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<!DOCTYPE html>
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<head>
<title>MathJax Example</title>
<script>
MathJax = {
tex: {
inlineMath: [['$', '$'], ['\(', '\)']]
},
svg: {
fontCache: 'global'
}
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<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
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<body>
<p> $O(n^{2})$ </p>
<p> $f$ </p>
<p> $n$ </p>
<p> $G(v)$ </p>
<p> $s_{o}\oplus s_{a}\in\mathbb{V}^{n+m}$ </p>
<p> $Z\in\mathbb{R}^{m\times d_{\text{token}}}$ </p>
<p> $E_{\psi}(s)$ </p>
<p> $\displaystyle=F^{i}(E_{\psi}(s_{o})\oplus\text{Proj}_{\psi}(Z)).$ </p>
<p> $\displaystyle\cos(v,v_{t}^{image})+\lambda\cos(v,v_{t}^{text})$ </p>
<p> $\cos(\psi_{i},\psi_{j})$ </p>
<p> ${}^{4}$ </p>
<p> $v_{t}^{text}=F^{t}(E_{\psi}(s^{\prime}))$ </p>
<p> ${}^{*}$ </p>
<p> $\displaystyle\text{argmax}_{Z}$ </p>
<p> $\rightarrow$ </p>
<p> $\mathcal{A}(x,t,s_{o})$ </p>
<p> $\displaystyle=F^{i}(E_{\psi}(s_{o}\oplus s_{a}))$ </p>
<p> ${}^{1}$ </p>
<p> $\text{Proj}_{\psi}(Z)_{i}=Z_{i}+\text{sg}(\psi_{j}-Z_{i})$ </p>
<p> $x_{t}$ </p>
<p> $500\times 20=10000$ </p>
<p> $w_{i},w_{j}$ </p>
<p> $v_{t}^{image}\leftarrow F^{i}(x_{t})$ </p>
<p> $m=4$ </p>
<p> $s_{a}=E_{\psi}^{-1}(\text{Proj}_{\psi}(Z))$ </p>
<p> ${}^{5}$ </p>
<p> $Z_{i}$ </p>
<p> ${}^{1,*}$ </p>
<p> $\text{Proj}_{\psi}(Z)$ </p>
<p> $s$ </p>
<p> $\displaystyle\text{argmax}_{s_{a}}$ </p>
<p> $t$ </p>
<p> $s^{\prime}\leftarrow$ </p>
<p> $v_{t}^{image}$ </p>
<p> $5\times 4\times 100=2000$ </p>
<p> ${}^{1,2}$ </p>
<p> $\psi\in\mathbb{R}^{|\mathbb{V}|\times d_{\text{token}}}$ </p>
<p> $bestloss\leftarrow\mathcal{L},bestZ\leftarrow Z$ </p>
<p> $G$ </p>
<p> $\lambda=0$ </p>
<p> $\text{Proj}_{\psi}:\mathbb{R}^{m\times d_{\text{token}}}\rightarrow\mathbb{R}^%
{m\times d_{\text{token}}}$ </p>
<p> $i\leftarrow 1$ </p>
<p> $s\in\mathbb{V}^{*}$ </p>
<p> $\displaystyle\text{argmax}_{s_{a}}\mathbb{E}_{x\sim G(F^{t}(E_{\psi}(s_{o}%
\oplus s_{a})))}\mathcal{A}(x,t,s_{o})~{},$ </p>
<p> $\displaystyle\cos(v,v_{t}^{image})+\lambda\cos(v,v_{t}^{text}),$ </p>
<p> $\cos(a,b)=\frac{a^{T}b}{\|a\|\|b\|}$ </p>
<p> $\eta$ </p>
<p> $512\times 512$ </p>
<p> $x$ </p>
<p> $E_{\psi}(s_{o}\oplus s_{a})=E_{\psi}(s_{o})\oplus E_{\psi}(s_{a})$ </p>
<p> $N$ </p>
<p> $bestloss>\mathcal{L}$ </p>
<p> $v_{t}^{image}=F^{i}(x_{t})$ </p>
<p> $d_{\text{emb}}$ </p>
<p> $\displaystyle\text{argmax}_{s_{a}}\cos(F^{i}(E_{\psi}(s_{o}\oplus s_{a})),v_{t%
}).$ </p>
<p> $s^{\prime}=$ </p>
<p> ${}^{3,*}$ </p>
<p> $Z\leftarrow Z-\eta\nabla_{Z}\mathcal{L}$ </p>
<p> $100$ </p>
<p> $s_{a}$ </p>
<p> $s_{o}\oplus s_{a}$ </p>
<p> $m$ </p>
<p> $v$ </p>
<p> $\displaystyle\text{s.t.}\quad v=F^{i}(E_{\psi}(s_{o}\oplus s_{a})),$ </p>
<p> $\mathbb{V}=\{w_{1},w_{2},\cdots,w_{|\mathbb{V}|}\}$ </p>
<p> $F^{i}$ </p>
<p> $\psi$ </p>
<p> $\displaystyle\text{s.t.}\quad v$ </p>
<p> $s_{o}$ </p>
<p> $F^{t}$ </p>
<p> ${}^{2}$ </p>
<p> $\oplus$ </p>
<p> $E_{\psi}(s)_{i}=\psi_{j}$ </p>
<p> $5\times 4=20$ </p>
<p> $3\times 100$ </p>
<p> ${}^{3}$ </p>
<p> $v\leftarrow F^{t}(E_{\psi}(s_{o})\oplus\text{Proj}_{\psi}(Z))$ </p>
<p> $\mathcal{L}=-\cos(v,v_{t}^{image})-\lambda\cos(v,v_{t}^{text})$ </p>
<p> $s_{o}\in\mathbb{V}^{n}$ </p>
<p> $s_{a}\leftarrow E_{\psi}^{-1}(\text{Proj}_{\psi}(bestZ))$ </p>
<p> $bestloss\leftarrow\infty,bestZ\leftarrow Z$ </p>
<p> $\displaystyle=F^{i}(E_{\psi}(s_{o}\oplus E_{\psi}^{-1}(\text{Proj}_{\psi}(Z))))$ </p>
<p> $t\in\mathbb{V}$ </p>
<p> $Z$ </p>
<p> $(\cdot)$ </p>
<p> $x\sim G(v)$ </p>
<p> $d_{\text{token}}$ </p>
<p> $s_{a}\in\mathbb{V}^{m}$ </p>
<p> $v_{t}$ </p>
<p> $\lambda$ </p>
<p> $\mathbb{V}$ </p>
<p> $w_{j}=s_{i}$ </p>
<p> $t\in\mathcal{V}$ </p>
<p> $x\sim G(F^{t}(E_{\psi}(s)))$ </p>
<p> $E_{\psi}$ </p>
<p> $j=\text{argmin}_{j^{\prime}}\|\psi_{j^{\prime}}-Z_{i}\|_{2}^{2}$ </p>
<p> $|s|\times d_{\text{token}}$ </p>
<p> $\displaystyle\text{argmax}_{v_{t}}\mathbb{E}_{x\sim G(v_{t})}\mathcal{A}(x,t,s%
_{o})~{}.$ </p>
<p> $E_{L}\cup E_{R}$ </p>
<p> $E_{L}=\{(u,w)|(u,w)\in E,w\neq v\}$ </p>
</body>
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