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$$\begin{align*} +%x^{\text{padded}} &= ([\text{PAD}]\times P) + x_{0:(T-P)}\\ +h^\text{embed}_{t}\,\,\,\,\,\,\,\,&= E^\text{global-embed}_{x_t}+E^\text{pos}_t & t\in [0..T), E^{\text{global-embed}}\in\mathbb{R}^{V\times D_G},\\& &E^{\text{pos}}\in\mathbb{R}^{T\times D_G}, h^\text{embed}\in\mathbb{R}^{T\times D_G} \\ +% \h{global-in}_{0} &= [E^{\text{pad}}_0,..,E^{\text{pad}}_{P-1}] & E^{\text{pad}}\inR{D_G}, K=\frac{T}{P}\\ +% \h{global-in}_{k} &= \h{embed}_{((k-1)\cdot P):(k\cdot P)} & k\in [1,..,K) \\ +h^\text{global-in}_{k} \,\,\,\,\,&= +\begin{cases} +E^{\text{global-pad}}, & \text{if } k = 0, \\ +h^\text{embed}_{((k-1)\cdot P):(k \cdot P)}, & k\in [1,..,K), \\ +\end{cases} & E^{\text{global-pad}}\in\mathbb{R}^{P\times D_G}, K=\frac{T}{P} +\\ +h^\text{global-out}_{0:K} \,\,\,&= \text{transformer}^{\text{global}}(h^\text{global-in}_{0:K}) & h^\text{global-out}\in\mathbb{R}^{K\times P\cdot D_G},h^\text{global-in}\in\mathbb{R}^{K\times P\cdot D_G}\\ +%\h{local-in}_{k,0} &= \h{global-out}_{k,0:D_G} & \h{local-in}\inR{K\times P \times D_G} \\ +%\h{local-in}_{k,p} &= w^{\text{GL}}\h{global-out}_{k,(p\cdot D_G):((p+1)\cdot D_G)} + E^\text{local-embed}_{x_{(k\cdot P + p - 1)}} & w^{\text{GL}}\inR{D_G\times D_L}, E^{\text{local-embed}}\inR{V\times D_L}, p\in [0,..,P), \\ +h^\text{local-in}_{k,p} \,\,\,\,\,\,\,&= w^{\text{GL}}h^\text{global-out}_{k,(p\cdot D_G):((p+1)\cdot D_G)} + +\begin{cases} +E^\text{local-pad}, & \text{if } p=0 \\ +E^\text{local-embed}_{x_{(k\cdot P + p - 1)}}, & p\in [1,..,P) \\ +\end{cases} & +\begin{array}{r} +E^{\text{local-pad}}\in\mathbb{R}^{D_L} , w^{\text{GL}}\in\mathbb{R}^{D_G\times D_L}\\ E^{\text{local-embed}}\in\mathbb{R}^{V\times D_L} +\end{array}\hspace{-2mm} +\\ +h^\text{local-out}_{k,0:P} \,\,\,\,\,&= \text{transformer}^{\text{local}}(h^\text{local-in}_{k,0:P}) & h^\text{local-in}_{k,p}\in\mathbb{R}^{D_L}, h^\text{local-out}\in\mathbb{R}^{K\times P\cdot D_L} \\ +p(x_{t}|x_{0:t}) &= \text{softmax}{(E^\text{local-embed}h^\text{local-out}_{k,p})}_{x_t} & t=k\cdot P+p\, +\end{align*}$$
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