$O(n^{2})$

$f$

$n$

$G(v)$

$s_{o}\oplus s_{a}\in\mathbb{V}^{n+m}$

$Z\in\mathbb{R}^{m\times d_{\text{token}}}$

$E_{\psi}(s)$

$\displaystyle=F^{i}(E_{\psi}(s_{o})\oplus\text{Proj}_{\psi}(Z)).$

$\displaystyle\cos(v,v_{t}^{image})+\lambda\cos(v,v_{t}^{text})$

$\cos(\psi_{i},\psi_{j})$

${}^{4}$

$v_{t}^{text}=F^{t}(E_{\psi}(s^{\prime}))$

${}^{*}$

$\displaystyle\text{argmax}_{Z}$

$\rightarrow$

$\mathcal{A}(x,t,s_{o})$

$\displaystyle=F^{i}(E_{\psi}(s_{o}\oplus s_{a}))$

${}^{1}$

$\text{Proj}_{\psi}(Z)_{i}=Z_{i}+\text{sg}(\psi_{j}-Z_{i})$

$x_{t}$

$500\times 20=10000$

$w_{i},w_{j}$

$v_{t}^{image}\leftarrow F^{i}(x_{t})$

$m=4$

$s_{a}=E_{\psi}^{-1}(\text{Proj}_{\psi}(Z))$

${}^{5}$

$Z_{i}$

${}^{1,*}$

$\text{Proj}_{\psi}(Z)$

$s$

$\displaystyle\text{argmax}_{s_{a}}$

$t$

$s^{\prime}\leftarrow$

$v_{t}^{image}$

$5\times 4\times 100=2000$

${}^{1,2}$

$\psi\in\mathbb{R}^{|\mathbb{V}|\times d_{\text{token}}}$

$bestloss\leftarrow\mathcal{L},bestZ\leftarrow Z$

$G$

$\lambda=0$

$\text{Proj}_{\psi}:\mathbb{R}^{m\times d_{\text{token}}}\rightarrow\mathbb{R}^% {m\times d_{\text{token}}}$

$i\leftarrow 1$

$s\in\mathbb{V}^{*}$

$\displaystyle\text{argmax}_{s_{a}}\mathbb{E}_{x\sim G(F^{t}(E_{\psi}(s_{o}% \oplus s_{a})))}\mathcal{A}(x,t,s_{o})~{},$

$\displaystyle\cos(v,v_{t}^{image})+\lambda\cos(v,v_{t}^{text}),$

$\cos(a,b)=\frac{a^{T}b}{\|a\|\|b\|}$

$\eta$

$512\times 512$

$x$

$E_{\psi}(s_{o}\oplus s_{a})=E_{\psi}(s_{o})\oplus E_{\psi}(s_{a})$

$N$

$bestloss>\mathcal{L}$

$v_{t}^{image}=F^{i}(x_{t})$

$d_{\text{emb}}$

$\displaystyle\text{argmax}_{s_{a}}\cos(F^{i}(E_{\psi}(s_{o}\oplus s_{a})),v_{t% }).$

$s^{\prime}=$

${}^{3,*}$

$Z\leftarrow Z-\eta\nabla_{Z}\mathcal{L}$

$100$

$s_{a}$

$s_{o}\oplus s_{a}$

$m$

$v$

$\displaystyle\text{s.t.}\quad v=F^{i}(E_{\psi}(s_{o}\oplus s_{a})),$

$\mathbb{V}=\{w_{1},w_{2},\cdots,w_{|\mathbb{V}|}\}$

$F^{i}$

$\psi$

$\displaystyle\text{s.t.}\quad v$

$s_{o}$

$F^{t}$

${}^{2}$

$\oplus$

$E_{\psi}(s)_{i}=\psi_{j}$

$5\times 4=20$

$3\times 100$

${}^{3}$

$v\leftarrow F^{t}(E_{\psi}(s_{o})\oplus\text{Proj}_{\psi}(Z))$

$\mathcal{L}=-\cos(v,v_{t}^{image})-\lambda\cos(v,v_{t}^{text})$

$s_{o}\in\mathbb{V}^{n}$

$s_{a}\leftarrow E_{\psi}^{-1}(\text{Proj}_{\psi}(bestZ))$

$bestloss\leftarrow\infty,bestZ\leftarrow Z$

$\displaystyle=F^{i}(E_{\psi}(s_{o}\oplus E_{\psi}^{-1}(\text{Proj}_{\psi}(Z))))$

$t\in\mathbb{V}$

$Z$

$(\cdot)$

$x\sim G(v)$

$d_{\text{token}}$

$s_{a}\in\mathbb{V}^{m}$

$v_{t}$

$\lambda$

$\mathbb{V}$

$w_{j}=s_{i}$

$t\in\mathcal{V}$

$x\sim G(F^{t}(E_{\psi}(s)))$

$E_{\psi}$

$j=\text{argmin}_{j^{\prime}}\|\psi_{j^{\prime}}-Z_{i}\|_{2}^{2}$

$|s|\times d_{\text{token}}$

$\displaystyle\text{argmax}_{v_{t}}\mathbb{E}_{x\sim G(v_{t})}\mathcal{A}(x,t,s% _{o})~{}.$

$E_{L}\cup E_{R}$

$E_{L}=\{(u,w)|(u,w)\in E,w\neq v\}$