Flow Matching with Arbitrary Auxiliary Paths

We introduce **Flow Matching with Arbitrary Auxiliary Paths (AuxPath-FM)**, a generative modeling framework that makes the auxiliary distribution an explicit choice in probability-path design. AuxPath-FM constructs trajectories $X_t=a(t)X_1+b(t)X_0+c(t)\eta$, allowing $\eta$ to be continuous, discrete, or coupled to the endpoints without requiring Gaussianity or independence. Under explicit moment and coefficient regularity conditions, we establish endpoint preservation, a weak continuity equation, and a tractable regression objective equivalent to marginal flow matching for each fixed path. This formulation accommodates Gaussian, Uniform, Laplace, Rademacher, and learned semantic auxiliaries within a common training framework. Semantic auxiliaries encode labels in the path and enable trajectory-level guidance with one backbone evaluation per sampling step. Experiments reveal how auxiliary choice affects generation across datasets and how semantic guidance improves label alignment. On ImageNet-1k, auxiliary guidance reduces FID from 23.25 to 18.01 at approximately the one-pass CFM cost and roughly half the computation of two-pass classifier-free guidance (CFG) at the same step count. AuxPath-FM connects flexible path design with a practical quality–cost tradeoff in conditional generation.