Question: What are the two decoding functions?
Evidence: In this case, the decoding function is a linear projection, which is $= f_{\text{de}}()=+ $ , where $\in ^{d_\times d_}$ is a trainable weight matrix and $\in ^{d_\times 1}$ is the bias term.
A family of bijective transformation was designed in NICE BIBREF17 , and the simplest continuous bijective function $f:^D\rightarrow ^D$ and its inverse $f^{-1}$ is defined as:
$$h: \hspace{14.22636pt} _1 &= _1, & _2 &= _2+m(_1) \nonumber \\ h^{-1}: \hspace{14.22636pt} _1 &= _1, & _2 &= _2-m(_1) \nonumber $$ (Eq. 15)
where $_1$ is a $d$ -dimensional partition of the input $\in ^D$ , and $m:^d\rightarrow ^{D-d}$ is an arbitrary continuous function, which could be a trainable multi-layer feedforward neural network with non-linear activation functions. It is named as an `additive coupling layer' BIBREF17 , which has unit Jacobian determinant. To allow the learning system to explore more powerful transformation, we follow the design of the `affine coupling layer' BIBREF24 :
$$h: \hspace{5.69046pt} _1 &= _1, & _2 &= _2 \odot \text{exp}(s(_1)) + t(_1) \nonumber \\ h^{-1}: \hspace{5.69046pt} _1 &= _1, & _2 &= (_2-t(_1)) \odot \text{exp}(-s(_1)) \nonumber $$ (Eq. 16)
where $s:^d\rightarrow ^{D-d}$ and $t:^d\rightarrow ^{D-d}$ are both neural networks with linear output units.
The requirement of the continuous bijective transformation is that, the dimensionality of the input $$ and the output $$ need to match exactly. In our case, the output $\in ^{d_}$ of the decoding function $f_{\text{de}}$ has lower dimensionality than the input $\in ^{d_}$ does. Our solution is to add an orthonormal regularised linear projection before the bijective function to transform the vector representation of a sentence to the desired dimension.
In this case, the decoding function is a linear projection, which is $= f_{\text{de}}()=+ $ , where $\in ^{d_\times d_}$ is a trainable weight matrix and $\in ^{d_\times 1}$ is the bias term.
A family of bijective transformation was designed in NICE BIBREF17 , and the simplest continuous bijective function $f:^D\rightarrow ^D$ and its inverse $f^{-1}$ is defined as:

$$h: \hspace{14.22636pt} _1 &= _1, & _2 &= _2+m(_1) \nonumber \\ h^{-1}: \hspace{14.22636pt} _1 &= _1, & _2 &= _2-m(_1) \nonumber $$ (Eq. 15)

where $_1$ is a $d$ -dimensional partition of the input $\in ^D$ , and $m:^d\rightarrow ^{D-d}$ is an arbitrary continuous function, which could be a trainable multi-layer feedforward neural network with non-linear activation functions. It is named as an `additive coupling layer' BIBREF17 , which has unit Jacobian determinant. To allow the learning system to explore more powerful transformation, we follow the design of the `affine coupling layer' BIBREF24 :

$$h: \hspace{5.69046pt} _1 &= _1, & _2 &= _2 \odot \text{exp}(s(_1)) + t(_1) \nonumber \\ h^{-1}: \hspace{5.69046pt} _1 &= _1, & _2 &= (_2-t(_1)) \odot \text{exp}(-s(_1)) \nonumber $$ (Eq. 16)

where $s:^d\rightarrow ^{D-d}$ and $t:^d\rightarrow ^{D-d}$ are both neural networks with linear output units.

The requirement of the continuous bijective transformation is that, the dimensionality of the input $$ and the output $$ need to match exactly.
Answer: a linear projection and a bijective function with continuous transformation though  ‘affine coupling layer’ of (Dinh et al.,2016).