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O(n^{2})
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f
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n
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G(v)
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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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{}^{4}
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v_{t}^{text}=F^{t}(E_{\psi}(s^{\prime}))
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{}^{*}
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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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{}^{1}
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\text{Proj}_{\psi}(Z)_{i}=Z_{i}+\text{sg}(\psi_{j}-Z_{i})
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x_{t}
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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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m=4
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s_{a}=E_{\psi}^{-1}(\text{Proj}_{\psi}(Z))
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{}^{5}
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Z_{i}
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{}^{1,*}
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\text{Proj}_{\psi}(Z)
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s
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\displaystyle\text{argmax}_{s_{a}}
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t
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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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G
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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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\eta
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512\times 512
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x
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E_{\psi}(s_{o}\oplus s_{a})=E_{\psi}(s_{o})\oplus E_{\psi}(s_{a})
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N
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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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100
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s_{a}
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s_{o}\oplus s_{a}
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m
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v
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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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F^{i}
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\psi
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\displaystyle\text{s.t.}\quad v
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s_{o}
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F^{t}
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{}^{2}
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\oplus
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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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{}^{3}
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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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Z
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(\cdot)
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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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v_{t}
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\lambda
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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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TeX data from arXiv
- Using https://github.com/KyuDan1/TeX2Image code.
- We have categories Math, Physics, Statistics, ComputerScience.
| Domain | Size |
|---|---|
| Mathematics | 4.22M |
| Computer Science | 2.76M |
| Statistics | 0.89M |
| Physics | 0.78M |
| Total (unique) | 7.17M |
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