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I∗,MD,t1,τD)‖22]+𝔼x,t2[‖vθ(xt2⟨A,B⟩,MC,t2,τC)−vθ(xt2⟨A′,B′⟩,MC,t2,τC)‖22],\begin{split}\mathcal{L}_{cyc}=\mathbb{E}_{x,t_{1}}\!\Big[\|v_{\theta}(x_{t_{1}}^{I},M_{D},t_{1},\tau_{D})-v_{\theta}(x_{t_{1}}^{I^{*}},M_{D},t_{1},\tau_{D})\|_{2}^{2}\Big]\\ +\mathbb{E}_{x,t_{2}}\!\Big[\|v_{\theta}(x_{t_{2}}^{\langle A,B\rangle},M_{C},t_{2},\tau_{C})-v_{\theta}(x_{t_{2}}^{\langle A^{\prime},B^{\prime}\rangle},M_{C},t_{2},\tau_{C})\|_{2}^{2}\Big],\end{split} (5) where t1t_{1} and t2t_{2} are two different timesteps.
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Cycle-consistent training unlocks robust layered image decomposition in diffusion models, even with complex interactions like shading and reflections.
