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parse/train/ujmgfuxSLrO/ujmgfuxSLrO.md

@@ -48,7 +48,7 @@ $$
 \mathbf { Y } ^ { l } = \left\{ \begin{array} { l l } { \mathcal { F } \left( \mathbf { X } , \mathbf { W } ^ { l } , \mathbf { b } ^ { l } , g ^ { l } \right) , } & { l = 1 } \\ { \mathcal { F } \left( \mathcal { H } \left( \mathbf { X } , \mathbf { Y } ^ { l - 1 } \right) , \mathbf { W } ^ { l } , \mathbf { b } ^ { l } , g ^ { l } \right) , } & { \mathrm { O t h e r w i s e } } \end{array} \right.
 $$
 
-where $\mathbf { W } ^ { l } = \left\{ \mathbf { W } _ { 1 } ^ { l } , \cdot \cdot \cdot , \mathbf { W } _ { g ^ { l } } ^ { l } \right\}$ and $\mathbf { b } ^ { l } = \left\{ \mathbf { b } _ { 1 } ^ { l } , \cdot \cdot \cdot , \mathbf { b } _ { g ^ { l } } ^ { l } \right\}$ are the learnable weights and biases of group linear transformation $\mathcal { F }$ with $g ^ { l }$ groups at the $l$ -th layer. Briefly, the $\mathcal { F }$ function takes the input $\mathbf { X }$ or $\mathcal { H } \left( \mathbf { X } , \mathbf { Y } ^ { l - 1 } \right) \Big )$ and splits into $g ^ { l }$ non-overlapping groups such that $\mathbf { X } = \left\{ \mathbf { X } _ { 1 } , \cdots , \mathbf { X } _ { g ^ { l } } \right\}$ . The function $\mathcal { F }$ then linearly transforms each $\mathbf { X } _ { i }$ with weights $\mathbf { W } _ { i } ^ { l }$ and bias $\mathbf { b } _ { i } ^ { l }$ to produce output $\mathbf { Y } _ { i } ^ { l } = \mathbf { X } _ { i } \mathbf { W } _ { i } ^ { l } + \mathbf { b } _ { i } ^ { l }$ . The outputs of each group $\mathbf { Y } _ { i } ^ { l }$ are then concatenated to produce the output $\mathbf { Y } ^ { l }$ . The function $\mathcal { H }$ first shuffles the output of each group in $\mathbf { Y } ^ { l - 1 }$ and then combines it with the input $\mathbf { X }$ using the input mixer connection of Mehta et al. (2020) to avoid vanishing gradient problems. Figure 2 visualizes the expansion phase in the DeLighT transformation with group linear transformation, feature shuffling, and the input mixer connection.
+where $\mathbf { W } ^ { l } = \left\{ \mathbf { W } _ { 1 } ^ { l } , \cdot \cdot \cdot , \mathbf { W } _ { g ^ { l } } ^ { l } \right\}$ and $\mathbf { b } ^ { l } = \left\{ \mathbf { b } _ { 1 } ^ { l } , \cdot \cdot \cdot , \mathbf { b } _ { g ^ { l } } ^ { l } \right\}$ are the learnable weights and biases of group linear transformation $\mathcal { F }$ with $g ^ { l }$ groups at the $l$ -th layer. Briefly, the $\mathcal { F }$ function takes the input $\mathbf { X }$ or $\mathcal { H } \left( \mathbf { X } , \mathbf { Y } ^ { l - 1 } \right) \Big )$ and splits into $g ^ { l }$ non-overlapping groups such that $\mathbf { X } = \left\{ \mathbf { X } _ { 1 } , \cdots , \mathbf { X } _ { g ^ { l } } \right\}$ . The function $\mathcal { F }$ then linearly transforms each $\mathbf { X } _ { i }$ with weights $\mathbf { W } _ { i } ^ { l }$ and bias $\mathbf { b } _ { i } ^ { l }$ to produce output $\mathbf { Y } _ { i } ^ { l } = \mathbf { X } _ { i } \mathbf { W } _ { i } ^ { l } + \mathbf { b } _ { i } ^ { l }$ . The outputs of each group $\mathbf { Y } _ { i } ^ { l }$ are then concatenated to produce the output $\mathbf { Y } ^ { l }$ . The function $\mathcal { H }$ first shuffles the output of each group in $\mathbf { Y } ^ { l - 1 }$ and then combines it with the input $\mathbf { X }$ using the input mixer connection of Mehta et al. (2020) to avoid vanishing gradient problems. Figure 2 visualizes the expansion phase in the DeLighT transformation with group linear transformation, feature shuffling, and the input mixer connection.
 
 The number of groups at the $l$ -th GLT in DeLighT transformation are computed as: