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| <p> $L(\cdot,\cdot)$ </p> |
| <p> $\displaystyle\int_{0}^{1}\mathbb{E}[||\partial\rho_{t}(Z)/\partial t-g_{v_{% |
| \theta}}(Z_{t},t)||]dt.$ </p> |
| <p> $g_{v_{\theta}}(\cdot)$ </p> |
| <p> ${}_{\textrm{75}}$ </p> |
| <p> $\hat{Z}_{0}$ </p> |
| <p> ${Z}_{i/N}$ </p> |
| <p> $\displaystyle=\mathbb{E}_{q(Z_{1/N:1}|Z_{0})}\left[\log\frac{q(Z_{1}|Z_{0})}{p% |
| _{v_{\theta}}(Z_{1})p_{v_{\theta}}(Z_{0}|Z_{1/N})}\right]+\sum_{i=1}^{N}% |
| \mathbb{E}_{q(Z_{i/N}|Z_{0})}\mathbb{E}_{q(Z_{(i-1)/N}|Z_{i/N},Z_{0})}\left[% |
| \log\frac{q(Z_{(i-1)/N}|Z_{i/N},Z_{0})}{p_{v_{\theta}}(Z_{(i-1)/N}|Z_{i/N})}\right]$ </p> |
| <p> $\displaystyle\mathcal{L}_{\textrm{FM-KT}}\!=\!\mathbb{E}[\frac{1}{N}\sum_{i=0}% |
| ^{N-1}\!L(\mathcal{T}((\nabla_{t}\alpha_{t}Z_{1}\!\!-\!\!g_{v_{\theta}}(Z_{1\!% |
| -\!i/N},1-i/N))/\!-\!\nabla_{t}\sigma_{t})$ </p> |
| <p> $\sigma_{t}=\sqrt{1-\alpha_{t}^{2}},\ s..t.\quad a=19.9,b=0.1$ </p> |
| <p> $\displaystyle=X_{1-i/K}+\mathcal{E}(Z_{1-i/K})-(1/K)\left[g_{v_{\theta}}(X_{1-% |
| i/K},1-i/K)+\mathcal{E}(Z_{1-i/K})\psi(1-i/K)\right],$ </p> |
| <p> $\hat{Z}_{i/N}$ </p> |
| <p> $\alpha^{\Theta}$ </p> |
| <p> $\textrm{Law}(Z_{(i-1)/N})\stackrel{{\scriptstyle\sim}}{{=}}\textrm{Law}(\hat{Z% |
| }_{(i-1)/N})$ </p> |
| <p> $\displaystyle+\underbrace{L(\mathcal{T}((\nabla_{t}\alpha_{t}Z_{1}-g_{v_{% |
| \theta}}(Z_{1-i/N},1-i/N))/-\nabla_{t}\sigma_{t}),Y)}_{\textrm{match the % |
| ground truth label (optional)}},$ </p> |
| <p> $\alpha_{t}=\textrm{exp}(-\frac{1}{4}a(1-t)^{2}-\frac{1}{2}b(1-t))$ </p> |
| <p> $\lim_{t\rightarrow 1}\alpha_{t}=1$ </p> |
| <p> $\{Z_{t}\}_{t}$ </p> |
| <p> $\sigma(t)=1$ </p> |
| <p> $X^{S}-X^{T}=\frac{dX_{t}}{dt}$ </p> |
| <p> $\displaystyle\mathcal{E}(Z_{1-1/K})=(1/K)\mathcal{K}(1)$ </p> |
| <p> $\sigma_{t}=\sqrt{1-\alpha_{t}^{2}}$ </p> |
| <p> $\mathit{a=19.9}$ </p> |
| <p> $\displaystyle\textrm{where}\quad Z_{1-i/N}=Z_{1-(i-1)/N}$ </p> |
| <p> $\mathcal{H}(t)=\operatorname*{arg\,sup}_{X_{t}}\{||\frac{dX_{t}}{dt}-g_{v_{% |
| \theta}}(X_{t},t)||_{2}^{2}\}$ </p> |
| <p> $||\frac{dX_{t}}{dt}-g_{v_{\theta}}(X_{t},t)||_{2}^{2}$ </p> |
| <p> $t\sim\mathcal{U}[0,1]$ </p> |
| <p> $\mathcal{T}_{\textrm{vanilla}}(\cdot)$ </p> |
| <p> $\sim 10^{6}\times 10^{4}=10^{10}$ </p> |
| <p> $\sim 1\,\mathrm{KB}$ </p> |
| <p> $\displaystyle-~{}NCC(\mathcal{X}_{\mathrm{fx}},\mathcal{X}_{\mathrm{wp,n}})+% |
| \lambda\sum_{p\in\Omega}||\nabla\varphi(p)||^{2}$ </p> |
| <p> $\varphi=\sum_{i=0}^{n}\varphi_{i}$ </p> |
| <p> $\mathcal{X}_{\mathrm{wp}}$ </p> |
| <p> $w_{k,i}=|m(i\in\omega)|^{g}$ </p> |
| <p> $\mathcal{X}_{\mathrm{mv}}$ </p> |
| <p> $0.7\times 0.7\times 3.0$ </p> |
| <p> $\mathcal{X}_{\mathrm{fx}}$ </p> |
| <p> $0.837\pm 0.021$ </p> |
| <p> $1.2\times 1.2\times 3.0$ </p> |
| <p> $\displaystyle\begin{split}\mathcal{X}_{\mathrm{wp,n}}=\mathcal{X}_{\mathrm{mv}% |
| }\circ\sum_{i=0}^{n}\varphi_{i}.\end{split}$ </p> |
| <p> $\varphi:\mathbb{R}^{3}\rightarrow\mathbb{R}^{3}$ </p> |
| <p> $\mathcal{X}_{\mathrm{wp,1}}$ </p> |
