of the decoder. So the inverse image $z'$ of $x$ can be regarded as the effectiveness label $x$ in the continuous latent space. And if $z'$ has been obtained, then we are reasonably motivated to pull the encoder distribution $p(z|x)$ to close $z'$ to maintain the expressiveness of the encoder.

However, finding the inverse image of the decoder exactly is not an easy task. To overcome this, we introduce an extra network to estimate the inverse image of $x$ of the decoder, i.e. the effectiveness label. Thus we call this network the *labeling network* and denote the output of it as $z_{label}$. The *labeling network* can also be considered as an approximation to the ideal encoder for current decoder. Specifically, the *labeling network* shares the same network structure with the original encoder of VAE, but it only outputs the variable $z_{label}$ rather than the reparameterized distribution. As the output of the VAE encoder is a distribution $q(z|x)$, we put the expressive constraint on the expectation of the $L_2$ distance ${||z-z_{label}||}^2$ between encoded latent variable and $z_{label}$ over the encoder distribution $q(z|x)$. Thus an expressiveness objective function is defined as follows: $$\mathcal{L}_\mathsf{exp} = \mathbb{E}_{q(z|x)}[{||z-z_{label}||}^2]\label{eq:vae_pre}$$ which is minimized to encourage the encoder 