Algorithm 1 Traing NVP-MC

Input: Energy function $U\left(x\right)$, batch size M, learning rate $\alpha$, number of iterations N, empty buffer B, transition kernels $K_{\theta }$ and $K_{\theta }^{-1}$.

1: Initialize B using HMC without the MH step. Initialize the parameters of the transition kernel $K_{\theta }$ and parameters of the discriminator $D_{\varphi }$.

2: for $i=1\to i=N$ do

3:       Sample a batch ${\left\{{(x,v)}_{\left(i\right)}\right\}}_{i⩽N}$ of Gaussian noise as the start points.

4:       for $i=1\to i=M$ do

5:             Randomly sample a number u in open interval $(0,1)$.

6:             Choose transition kernel: $K_{\theta }(x,v)=\left\{\begin{array}{cc}{K}_{\theta }(x,v),\hfill & 0<u<0.5\hfill \\ \hfill K_{\theta }^{-1}(x,v),& \hfill 1>u>0.5.\end{array}\right.$

7:             Generate the new sample $\left\{{(x^{\prime },v^{″})}_{\left(i\right)}\right\}$ through $K_{\theta }.$

8:             Accept the new sample with probability computed by Equation (4),

9:             and replace the samples in B with the accepted samples.

end for

10:           Sample a batch ${\left\{{\left(x^{″}\right)}_{\left(i\right)}\right\}}_{i⩽N}$ from B as the correct samples.

11:           Update the discriminator:

12:           $\varphi \leftarrow \varphi -\alpha {\nabla }_{\varphi }\frac{1}{M}{\sum }_{i=1}^M\left[\log D_{\varphi }\left(x_{\left(i\right)}^{″}\right)+\log \left(1-D_{\varphi }\left(K_{\theta }\left(x_{\left(i\right)}\right)\right)\right)\right].$

13:           Update the transition kernel:

14:           $\theta \leftarrow \theta -\alpha {\nabla }_{\theta }\frac{1}{M}{\sum }_{i=1}^M\log \left(1-D_{\varphi }\left(K_{\theta }\left(x_{\left(i\right)}\right)\right)\right).$

end for