Signed Rectified Flow: Negativity-Controlled Generation
We introduce Signed Rectified Flow (Signed RF), a generalization of Rectified Flow that targets the signed measure $\pi^{sign} = (1+\alpha)\pi^+ - \alpha\pi^-$, where $\alpha>0$, $\pi^+$ is the distribution to promote, and $\pi^-$ is the distribution to suppress. Although direct sampling from a signed measure is not well-defined, Signed RF induces a valid generative process that concentrates probability in regions where the signed measure is positive while provably excluding regions dominated by its negative component. It therefore provides a principled framework for incorporating negative information and exclusion constraints into generative modeling. We analyze the signed continuity equation underlying Signed RF and use a charged-particle interpretation to explain how negative mass forms exclusion barriers. This theory further motivates practical adaptive guidance algorithms. Across several applications, Signed RF improves the fidelity-diversity trade-off on ImageNet, reduces nearest-neighbor similarity in anti-memorization experiments, and reduces nudity induced by adversarial prompts in Stable Diffusion 3.5 while preserving CLIP and aesthetic scores.