  VAE requires the standard Gaussian distribution as a prior in the latent
space. Since all codes tend to follow the same prior, it often suffers the
so-called "posterior collapse". To avoid this, this paper introduces the class
specific distribution for the latent code. But different from CVAE, we present
a method for disentangling the latent space into the label relevant and
irrelevant dimensions, $\bm{\mathrm{z}}_s$ and $\bm{\mathrm{z}}_u$, for a
single input. We apply two separated encoders to map the input into
$\bm{\mathrm{z}}_s$ and $\bm{\mathrm{z}}_u$ respectively, and then give the
concatenated code to the decoder to reconstruct the input. The label irrelevant
code $\bm{\mathrm{z}}_u$ represent the common characteristics of all inputs,
hence they are constrained by the standard Gaussian, and their encoder is
trained in amortized variational inference way, like VAE. While
$\bm{\mathrm{z}}_s$ is assumed to follow the Gaussian mixture distribution in
which each component corresponds to a particular class. The parameters for the
Gaussian components in $\bm{\mathrm{z}}_s$ encoder are optimized by the label
supervision in a global stochastic way. In theory, we show that our method is
actually equivalent to adding a KL divergence term on the joint distribution of
$\bm{\mathrm{z}}_s$ and the class label $c$, and it can directly increase the
mutual information between $\bm{\mathrm{z}}_s$ and the label $c$. Our model can
also be extended to GAN by adding a discriminator in the pixel domain so that
it produces high quality and diverse images.
