Adaptively Pivoted Flow Matching for Composition-Consistent Translation

Controlled image translation requires jointly achieving target fidelity, source equivariance under appearance interventions, and compositional consistency across sequential paths. Factoring conditional transport through a shared latent pivot ($T_{s\to t}=D_t\circ E_s$) naturally induces a translation groupoid structure: this algebraic formulation inherently ensures compositional consistency under exact integration, guarantees bi-Lipschitz source sensitivity against information collapse, and introduces a structural inductive bias for source equivariance. While this architecture is coupled with a standard Gaussian prior, overlapping interpolation distributions associate the same state, time, and condition with different target velocities, producing conditional velocity ambiguity. To reduce this ambiguity, we introduce Adaptively Pivoted Flow Matching (AP-FM), which replaces the single Gaussian with an adaptive prior comprising trainable multi-Gaussian modes optimized with a prior mode adaptation loss defined using a frozen reference field. At inference, the observed source determines the latent pivot; neither identity labels nor mode lookup is required. On SRN Cars and DeepFashion, AP-FM preserves source appearance under interventions and composed translations, with improved target fidelity over a standard Gaussian prior.