Flow Matching with Arbitrary Auxiliary Paths

TL;DR

AuxPath-FM introduces a flexible generative modeling framework that incorporates arbitrary auxiliary distributions into probability paths, enabling diverse and task-specific flow designs.

Contribution

It generalizes conditional flow matching by allowing any auxiliary distribution, expanding the design space for probability paths in generative models.

Findings

Supports various priors like Gaussian, Uniform, Laplace, and Rademacher.

Maintains a consistent training objective with the marginal formulation.

Enables label-guided generation through structured auxiliary distributions.

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 $\eta$ to follow any distribution, producing trajectories of the form $X_t = a(t)X_1 + b(t)X_0 + c(t)\eta$. 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.