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| <p> $s+r\approx o$ </p> |
| <p> $\mathbf{I}_{t}^{s,r}\in\mathbb{Z}^{|\mathcal{E}|\times|\mathcal{R}|\times|% |
| \mathcal{E}|}$ </p> |
| <p> $\textbf{W}^{(l)}_{6}$ </p> |
| <p> $\textbf{t}_{i}^{\prime\prime}=\sigma(\sum_{r\in\mathcal{R}_{TG}}\frac{1}{% |
| \mathcal{N}_{i}}\textbf{W}_{r}\textbf{t}_{j}^{\prime}+\textbf{W}_{11}\textbf{t% |
| }_{i}^{\prime})\textnormal{,}$ </p> |
| <p> $y^{e}_{t}$ </p> |
| <p> $\textbf{W}_{r}\in\mathbb{R}^{d\times d}$ </p> |
| <p> $\textbf{W}_{9}\in\mathbb{R}^{d\times 32}$ </p> |
| <p> $\alpha_{o,s}$ </p> |
| <p> $\textbf{o}^{(l+1)}_{t}=\sigma(\sum_{(s,r,o)\in\mathcal{F}_{t}}\textbf{W}^{(l)}% |
| _{1}(\psi(\textbf{s}^{(l)}_{t}||\textbf{r}_{t}))+\textbf{W}^{(l)}_{2}\textbf{o% |
| }^{(l)}_{t})\textnormal{,}$ </p> |
| <p> $\textbf{UE}_{t-k:t-1}$ </p> |
| <p> $\textbf{W}_{11}\in\mathbb{R}^{d\times d}$ </p> |
| <p> $f_{q}\in\{G_{t-k:t-1}\}$ </p> |
| <p> $\textbf{R}_{t}=\mathrm{GRU}(\textbf{R}_{t-1},[pooling(\textbf{E}^{\mathcal{R}}% |
| _{t-1})||\textbf{R}])\textnormal{,}$ </p> |
| <p> $(s,r,o,t)$ </p> |
| <p> $\mathcal{R}_{TG}$ </p> |
| <p> $\textbf{E}^{\mathcal{R}}$ </p> |
| <p> $t>t_{T}$ </p> |
| <p> $\alpha^{(l)}_{o,s}=\frac{exp(\textbf{W}^{(l)}_{3}\sigma(\textbf{W}^{(l)}_{4}[% |
| \textbf{s}^{(l)}||\textbf{r}||\textbf{o}^{(l)}||\textbf{t}^{\prime\prime}]))}{% |
| \sum_{s^{\prime}\in\mathcal{N}_{(o)}}exp(\textbf{W}^{(l)}_{3}\sigma(\textbf{W}% |
| ^{(l)}_{4}[\textbf{s'}^{(l)}||\textbf{r}||\textbf{o}^{(l)}||\textbf{t}^{\prime% |
| \prime}]))}\textnormal{,}$ </p> |
| <p> $q=(s,r,?,t)$ </p> |
| <p> $p_{R}(o|s,r,t,G_{t-1})=softmax(\mathbf{ConvTransE}(\textbf{s},\textbf{r},% |
| \textbf{t}^{\prime\prime})\textbf{GE}^{\top}_{t})\textnormal{.}$ </p> |
| <p> $\textbf{R}_{t}$ </p> |
| <p> $\textbf{t}^{\prime}=\textbf{W}_{8}(\textbf{W}_{9}\textbf{t}||\sigma(\textbf{W}% |
| _{10}\textbf{t}))\textnormal{,}$ </p> |
| <p> $\mathbf{W}^{(l)}_{1}$ </p> |
| <p> $\displaystyle=\beta\sum_{(s,r,t)\in\mathcal{Q}^{e}_{t}}y^{e}_{t}\log p(o|s,r,t% |
| ,G_{t-1})$ </p> |
| <p> $\textbf{W}_{10}\in\mathbb{R}^{d\times 32}$ </p> |
| <p> $\textbf{s},\textbf{o}\in\textbf{UE}_{t-k:t-1}$ </p> |
| <p> $\textbf{W}^{(l)}_{3}\in\mathbb{R}^{4d}$ </p> |
| <p> $p_{H}(o|s,r,t,G_{t-1})=softmax(\mathbf{ConvTransE}(\textbf{s},\textbf{r},% |
| \textbf{t}^{\prime\prime})\textbf{GE}^{\top}_{t}\odot\mathbf{I}_{t}^{s,r})% |
| \textnormal{,}$ </p> |
| <p> $\mathcal{Q}^{r}_{t}$ </p> |
