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O(n^{2})
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s_{o}\oplus s_{a}\in\mathbb{V}^{n+m}
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Z\in\mathbb{R}^{m\times d_{\text{token}}}
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E_{\psi}(s)
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\displaystyle=F^{i}(E_{\psi}(s_{o})\oplus\text{Proj}_{\psi}(Z)).
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\displaystyle\cos(v,v_{t}^{image})+\lambda\cos(v,v_{t}^{text})
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\cos(\psi_{i},\psi_{j})
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v_{t}^{text}=F^{t}(E_{\psi}(s^{\prime}))
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\displaystyle\text{argmax}_{Z}
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\rightarrow
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\mathcal{A}(x,t,s_{o})
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\displaystyle=F^{i}(E_{\psi}(s_{o}\oplus s_{a}))
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\text{Proj}_{\psi}(Z)_{i}=Z_{i}+\text{sg}(\psi_{j}-Z_{i})
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500\times 20=10000
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w_{i},w_{j}
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v_{t}^{image}\leftarrow F^{i}(x_{t})
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s_{a}=E_{\psi}^{-1}(\text{Proj}_{\psi}(Z))
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{}^{1,*}
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\text{Proj}_{\psi}(Z)
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\displaystyle\text{argmax}_{s_{a}}
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s^{\prime}\leftarrow
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v_{t}^{image}
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5\times 4\times 100=2000
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{}^{1,2}
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\psi\in\mathbb{R}^{|\mathbb{V}|\times d_{\text{token}}}
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bestloss\leftarrow\mathcal{L},bestZ\leftarrow Z
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\lambda=0
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\text{Proj}_{\psi}:\mathbb{R}^{m\times d_{\text{token}}}\rightarrow\mathbb{R}^%
{m\times d_{\text{token}}}
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i\leftarrow 1
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s\in\mathbb{V}^{*}
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\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})~{},
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\displaystyle\cos(v,v_{t}^{image})+\lambda\cos(v,v_{t}^{text}),
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\cos(a,b)=\frac{a^{T}b}{\|a\|\|b\|}
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512\times 512
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E_{\psi}(s_{o}\oplus s_{a})=E_{\psi}(s_{o})\oplus E_{\psi}(s_{a})
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bestloss>\mathcal{L}
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v_{t}^{image}=F^{i}(x_{t})
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d_{\text{emb}}
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\displaystyle\text{argmax}_{s_{a}}\cos(F^{i}(E_{\psi}(s_{o}\oplus s_{a})),v_{t%
}).
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s^{\prime}=
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{}^{3,*}
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Z\leftarrow Z-\eta\nabla_{Z}\mathcal{L}
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s_{o}\oplus s_{a}
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\displaystyle\text{s.t.}\quad v=F^{i}(E_{\psi}(s_{o}\oplus s_{a})),
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\mathbb{V}=\{w_{1},w_{2},\cdots,w_{|\mathbb{V}|}\}
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\displaystyle\text{s.t.}\quad v
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E_{\psi}(s)_{i}=\psi_{j}
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5\times 4=20
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3\times 100
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v\leftarrow F^{t}(E_{\psi}(s_{o})\oplus\text{Proj}_{\psi}(Z))
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\mathcal{L}=-\cos(v,v_{t}^{image})-\lambda\cos(v,v_{t}^{text})
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s_{o}\in\mathbb{V}^{n}
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s_{a}\leftarrow E_{\psi}^{-1}(\text{Proj}_{\psi}(bestZ))
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bestloss\leftarrow\infty,bestZ\leftarrow Z
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\displaystyle=F^{i}(E_{\psi}(s_{o}\oplus E_{\psi}^{-1}(\text{Proj}_{\psi}(Z))))
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t\in\mathbb{V}
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x\sim G(v)
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d_{\text{token}}
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s_{a}\in\mathbb{V}^{m}
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\mathbb{V}
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w_{j}=s_{i}
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t\in\mathcal{V}
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x\sim G(F^{t}(E_{\psi}(s)))
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E_{\psi}
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j=\text{argmin}_{j^{\prime}}\|\psi_{j^{\prime}}-Z_{i}\|_{2}^{2}
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|s|\times d_{\text{token}}
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\displaystyle\text{argmax}_{v_{t}}\mathbb{E}_{x\sim G(v_{t})}\mathcal{A}(x,t,s%
_{o})~{}.
