Signed Rectified Flow: Negativity Controlled Generation

We introduce \emph{Signed Rectified Flow (Signed RF)}, a generalization of Rectified Flow that targets a signed measure $\pi^{\mathtt{sign}} = (1+\alpha)\,\pi^+ - \alpha \,\pi^-$, where $\alpha>0$, $\pi^+$ represents the distribution to promote, and $\pi^-$ represents the distribution to suppress. Although sampling from a signed measure is not well-defined, Signed RF induces a valid generative process that concentrates on the positive region of $\pi^{\mathtt{sign}}$ while provably excluding regions dominated by the negative component. This yields a principled framework for incorporating negative information and exclusion constraints into generative modeling. Theoretically, we analyze the signed continuity equation underlying Signed RF and explain how negative mass creates exclusion barriers through a charged-particle interpretation. Empirically, Signed RF leads to practical adaptive guidance algorithms. Across applications, Signed RF improves the fidelity--diversity trade-off on ImageNet, reduces nearest-neighbor similarity in anti-memorization stress tests, and mitigates adversarial-prompt nudity in SD 3.5 while preserving CLIP and aesthetic scores.