| <p> $\textbf{UE}_{t}$ </p> |
| <p> $\Theta\in\mathbb{R}^{|\mathcal{E}|\times d}$ </p> |
| <p> $\{\textbf{E}_{t-k+1},\textbf{E}_{t-k+2},...,\textbf{E}_{t}\}$ </p> |
| <p> $\mathcal{N}_{(o)}$ </p> |
| <p> $y^{r}_{t}$ </p> |
| <p> $G=\{\mathcal{E},\mathcal{R},\mathcal{F},\mathcal{T}\}$ </p> |
| <p> $TG=\{\mathcal{E}_{TG},\mathcal{R}_{TG}\}$ </p> |
| <p> $\textbf{W}_{8}\in\mathbb{R}^{d\times 2d}$ </p> |
| <p> $\mathcal{Q}^{e}_{t}$ </p> |
| <p> $\textbf{W}^{(l)}_{4}\in\mathbb{R}^{4d\times 4d}$ </p> |
| <p> $\mathsf{LMS}$ </p> |
| <p> $\textbf{W}_{7}\in\mathbb{R}^{1\times d}$ </p> |
| <p> $\textbf{UE}_{t-k:t-1}=\mathrm{MEAN}(\sum_{i=t-k+1}^{t}\textbf{E}_{i})% |
| \textnormal{.}$ </p> |
| <p> $\displaystyle+(1-\beta)\sum_{(s,o,t)\in\mathcal{Q}^{r}_{t}}y^{r}_{t}\log p(r|s% |
| ,o,t,G_{t-1})\textnormal{,}$ </p> |
| <p> $\textbf{E}_{t}=\mathrm{GRU}(\textbf{E}_{t-1},\textbf{G}_{t-1})\textnormal{.}$ </p> |
| <p> $\textbf{W}^{(l)}_{5}$ </p> |
| <p> $\mathbf{I}_{t}^{s,r}$ </p> |
| <p> $p(o|s,r,t,G_{t-1})=\alpha p_{H}(o|s,r,t,G_{t-1})+(1-\alpha)p_{R}(o|s,r,t,G_{t-% |
| 1})\textnormal{.}$ </p> |
| <p> $\mathbf{W}^{(l)}_{2}$ </p> |
| <p> $\textbf{o}^{(l+1)}=\sigma(\sum_{(s,r,o)\in UG}\alpha^{(l)}_{o,s}\textbf{W}^{(l% |
| )}_{5}\psi(\textbf{s}^{(l)}||\textbf{r})+\textbf{W}^{(l)}_{6}\textbf{o}^{(l)})% |
| \textnormal{,}$ </p> |
| <p> $\textbf{GE}_{t}=\sigma(\textbf{W}_{7}\Theta_{e})\textbf{E}_{t}+(1-\sigma(% |
| \textbf{W}_{7}\Theta_{e}))\textbf{UE}_{t}\textnormal{,}$ </p> |
| <p> $\mathcal{E}_{TG}$ </p> |
| <p> $G=\{G_{1},G_{2},...,G_{\mathcal{T}}\}$ </p> |
| <p> $p(o|s,r,t,G_{t-1})$ </p> |
| <p> $1-A_{h}$ </p> |
| <p> ${\mathcal{X}_{1}}\in\mathbb{R}^{{1024}\times{{3}}}$ </p> |
| <p> $P=\{{\mathcal{P}_{1}}\in\mathbb{R}^{{2}\times{{N}\times{C}}},{\mathcal{P}_{2}}% |
| \in\mathbb{R}^{{2}\times{{N_{1}}}\times{C_{1}}}$ </p> |
| <p> ${\mathcal{X}_{3}}\in\mathbb{R}^{{128}\times{{3}}}$ </p> |
| <p> $\displaystyle+||(\Pi(\mathcal{J}(\mathcal{M}^{h}_{MANO}))-\Pi(\mathcal{J}(\hat% |
| {\mathcal{M}}^{h}_{MANO})))||_{2}.$ </p> |
| <p> $X_{h}=\{X_{l}\in\mathbb{R}^{{N}\times{C}},X_{r}\in\mathbb{R}^{{N}\times{C}}\}$ </p> |
| <p> $i<num\_layers$ </p> |
| <p> $\mathcal{\hat{P}}=\mathcal{P}\odot\bm{\alpha}+\bm{\beta},(\bm{\alpha},\bm{% |
| \beta})=\psi(\mathcal{F}).$ </p> |
| <p> $\mathcal{L}_{V}=\sum_{h\in\{L,R\}}||\mathcal{M}^{h}_{GCN}-\hat{\mathcal{M}}^{h% |
| }_{GCN}||_{1}+||\mathcal{M}^{h}_{MANO}-\hat{\mathcal{M}}^{h}_{MANO}||_{1}.$ </p> |