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E_{L}\cup E_{R}
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E_{L}=\{(u,w)|(u,w)\in E,w\neq v\}
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v_{1},v_{2}\in\overline{V_{m}}
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|\Delta E|^{\prime}=w^{*}
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1\leq j\leq k
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{\mathcal{R}}
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\pi\left(G^{\prime}\right)\leq\pi(G)-d
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\displaystyle\underbrace{n_{R}\times|y-C|\times k^{\prime}}_{\text{between the%
subpath of }w_{2}\text{ and the vertices in }u}+\underbrace{n_{R}\times|y-B-C%
|\times k}_{{\text{between the subpath of }w_{2}\text{ and the vertices in }v}}+
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\operatorname{min}(\mathcal{E}(M_{L}^{*}),\mathcal{E}(M_{R}^{*}))
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E_{m}=\{e^{*}\}
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\displaystyle n_{L}\times n_{R}\times|x+y-A-B-C|\xrightarrow{n_{R}=0}
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v_{i}\in V_{L}
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M_{R}^{*}\xleftarrow[]{}\emptyset
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\alpha_{1}=A-x+x-A-B<B\xrightarrow{}0<2B
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u,v\in V
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\left|\sum_{k=i}^{j-2}(w_{k}+\epsilon_{k})-w^{*}-\sum_{k=i}^{j-2}w_{k}\right|=%
\left|w^{*}-\sum_{k=i}^{j-2}\epsilon_{k}\right|
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n_{L}(|x-A|+|x-A-B|)
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\mathcal{E}_{L}^{(i+1)}-\mathcal{E}_{L}^{(i)}=n_{L}\times\big{(}(i+1)\;w_{i+1}%
-(k-i)\;w_{i+1}\big{)}
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w^{\prime}(e_{i})=w(e_{i})+w(e^{*})
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S_{R}\xleftarrow{}\sum_{\forall e_{i}\in E_{R}}R_{i}
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u,v\in G_{1}
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\Delta({\text{MARK\_RIGHT}})
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\operatorname{min}(\mathcal{E}(M_{R}^{*}),\mathcal{E}(M_{L}^{*}))\leq\mathcal{%
E}(M)
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{\mathcal{L}}-(\frac{{\mathcal{L}}{n_{L}}+{n_{R}}(1-{\mathcal{R}})}{2{n_{L}}})%
=\frac{{\mathcal{L}}{n_{L}}-{n_{R}}(1-{\mathcal{R}})}{2{n_{L}}}
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T_{j}^{R}
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\varphi(x)=x/\alpha-\beta
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\mathcal{E}=\mathcal{E}_{R}
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S_{R}-S_{L}\leq R_{i}
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n_{L}\geq 0
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c_{2}\geq\epsilon
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{n^{2}_{L}}\times((i+1)({\mathcal{L}}-i-1)-i({\mathcal{L}}-i))={n^{2}_{L}}%
\times(i{\mathcal{L}}-i^{2}-i+{\mathcal{L}}-i-1-i{\mathcal{L}}+i^{2})={n^{2}_{%
L}}\times({\mathcal{L}}-2i-1)
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e^{*}_{k}=(u_{j},u_{j+1})
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|\Delta E|=|\overline{V_{m}}|(w^{*}_{1}+\dots+w^{*}_{k})=(n-2k)(w^{*}_{1}+%
\dots+w^{*}_{k})
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