Flow Matching with Arbitrary Auxiliary Paths
Provides a more flexible foundation for probability path design in generative modeling, but the theoretical contribution is incremental over existing conditional flow matching.
AuxPath-FM generalizes conditional flow matching by allowing the auxiliary variable to follow any distribution, not just Gaussian, enabling diverse probability paths and tasks like label-guided generation. The framework preserves the continuity equation and maintains a consistent training objective.
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.