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