory, the method integrates the continuity equation with marginal consistency and introduces learnable time-scaling functions \(a(t)\), \(b(t)\), and \(c(t)\) to modulate the generation trajectory. Experiments demonstrate that AuxPath-FM achieves high-quality generation across various priors and effectively supports structured semantic tasks like label-guided synthesis, confirming both its theoretical generality and practical adaptability.

Abstract

We introduce a new generative modeling framework, \textbf{Flow Matching with Arbitrary Auxiliary Paths (AuxPath-FM)}, which generalizes conditional flow matching by incorporating an auxiliary variable drawn from an arbitrary distribution into the probability path. Unlike prior methods that restrict auxiliary components to Gaussian noise, AuxPath-FM allows the variable $η$ to follow any distribution, producing trajectories of the form $X_t = a(t)X_1 + b(t)X_0 + c(t)η$. We theoretically demonstrate that this construction preserves the continuity equation and maintains a training objective consistent with the marginal formulation. This flexibility enables the design of diverse probability paths using various priors, including Gaussian, Uniform, Laplace, and discrete Rademacher distributions, each offering unique geometric properties for generative flows. Furthermore, our framework allows for specialized tasks such as label-guided generation by encoding structured semantic information into the auxiliary distribution. Overall, AuxPath-FM provides a principled and general foundation for probability path design, offering both theoretical generality and practical flexibility for diverse generative modeling tasks.

Problem

• Flow Matching
• Auxiliary Paths
• Generative Modeling
• Probability Path
• Conditional Generation

Innovation

• Flow Matching
• Arbitrary Auxiliary Paths
• Generative Modeling
• Probability Path Design
• Conditional Flow Matching