| <p> $\mathcal{X}_{\mathrm{wp,n}}$ </p> |
| <p> $0.926\pm 0.012$ </p> |
| <p> $0.847\pm 0.008$ </p> |
| <p> $\centering\mathcal{X}_{\mathrm{wp}}=\mathcal{X}_{\mathrm{mv}}\circ\varphi% |
| \approx\mathcal{X}_{\mathrm{fx}}.\@add@centering$ </p> |
| <p> $\mathcal{L}_{\mathrm{sim}}$ </p> |
| <p> $0.866\pm 0.020$ </p> |
| <p> $0.915\pm 0.012$ </p> |
| <p> $230\times 230$ </p> |
| <p> $\mathcal{X}_{\mathrm{wp,0}}$ </p> |
| <p> $0.811\pm 0.023$ </p> |
| <p> $0.807\pm 0.23$ </p> |
| <p> $290\times 250$ </p> |
| <p> $lr_{\mathrm{epoch}}=3\cdot 10^{-4}\cdot e^{-3\mathrm{epoch}/500}$ </p> |
| <p> $H\times W\times L$ </p> |
| <p> $0.920\pm 0.014$ </p> |
| <p> $\%|J_{\phi}|\leq 0$ </p> |
| <p> $\%|J_{\varphi}|<0$ </p> |
| <p> $\mathcal{L}_{\mathrm{smooth}}$ </p> |
| <p> $\displaystyle~{}\mathcal{L}_{\mathrm{sim}}+\lambda\mathcal{L}_{\mathrm{smooth}}$ </p> |
| <p> $0.866\pm 0.013$ </p> |
| <p> $94.65\%$ </p> |
| <p> $\langle search\rangle$ </p> |
| <p> $P(\hat{a}=\langle search\rangle|\theta^{\prime},q)$ </p> |
| <p> $LM_{\theta^{\prime}}$ </p> |
| <p> $LM_{\theta^{\prime}}:Q\mapsto\Omega\cup\{\langle search\rangle\}$ </p> |
| <p> $LM_{\theta}:Q\mapsto\Omega$ </p> |
| <p> $\psi\circ LM_{\theta}:Q\mapsto\Omega\cup\{\langle search\rangle\}$ </p> |
| <p> $alpha=32$ </p> |
| <p> $\underset{\theta}{\text{argmin}}\prod_{q\in Q}\left[P(\hat{a}=\langle search% |
| \rangle|\theta^{\prime},q)+\lambda P(\hat{a}\notin A|\theta^{\prime},q)\right]$ </p> |
| <p> $(\langle search\rangle)$ </p> |
| <p> $7e-5$ </p> |
| <p> $LM_{\theta}$ </p> |
| <p> $\psi(LM_{\theta}(q))=\begin{cases}\mathds{1}(\hat{a}),&\text{if }\hat{a}\in A% |
| \\ |
| \langle search\rangle,&\text{otherwise}\end{cases}$ </p> |
| <p> $P(\hat{a}\notin A|\theta^{\prime},q)$ </p> |
| <p> $733.5$ </p> |
| <p> $TR^{(1)},TR^{(2)},\cdots,TR^{(m)}$ </p> |
| <p> $\mathcal{V}_{\mathrm{target}}$ </p> |
| <p> $\hat{P}_{v}$ </p> |
| <p> $\mathcal{V}_{target}$ </p> |
| <p> $\operatorname*{arg\,min}_{\theta}\sum_{G}\mathcal{L}(G,\theta).$ </p> |
| <p> $p^{(1)},p^{(2)},\cdots,p^{(m)}$ </p> |
| <p> $q^{(1)},q^{(2)},\cdots,q^{(n)}$ </p> |
| <p> $\hat{\mathcal{V}}_{target}$ </p> |
| <p> $\in TR^{i}$ </p> |
| <p> $\displaystyle-\frac{1}{|\mathcal{V}|}\sum_{v\notin\mathcal{V}_{\textrm{target}% |
| }}\log P(\hat{P}_{v}=0|\mathcal{G},\theta)$ </p> |
| <p> $\displaystyle\mathcal{L}(G,\theta)=$ </p> |
| <p> $\displaystyle-\frac{1}{|\mathcal{V}|}\sum_{v\in\mathcal{V}_{\textrm{target}}}% |
| \log P(\hat{P}_{v}=1|\mathcal{G},\theta)$ </p> |
| <p> $1,2,\cdots,m$ </p> |
| <p> $TR^{i}$ </p> |
| <p> $N=1923$ </p> |
| <p> $p^{(1)}_{ref},p^{(2)}_{ref},\cdots,p^{(m)}_{ref}$ </p> |
| <p> $\mathcal{V}_{\mathrm{target}}\subset\mathcal{V}$ </p> |
| <p> $[0.1,0.01,0.001,0.0001]$ </p> |
| <p> $\mathrm{EE}_{y}$ </p> |
| <p> $\left[\mathcal{D}_{t},\mathcal{D}_{r}\right]$ </p> |
| <p> $\left[\mathcal{D}_{r},\mathcal{H}_{r}\right]$ </p> |
| <p> $W=F\cdot S$ </p> |
| <p> $\mathcal{L}_{GPs}=\mathcal{L}_{G}+w_{P}\mathcal{L}_{P}+w_{S}\mathcal{L}_{s}$ </p> |
| <p> $\eta_{t}>6$ </p> |
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