| <p> ${\mathcal{F}_{3}}\in\mathbb{R}^{{\frac{H}{4}}\times{\frac{W}{4}}\times{256}}\}$ </p> |
| <p> ${G_{out}}=\sum_{k=0}^{K-1}C_{k}(\hat{L})G_{in}W_{k}.$ </p> |
| <p> $M=\{M_{l}\in\mathbb{R}^{{H}\times{{W}}},M_{r}\in\mathbb{R}^{{H}\times{{W}}}\}$ </p> |
| <p> $\hat{P_{i}},NumPoints\_{i},BallRadius\_{i}$ </p> |
| <p> $\mathcal{I}_{c}\in\mathbb{R}^{{H}\times{W}\times{3}}$ </p> |
| <p> $num\_layers$ </p> |
| <p> $\psi_{i}(\hat{F_{i}},P_{i})$ </p> |
| <p> $\displaystyle\mathcal{L}_{rep}$ </p> |
| <p> $cat(P_{i},PointNet(group(S_{i})))$ </p> |
| <p> $BallRadius$ </p> |
| <p> $(G\odot(\alpha+1)+\beta)$ </p> |
| <p> $NumPoints$ </p> |
| <p> $(P_{i}\odot(\alpha+1)+\beta)$ </p> |
| <p> $\hat{P_{i}}$ </p> |
| <p> $A_{h}\in[0,1]$ </p> |
| <p> $F=\{{\mathcal{F}_{1}}\in\mathbb{R}^{{H}\times{{W}\times{3}}},{\mathcal{F}_{2}}% |
| \in\mathbb{R}^{{\frac{H}{2}}\times{\frac{W}{2}}\times{64}}$ </p> |
| <p> $P_{ct}=\{P_{l}\in\mathbb{R}^{2},P_{r}\in\mathbb{R}^{2}\}$ </p> |
| <p> $\psi_{i+1}(C,G)$ </p> |
| <p> $\mathcal{G}\in\mathbb{R}^{{2}\times{1024}\times{1}}$ </p> |
| <p> $\mathcal{I}_{d}\in\mathbb{R}^{{H}\times{W}\times{1}}$ </p> |
| <p> $Fetch(F_{i}|u,v)$ </p> |
| <p> $\mathcal{L}_{J}=\sum_{h\in\{L,R\}}||\mathcal{J}(\mathcal{M}^{h}_{MANO})-% |
| \mathcal{J}(\hat{\mathcal{M}}^{h}_{MANO})||_{1}.$ </p> |
| <p> $K^{-1}X_{i}$ </p> |
| <p> $\mathcal{L}_{m}=||M-\hat{M}||_{1},$ </p> |
| <p> $[1,num\_layers]$ </p> |
| <p> $W_{k}\in\mathbb{R}^{{C_{in}}\times{C_{out}}}$ </p> |
| <p> $G_{in}\in\mathbb{R}^{{N}\times{C_{in}}}$ </p> |
| <p> $PointNet(\hat{P_{i}})$ </p> |
| <p> $\hat{L}\in\mathbb{R}^{{N}\times{N}}$ </p> |
| <p> $\mathcal{L}_{root}=\sum_{h\in\{L,R\}}||Root^{h}-\hat{Root^{h}}||_{1}.$ </p> |
| <p> $\mathcal{L}_{smooth}=\sum_{i=1}^{3}||e_{i}\cdot\hat{n}||_{1}+||e-\hat{e}||_{1},$ </p> |
| <p> ${\mathcal{P}_{3}}\in\mathbb{R}^{{2}\times{{N_{2}}}\times{C_{2}}}\}$ </p> |
| <p> $\mathcal{G_{V}}\in\mathbb{R}^{{N}\times{C}},(N=63,126,252),(C=512,256,128)$ </p> |
| <p> $\mathcal{L}_{c}=\sum_{h\in\{L,R\}}(1-A_{h})^{\gamma}\log(A_{h}),$ </p> |
| <p> ${\mathcal{X}_{2}}\in\mathbb{R}^{{512}\times{{3}}}$ </p> |
| <p> $G_{out}\in\mathbb{R}^{{N}\times{C_{out}}}$ </p> |
| <p> $\displaystyle=\sum_{h\in\{L,R\}}||(\Pi(\mathcal{M}^{h}_{MANO})-\Pi(\hat{% |
| \mathcal{M}}^{h}_{MANO}))||_{2}$ </p> |
| <p> $F(t)=\frac{(1-t)(1-2t)-\sqrt{(1-t)(1-5t)}}{2t(2-t)}.$ </p> |
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